Chapters 2-3

Transcription

Chapters 2-3

O In Chapter 1 , you
analyzed functions
and their graphs and
determined whether
inverse functions
existed.
O In Chapter 2, you will:
O ARCHITECTURE Polynomial functions are often used when
■ Model real-world data with
polynomial functions.
■ Use the Remainder and Factor
Theorems.
■ Find real and complex zeros of
designing and building a new structure. Architects use functions to
determine the w eight and strength of the materials, analyze costs,
estimate deterioration of materials, and determine the proper labor
force.
PREREAD Scan the lessons of Chapter 2, and use what you
already know about functions to make a prediction of the purpose
of this chapter.
■ Analyze and graph rational
functions.
■ Solve polynomial and rational
inequalities.
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Prerequisite Skills.
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Textbook Option Take the Quick Check below.
QuickCheck
Factor each polynomial. (Lesson 0-3)
power function
p. 5
funcion potencia
monomial function
p. 86
funcion monomio
radical function
p. 89
funcion radical
extraneous solutions
p. 91
solucion extrana
polynomial function
p. 97
funcion polinomial
1.
x 2 + x - 20
2. x 2 + 5x — 24
leading coefficient
p. 97
coeficiente llder
3.
2x2 - 1 7 x + 2 1
4. 3 x 2 — 5x — 12
leading-term test
p. 98
conduccion de prueba
de termino
quartic function
p. 99
funcion quartic
5. 12x 2 + 13x — 35
6. 8 x 2 - 4 2 x + 27
7. GEOMETRY Thearea of a square can be represented by
16x2 + 5 6 x + 49. Determine the expression that represents
the width of the square.
quadratic form
p. 100 forma de ecuacion
cuadratica
repeated zero
p. 101
cero repetido
Use a table to graph each function. (Lesson 0-3)
lower bound
p. 121
mas abajo ligado
9. f(x) = —2
upper bound
p. 121
superior ligado
8
.
f(x) = j x
10.
f(x) = x 2 + 3
1 1
12.
f(x) = 2 x 2 — 5x - 3
13. f(x) = 3 x 2 - x — 2
. f(x) = - x 2 + x -
6
14. TELEVISIONS An electronics magazine estimates that the total
number of plasma televisions sold worldwide can be represented by
f(x) = 2 t + 0.5t2, where t is the number of days after their release
date. Graph this function for 0 < t < 40.
rational function
asymptotes
p. 130 funcion racional
p. 130 asTntota
vertical asymptote
p. 131
asintota vertical
horizontal asymptote
p. 131
asTntota horizontal
polynomial inequality
p. 141
desigualdad de polinomio
sign chart
p. 141
carta de signo
rational inequality
p. 143 desigualdad racional
Write each set of numbers in set-builder and interval notation, if
possible. (Lesson 1-1)
15. x
< 6
17. —2 < x < 9
19. x < - 4 o r x > 5
16.
{—2, —1 ,0 ,...}
18. 1 < x < 4
20. x < - 1 o r x > 7
21. MUSIC At a music store, all of the compact discs are between $9.99
and $19.99. Describe the prices in set-builder and interval notation.
ReviewVocabulary
com p lex con jug ate s p. P7 conjugados complejos a pair of
complex numbers in the form a + bi and a — bi
reciprocal fu n c tio n s p. 45 funciones reciprocas functions of
the form f(x) = |
/W = l
2
Online Option Take an online self-check Chapter
Readiness Quiz at connectED.mcaraw-hill.com.
1
Tx
-a
85
• You analyzed parent
functions and their
families of graphs.
(Lesson 1-5)
NewVocabulary
power function
monomial function
radical function
extraneous solution
Graph and analyze
power functions.
Suspension bridges are used to span long
distances by hanging, or suspending, the main
deck using steel cables. iThe am ount of w eight
that a steel cable can support is a function of
the cable’s diam eter and can be modeled by a
power function.
\ Graph and analyze
■radical functions,
and solve radical
equations.
P ow er Functions In Lesson 1-5, you studied several parent functions that can be classified
as power functions. A pow er fu nction is any function of the form/(x) = ax", where a and n are
nonzero constant real numbers.
1
A power function is also a type of monom ial function. A m onom ial fu nction is any function that
can be written as f(x ) = a or f(x ) = a x n, where a and n are nonzero constant real numbers.
K eyConcept Monomial Functions
Let f be the power function f(x) = ax", where n is a positive integer.
n Even, a Positive
Domain: (—oo, oo)
Range: [0, oo)
Domain: (— oo, oo)
Range: (-o o , 0]
x- and y-lntercept: 0
Continuity: continuous for x e R
Symm etry: y-axis
Minim um : (0 ,0 )
Symm etry: y-axis
Decreasing: (-o o , 0)
Increasing: (0, oo)
Decreasing: (0, oo)
End behavior:
lim
f(x) = oo and
M axim um : (0 ,0 )
Increasing: ( - o o , 0)
End behavior: lim fix) =
. X—
►
—
oo
lim fix) = oo
X-*oo
-o o
and
lim f(x) = - o o
n Odd, a Positive
Domain and Range: ( - o o , oo)
Domain and Range:
Continuity: continuous on (-o o , oo)
Symmetry: origin
Symm etry: origin
Extrema: none
End Behavior:
Increasing: (—oo, oo)
lim
f(x) = - o o and
lim fix) = oo
X—too
86
Lesson 2-1
(-oo, oo)
Extrema: none
End Behavior:
Decreasing:
lim fix)
X—
>—
oo
=
oo
lim f(x) = — oo
and
(-oo, oo)
ReviewVocabulary
Degree of a Monomial The sum
of the exponents of the variables
of a monomial.
>Monomial functions with an even degree are also even in the sense th a t/(—x) = /(x). Likewise,
monomial functions with an odd degree are also odd, or / ( —x) = —f i x ) .
: j.
Analyze Monomial Functions
Graph and analyze each function. Describe the domain, range, intercepts, end behavior,
continuity, and where the function is increasing or decreasing.
a.
f(x) = \x4
Evaluate the function for several x-values in its domain. Then use a smooth curve to connect
each of these points to complete the graph.
- 3
40.5
-2
-1
0
1
2
3
8
0.5
0
0.5
8
40.5
Domain: (—00, 00)
y 1
12
Range: [0, 00)
4
Intercept: 0
-i
End behavior: lim f i x ) = 00 and lim f( x ) = 00
X— >— 0 0
x—*0 0
O
-4
ix
I
Continuity: continuous on (—00, 00)
Decreasing: (—00, 0)
Increasing: (0, 00)
b. fix) - —x7
X
fix)
-3
-2
-1
0
1
2
3
2187
128
1
0
-1
-1 2 8
-2 1 8 7
Domain: ( — 00, 00)
Range: ( — 00, 00)
Intercept: 0
End behavior: lim f{ x ) = 00 and lim f i x ) = —00
X—
>—OCT
X—Ô
O
Continuity: continuous on ( — 00, 00)
Decreasing: ( — 00, 00)
GuidedPractice
1B. f ( x ) = - § x 5
1A. f ( x ) = 3x6
ReviewVocabulary
Reciprocal Functions
Reciprocal functions have the
form f(x) = | . (Lesson 1 -5)
>Recall th a t/(x ) = j or x ~ 1 is undefined at x = 0. Similarly,/(x) = x ~ 2 and f i x ) = x ~ 3 are undefined at
x = 0. Because power functions can be undefined when n < 0, the graphs of these functions will
contain discontinuities.
■ ■ ^ F u n c t i o n s with Negative Exponents
Graph and analyze each function. Describe the domain, range, intercepts, end behavior,
a.
fix) = 3x~2
-3
-2
-1
0
1
2
3
0.3
0.75
3
undefined
3
0.75
0.3
Domain: (—00, 0) U (0, 00)
Range: (0, 00)
Intercepts: none
End behavior: lim f ix ) = 0 and lim f ( x ) = 0
X— oo
X—
»oo
Continuity: infinite discontinuity at x = 0
Increasing: (—00, 0)
Decreasing: (0, 00)
±O
Tx
b.
f( x ) = - | x -
5
X
-3
-2
-1
0
1
2
3
fix)
0.0031
0.0234
0.75
undefined
-0 .7 5
- 0 .0 2 3 4
-0 .0 0 3 1
Domain: (—oo, 0) U (0, oo)
Range: (—oo, 0) U (0, oo)
. Y
Intercepts: none
End behavior: lim fix ) = 0 and lim f{x) = 0
X—>—oo
X—»oo
-5
0
ix
|
Increasing: (—oo, 0) and (0, oo)
f -
—I
GuidedPractice
2A. fix ) = - j x ~ 4
2B. fix ) —
= A4xv - 3
Review Vocabulary
Rational Exponents exponents
written as fractions in simplest
form. (Lesson 0-4)
Recall that x" indicates the nth root of x, and x n , where ^ is in sim plest form, indicates the nth root
of x p. If n is an even integer, then the dom ain m ust be restricted to nonnegative values.
v_
Rational Exponents
G raph and analyze each fu nction. D escribe the dom ain, range, intercepts, end behavior,
continuity, and w here the function is increasing or decreasing.
a.
fix ) = x 2
0
1
2
3
4
5
6
0
1
5.657
15.588
32
55.902
88.182
Domain: [0, oo)
Range: [0, oo)
x- and y-Intercepts:
0
End behavior: lim fix ) = oo
X—
>O0
Continuity: continuous on [0, oo)
Increasing: (0, oo)
_2
b.
f i x ) = 6x 3
-3
-2
-1
0
1
2
3
2.884
3.780
6
undefined
6
3.780
2.884
Domain: (—oo, 0) U (0, oo)
Range: (0, oo)
Intercepts: none
End behavior: lim fix ) = 0 and lim fix ) = 0
X—
>—oo'
X—
>00-'
Increasing: (—oo, 0)
f
Decreasing: (0, oo)
GuidedPractice
3
3A. fix ) = 2,t 4
88
Lesson 2-1
P o w e r a n d R a d ica l F u n c tio n s
3B. f(x) = 10x 3
Power Regression
in io [
:
BIOLOGICAL SCIENCE The following data represents the resting metabolic rate R in kilocalories
per day for the mass m in kilogram s of several selected animals.
0.3
0.4
0.7
0.8
0.85
2.4
2.6
5.5
6.4
28
35
54
66
46
135
143
331
293
7
7.9
8.41
8.5
13
29.3
29.8
39.5
83.6
265
327
346
363
520
956
839
1036
1948
'6
292
Source: American Journal o f Physical Anthropology
a. Create a scatter plot of the data.
The scatter plot appears to resem ble the square root
function, which is a pow er function. Therefore, test
a pow er regression model.
A Calorie is a unit of energy equal
to the amount of heat needed to
raise the temperature of one
kilogram of water by 1 °C. One
Calorie is equivalent to
4.1868 kilojoules. The average
apple contains 60 Calories.
[0 ,1 0 0 ] scl: 10 by [0, 20 0 0] scl: 200
b . Write a polynomial function to model the data set. Round each coefficient to the nearest
Source: Foods & Nutrition
Encyclopedia
thousandth, and state the correlation coefficient.
Using the PwrReg tool on a graphing calculator and rounding each coefficient to the nearest
thousandth yields/(x) = 69.582x0 759. The correlation coefficient r for the data, 0.995, suggests
that a pow er regression m ay accurately reflect the data.
We can graph the com plete (unrounded) regression by sending
it to the |Y=l menu. In the IY= I menu, pick up this regression
equation by entering I VARS i, S tatistics, EQ. Graph this function
and the scatter plot in the same view ing window. The function
appears to fit the data reasonably well.
StudyTip
Regression Model A polynomial
function with rounded coefficients
will produce estimates different
from values calculated using the
unrounded regression equation.
From this point forward, you can
assume that when asked to use a
model to estimate a value, you are
to use the unrounded regression
equation.
[0 ,1 0 0 ] scl: 10 by [0, 2000] scl: 200
C. Use the equation to predict the resting metabolic rate for a 60-kilogram animal.
Use the CALC feature on the calculator to find/(60). The value of/(60) is about 1554, so the
resting metabolic rate for a 60-kilogram animal is about 1554 kilocalories.
w
GuidedPractice
4. CARS The table shows the braking distance in feet at several speeds in m iles per hour for a
specific car on a dry, well-paved roadway.
Speed
10
20
30
40
50
60
70
Distance
4.2
16.7
37.6
66.9
104.5
150.5
204.9
A. Create a scatter plot of the data.
B. Determine a pow er function to model the data.
C. Predict the braking distance of a car going 80 miles per hour.
2
R adical Functions
An expression w ith rational exponents can be written in radical form.
Exponential Form
—
x«
=
Radical Form
rt/------------yxf
Power functions w ith rational exponents represent the m ost basic of radical functions. A radical
function is a function that can be written as f(x ) = V x ? , where n and p are positive integers greater
than 1 that have no com m on factors. Som e exam ples of radical functions are shown below.
f(x ) = 3 ^ 5 x 3
f(x ) = —5i / x 4 + 3x 2 - 1
f( x ) = \Jx + 1 2 + ^ x — 7
mb
-—
mssb11 a — ■
/H connectED .m cgraw -hill.com |
89
It is im portant to understand the characteristics of the graphs of radical functions as well.
KeyC oncept Radical Functions
Let f be the radical function fix) - Vx where n is a positive integer.
nOdd
nEven
Domain and Range: [0, oo)
Domain and Range:
Continuity: continuous on (— oo, oo)
Symmetry: none
Increasing: (0, oo)
(-oo, oo)
Symm etry: origin
Extrema: absolute minimum at (0, 0)
Extrema: none
End Behavior: lim f(x) = oo
End Behavior:
lim fix) =
X—
►
—
oo
Increasing:
-o o
(-oo, oo)
and
lim f(x) = oo
Graph Radical Functions
Graph and analyze each function. Describe the dom ain, range, intercepts, end behavior,
WatchOut!
a. f i x ) = i t f s x 3
m
Radical Functions Remember
that when n is even, the domain
and range will have restrictions.
fix )
0
1
2
3
4
5
0
2 .9 9
5 .0 3
6 .8 2
8 .4 6
10
Domain and Range: [0, oo)
x- and i/-Intercepts:
0
End behavior: lim fix ) = oo
X—
>ocr
Increasing: (0, oo)
b.
TV
V 6 .r — 8
fix ) - —
X
fix)
-3
-2
-1
0
1
2
3
- 0 .4 8
- 0 .4 6
- 0 .4 2
- 0 .3 8
- 0 .2 9
0 .3 3
0 .4 0
Domain and Range: (—oo, oo)
x-Intercept:
re
y-lntercept: about —0.38
End behavior: lim fix ) = —oo and lim fix ) = oo
X—
►
—OCT
X—KXT
Continuity: continuous on (—oo, oo)
Increasing: (—oo, oo)
f GuidedPractice
5A. fix ) = —\ /l2 x 2 - 5
90
| L esson 2-1 | P o w e r a n d R a d ica l F u n c tio n s
y
5B. fix) = j</2x3 - 16
x
Like radical functions, a radical equation is any equation in w hich a variable is in the radicand. To
solve a radical equation, first isolate the radical expression. Then raise each side of the equation to
a power equal to the index of the radical to elim inate the radical.
Raising each side of an equation to a power som etim es produces extraneous solutions, or solutions
that do not satisfy the original equation. It is im portant to check for extraneous solutions.
^ S S S ^ s° |ve Radical Equations
Solve each equation.
a. 2x = v lO O — 12x — 2
x = V l0 0 ^ l 2 x -
2
2
4x 2 +
8
Original equation
Isolate the radical.
2x + 2 = VlOO - 12x
Square each side to eliminate the radical.
x + 4 = 100 — 12x
4x 2 + 20x - 96 = 0
Subtract 100 — 12x from each side.
StudyTip
4(x 2 + 5x - 24) = 0
Factor.
Common Factors Remember that
you can sometimes factor out a
common multiple before using
any other factoring methods.
x +
4(x +
8
8
x=
Factor.
)(x - 3) = 0
= 0
or
Solve.
x = 3
— 8
CHECK x =
Zero Product Property
x —3 = 0
CHECK x = 3
- 8
2x = VlOO -
1 2
x -
- 1 6 = VlOO -
1 2
(—8 ) -
2
2
x = VlOO - 12x -
2
6
^ VlOO - 12(3) - 2
—16 = Vl9<5 — 2
6
= V 64 —2
- 1 6 + 12 X
6
=
2
6
✓
One solution checks and the other solution does not. Therefore, the solution is 3.
b. \/(x - 5 ) 2 + 14 = 50
Original equation
y J ( x - 5)2 + 14 = 50
Isolate the radical.
\]{x — 5 ) 2 = 36
Raise each side to the third power. (The index is 3.)
(x - 5 ) 2 = 46,656
Take the square root of each side.
x - 5 = ±216
x =
or
2 2 1
Add 5 to each side.
- 2 1 1
A check of the solutions in the original equation confirm s that the solutions are valid.
C. V x — 2 = 5 — V l5 — :
\Jx — 2 = 5 — V l 5 — x
WatchOut!
x -
Squaring Radical Expressions
Take extra care as you square
5 - V 1 5 - x. While similar to
using the FOIL method with
binomial expressions, there are
some differences. Be sure to
account for every term.
2
2
= 25 - 10V 15 - x + (15 - x)
x - 42 = —10V 15 - x
Original equation
Square each side.
isolate the radical.
4x 2 - 168x + 1764 = 100(15 - x)
Square each side.
4x 2 - 168x + 1764 = 1500 - lOOx
Distributive Property
4x 2 — 6 8 x + 264 = 0
Combine like terms.
4(x 2 - 17x +
Factor.
6 6
)= 0
Factor.
4(x — 6 )(x — 11) = 0
x —6 =
0
x =
6
or
x
—
11
=
x=
0
Solve.
11
A check of the solutions in the original equation confirms that both solutions are valid.
►GuidedPractice
6
A. 3x = 3 + V l 8 x - 18
6
B. V 4x + 8 + 3 = 7
6
C. V x + 7 = 3 + V 2 — x
l_conn ectED. m eg raw- hill~cornjj
91
G raph and analyze each function. D escribe the dom ain,
range, intercepts, end behavior, continuity, and where the
function is increasing or decreasing. (Examples 1 and 2)
II
2. g(x) =
5x 2
3. h(x) = - x
4. fix) =: —4X 4
3
5. g(x) = V
3*
6. /(*) =
‘ I*®
7. /(*) = - i x
2
8. g(x)-=4 * ‘
7
10. h(x) --= —3 x -:
11. /(*) = —8 x - 5
12. g(x) ~-= 7x~2
13. /(*) = - V
V 3
4
8 .8 5
8
1 2 .5 2
12
1 5 .3 4
16
17.71
20
1 9 .8 0
24
2 1 .6 9
28
2 3 .4 3
b. Determ ine a pow er function
to m odel the data.
C. Use the function to predict the speed at which a diver
would enter the w ater from a cliff dive of 30 meters.
16. /(*) =
33. WEATHER The wind chill
17. GEOMETRY The volume of a sphere is given by
temperature is the
apparent tem perature felt
on exposed skin, taking
into account the effect
of the wind. The table
shows the wind chill
tem perature produced at
winds of various speeds
when the actual
tem perature is 50°F.
(Example 4)
V(r) = | itr3, where r is the radius. (Exam ple 1)
a. State the domain and range of the function.
b. Graph the function.
-t*
OO
II
2
Graph and analyze each function. D escribe the dom ain,
range, intercepts, end behavior, continuity, and w here the
function is increasing or decreasing. Example 3)
I
i
18.
19. fix ) = —6 x 5
20. gix) = - j x
Speed
(m /s)
of the data.
14. h(x) =
9
Distance
(m)
a. Create a scatter plot
9. /(*) = 2 x - 4
15. h(x) =
diving, com petitors perform three
dives from a height of 28 meters.
Judges award divers a score from
0 to 1 0 points based on degree of
difficulty, take-off, positions, and
water entrance. The table shows
the speed of a diver at various
distances in the dive. (Exam ple 4)
in
X
OO
1.
32. CLIFF DIVING In the sport of cliff
21. fix ) = lOx
3
W ind Speed
(mph)
W ind Chill
5
4 8 .2 2
10
4 6 .0 4
15
4 4 .6 4
20
4 3 .6 0
25
4 2 .7 6
(°F)
30
4 2 .0 4
35
4 1 .4 3
40
4 0 .8 8
a. Create a scatter plot of
i
the data.
6
b. Determ ine a power function to model the data.
22. gix) = —3 x 8
23. hix) = | x 5
24. fix ) = - ± x ~ *
25. f ( x ) = x
C. Use the function to predict the wind chill temperature
when the wind speed is 65 m iles per hour.
5
_3
1
28. h(x) = —5x
29. hix) = j x
2
G raph and analyze each fu nction. D escribe the dom ain,
range, intercepts, end behavior, continuity, and w here the
fu nction is increasing or decreasing. (Example 5)
X
27.
II
26. hix) = 7x 3
3
5
Com plete each step.
b. D eterm ine a pow er function to m odel the data.
C. Calculate the value of each model at x = 30. (Exam ple 4)
34.
f{x ) = 3\/6 + 3x
35.g{x) = - 2 ^ 1 0 2 4 + 8x
36.
fix ) = -| \ Z l 6 x + 48 - 3
(3 7 ) h(x) = 4 + V 7x - 12
38.
g(x) = a/(1 - 4x ) 3 - 16
39.fi x ) = - y / i2 5 x - 7)2 - 49
40.
h(x) = ^ \ Jl7 - 2x -
8
41 . gix ) = V 2 2 — x — V 3x — 3
31.
92
42. FLUID MECHANICS The velocity of the water flowing
1
4
1
1
2
22
2
32
3
85
3
360
4
190
4
2000
(Exam ple 5)
5
370
5
7800
a. Graph the velocity through a nozzle as a function of
6
650
6
2 5 ,0 0 0
7
1000
7
6 0 ,0 0 0
8
1500
8
1 3 0 ,0 0 0
| Lesson 2-1
P o w e r a n d R a d ica l F u n c tio n s
through a hose with a nozzle can be m odeled using
V(P) = 12.1 y/P , where V is the velocity in feet per
second and P is the pressure in pounds per square inch.
pressure.
b. D escribe the domain, range, end behavior, and
continuity of the function and determ ine where it is
increasing or decreasing.
43. AGRICULTURAL SCIENCE The net energy N E m required
to maintain the body weight of beef cattle, in
megacalories (Meal) per day, is estim ated by the formula
4/
Solve each inequality.
63. ^ 1 0 4 0 +
8
x > 4
64. V 41 - 7x > - 1
n
N Em = 0.077 V m , where m is the anim al's mass in
kilograms. One m egacalorie is equal to one million
calories. (Example 6 )
65.
(1 - Ax)2 > 125
6 6
. \J6 + 3x < 9
6 8
. (2x — 6
5
67.
a. Find the net energy per day required to maintain
(19 - 4x ) 3 - 12 < - 1 3
> 64
8 )3
a 400-kilogram steer.
b. If 0.96 m egacalorie of energy is provided per pound of
whole grain corn, how m uch corn does a 400-kilogram
steer need to consume daily to m aintain its body
weight?
69) CHEMISTRY Boyle's Law states that, at constant
tem perature, the pressure of a gas is inversely
proportional to its volum e. The results of an experiment
to explore Boyle's Law are shown.
Volume
(liters)
Pressure
(atmospheres)
45. 0.5x = V 4 - 3x + 2
1.0
3.65
1.5
2.41
47. \j{2x - 5)3 - 10 = 17
2.0
1.79
2.5
1.46
3.0
1.21
3.5
1.02
4.0
0.92
Solve each equation. (Example 6 )
= V —6 - 2x + V31 - 3x
46. —3 = V 22 — x - V 3x - 3
48.
\j(Ax + 164)3 + 36 = 100
49. * = \j2x - 4 + 2
50.
7 + V (—36 - 5x ) 5 = 250
51. x = 5 + \Jx + \
r\
52.
V 6x - 11 + 4 = V l2 x + 1
53. V 4x - 40 = - 2 0
V* +
55. 7 + ^ 1 0 5 4 - 3x =
,, 54.
2
-
1
= V-
2
- 2x
11
b. Determ ine a power function to model the pressure P
as a function of volum e v.
Determine whether each function is a monom ial function
given that a and b are positive integers. Explain your
reasoning.
C. Based on the inform ation provided in the problem
statement, does the function you determ ined in part b
m ake sense? Explain.
56. y = y x 4a
J
b
57. G(x) = —2ax4
d. Use the model to predict the pressure of the gas if the
volum e is 3.25 liters.
58. F(b) = 3ab5x
59. y = h ab
J
3
e. Use the model to predict the pressure of the gas if the
volum e is 6 liters.
46
61. y = Aabx~2
60. H(t) = \ t 2
Uu
W ithout using a calculator, match each graph with the
appropriate function.
62. CHEMISTRY The function r = R 0(Aj^ can be used to
approximate the nuclear radius of an elem ent based on
its molecular mass, where r is length of the radius in
meters, R 0 is a constant (about 1 . 2 x 1 0 - 1 5 m eter), and
A is the molecular mass.
Carbon (C)
12.0
Helium (H)
4.0
Iodine (I)
126.9
Lead (Pb)
207.2
Sodium (Na)
Sulfur (S)
70.
73.
8
?
—
tyA
4
32.1
I
D---I
a. If the nuclear radius of sodium is about 3.412 x 10
meter, what is its m olecular mass?
b. The approximate nuclear radius of an elem ent is
6.030 x 10 - 1 5 meter. Identify the element.
C. The ratio of the m olecular m asses of two elements is
27:8. W hat is the ratio of their nuclear radii?
4 -4 -
15
\A
a. f( x ) = \ U 3 ^
b. g(x) = f x 6
_ Ax
Av-3
c. h{x) =
d. p(x) = 5\j2x + 1
connectED.m cgraw-hill.com
1
93
74. ELECTRICITY The voltage used by an electrical device such
as a DVD player can be calculated using V = V P R , where
V is the voltage in volts, P is the pow er in watts, and R is
IT
the resistance in ohms. The function I = y — can be used
80.
MULTIPLE REPRESENTATIONS In this problem , you will
investigate the average rates of change of power functions.
a. GRAPHICAL For pow er functions of the form/(x) = x ",
graph a function for two values of n such that
< n < 1 , n = 1 , and two values of n such that n >
to calculate the current, where I is the current in amps.
a. If a lamp uses 120 volts and has a resistance of
11
0
graphs from part a to analyze the average rates of
change of the functions as x approaches infinity.
D escribe this rate as increasing, constant, or decreasing.
b. If a DVD player has a current of 10 amps and
consumes 1 2 0 0 watts of power, what is the resistance
of the DVD player?
n
C. O hm 's Law expresses voltage in terms of current and
resistance. Use the equations given above to write
O hm 's Law using voltage, resistance, and amperage.
)l
1 1
2’ 24 c
0
Average Rate of
Change as x—>oo
f(x)
0 < n< 1
Use the points provided to determ ine the pow er function
represented by the graph.
►
I
.
b. TABULAR Copy and com plete the table, using your
ohms, what is the power consum ption of the lamp?
76.
1
n= 1
y
n> 1
C.
- j
VERBAL M ake a conjecture about the average rate of
change of a power function as x approaches infinity for
the intervals 0 < n < 1 , n = 1 , and n > 1 .
M )
X
H.O.T. Problem s
Use Higher-Order Thinking Skills
78.
81. CHALLENGE Show th a t\ j^ 0 - = 2 ln + 3V 2 " + 1.
82. REASONING Consider y = 2 T.
3' - 5
M l
79. OPTICS A contact lens w ith the appropriate depth ensures
proper fit and oxygen permeation. The depth of a lens can
be calculated using the formula S =
■tF W -
where S is the depth, r is the radius of curvature, and d is
the diameter, with all units in millimeters.
a.
D escribe
the value of y if x <
0.
b.
D escribe
the value of y if 0<
x < 1.
c.
D escribe
the value of y if x >
1.
d.
Write a conjecture about the relationship betw een the
value of the base and the value of the pow er if the
exponent is greater than or less than 1. Justify your
answer.
83. PREWRITE Your senior project is to tutor an
underclassm an for four sessions on power and radical
functions. M ake a plan for writing that addresses purpose
and audience, and has a controlling idea, logical
sequence, and time fram e for completion.
a_
84. REASONING Given/(x) = x b, where a and b are integers
with no com m on factors, determ ine whether each
statem ent is true o r false. Explain.
lens
eye
a. If the depth of the contact lens is 1.15 m illimeters and
the radius of curvature is 7.50 millim eters, what is the
diameter of the contact lens?
b. If the depth of the contact lens is increased by
millimeter and the diameter of the lens is
millimeters, what radius of curvature would
be required?
0 .1
8 .2
C. If the radius of curvature remains constant, does the
depth of the contact lens increase or decrease as the
diam eter increases?
94
| Lesson 2-1 | Power and Radical Functions
a. If the value of b is even and the value of a is odd, then
the function is undefined for x < 0 .
b. If the value of a is even and the value of b is odd, then
the function is undefined for x < 0 .
C. If the value of a is 1, then the function is defined for
all x.
\
85. REASONING Consider/(x) = x " + 5. How would you
expect the graph of the function to change as n increases
if n is odd and greater than or equal to 3?
86. WRITING IN MATH Use words, graphs, tables, and equations
to show the relationship betw een functions in exponential
form and in radical form.
Spiral Review
87.
FINANCE If you deposit $1000 in a savings account with an interest rate of r com pounded annually,
then the balance in the account after 3 years is given by B(r) = 1000(1 + r)3, where r is written as a
decimal. (Lesson 1-7)
a. Find a formula for the interest rate r required to achieve a balance of B in the account
after 3 years.
b. W hat interest rate will yield a balance of $1100 after 3 years?
Find ( / + g)(x), ( / — g)(x), ( / •g)(x), and
new function. (Lesson 1-6)
(x) for each fix ) and gix). State the domain of each
89. /(x) = ■
x+ 1
g(x) = X 2 - 1
B. f( x ) = x2 — 2x
g(x) = x + 9
90. f{ x ) =
g(x) = x 2 + 5x
Use the graph o f /(x ) to graph g(x) = \f(x)\ and h(x) =/(|x|). (Lesson 1-5)
91. f( x ) = —4x + 2
92. /(x) =
93. f( x ) = x 2 - 3x - 10
+ 3 —6
Use the graph of each function to estimate intervals to the nearest 0.5 unit on w hich the
function is increasing, decreasing, or constant. Support the answ er numerically. (Lesson 1-4)
94.
95.
y
A
4
-f
\
\
b
1
j
y
j
J
1
/
-4
-f
ix
1
8 x
'v
y —9,
-4 1\X = -----T
x+4
I
\
f(x) = 0 .5 (x + 4 )(x + 1 )(x -2 )
Simplify. (Lesson 0-2)
i + V3*
97.
1
98.
- y fli
2
-V
2
/
99.
3 + V 6i
(1
+
0 2
( - 3 + 2 i)2
Skills Review fo r Standardized Tests
100. SAT/ACT If m and n are both positive, then w hich of
the following is equivalent to
A 3m V ti
D
B
E 8 Vn
6
mVn
6
2 m\/ 18m ?
n\Jl
Vn
C 4V7T
101. REVIEW If f{x, y) = x 2 i/ 3 and Ha, b)
value of f(2 a , 2 b)?
F 50
G 100
H 160
J 320
K 640
1 0
, w hat is the
102. REVIEW The num ber of m inutes m it takes c children
to eat p pieces of pizza varies directly as the number
of pieces of pizza and inversely as the number of
children. If it takes 5 children 30 minutes to eat
1 0 pieces of pizza, how m any minutes should it take
15 children to eat 50 pieces of pizza?
A 30
C 50
B 40
D 60
103. If
5m + 2 = 3, then m = ?
F 3
H 5
G 4
J
^
...
6
/H c o n n e c tE D .m c g ra w -h ill.c o m |
95
p—
Graphing Technology Lab
j #§§i
•
I
oooo
oooo
oooo
Behavior of Graphs
VC D O O >
Objective
In Lesson 1 -3 , you analyzed the end behavior of functions by m aking a ta b le of values and graphing
G raph and a n a ly ze th e
th em . For a polynomial function, the behavior of the graph can be d eterm in ed by analyzing specific
b e h a v io r of poly n o m ia l
term s of the function.
fu n c tio n s .
Activity 1
Graph Polynomial Functions
Sketch each graph, and identify the end behavior of the function.
a.
f i x ) = x 3 + 6x2 - 4x + 2
Use a table of values to sketch the graph.
*
I
fix)
-1 0
-5
-2
0
2
5
10
-3 5 8
47
26
2
26
257
1562
In the graph of fix ), it appears that lim fix ) =
lim fix ) = oo.
b.
—oo
and
[ - 1 0 , 1 0 ] scl: 1 by [ - 4 0 , 60] scl: 10
g ix ) = - 2 x 3 + 6x2 - 4x + 2
Stud; Tip
Table of Values Be sure to use
enough points to get the overall
shape of the graph.
-8
-5
-2
0
2
5
8
1442
422
50
2
2
-1 1 8
-6 7 0
In the graph of g{x), it appears that lim g(x) =
lim g(x) = — oo.
x—>00 °
oo
and
-5 , 5] scl: 1 by [ - 4 0 , 60] scl: 10
C. hix) = —x 4 + x 3 + 6x2 — 4x + 2
-8
-5
-2
0
2
5
8
-4 1 9 0
-5 7 8
10
2
10
-3 6 8
-3 2 3 0
X
h{x)
In the graph of h(x), it appears that lim h{x) =
lim h(x) = - o o .
— oo
and
X —>oo
Analyze the Results
1. Look at the terms of each function above. W hat differences do you see?
2. How is the end behavior of the graphs of each function affected by these differences?
3. Develop a pattern for every possible type of end behavior of a polynom ial function.
4. Give an example of a polynom ial function w ith a graph that approaches positive infinity
when x approaches both negative infinity and positive infinity.
Exercises
Describe the end behavior of each function w ithout m aking a table of values or graphing.
5. f(x) = - 2 x 3 + 4x
8.
96
i
Lesson 2-2
g(x) =
6
x 6 - 2x 2 + 10x
6
. fix ) = 5x 4 + 3
9. g(x) = 3x — 4x 4
7. f(x ) = - x
10. h(x) =
6
5
+ 2x - 4
x 2 - 3x 3 - 2x 6
Polynomial Functions
: Then
: •Why?
Savings as a Percent
------------------------- —
of functions.
(Lesson 1-2)
1
functions.
( Model real-world
sdata with polynomial
functions.
personal savings as a percent of
disposable income in the United
States. Often data with multiple
relative extrema are best
modeled by a polynomial
function.
1966
1982
1998
2010
Year
E
NewVocabulary
polynomial function
polynomial function of
degree n
leading coefficient
leading-term test
quartic function
turning point
quadratic form
repeated zero
multiplicity
Graph Polynom ial Functions In Lesson 2-1, you learned about the basic characteristics
of monom ial functions. M onom ial functions are the m ost basic polynom ial functions. The
sums and differences of monom ial functions form other types of polynom ial functions.
1
Let n be a nonnegative integer and let a 0, a v a 2,
the function given by
f{ x ) = a nx n
a n _ v a n be real num bers w ith a n =/=0. Then
- 1 -x n
1
+ ■•• + a 2x 2 + a xx + a 0
is called a polynomial function of degree n. The leading coefficient of a polynom ial function
is the coefficient of the variable w ith the greatest exponent. The leading coefficient o ff( x ) is a n.
You are already fam iliar w ith the follow ing polynom ial functions.
Q u a d ra tic F u n c tio n s
C o n s ta n t F u n c tio n s
y
I il il
f(x) = c, c ± 0
y
0
X
D e g re e : 0
D e g re e : 1
The zero function is a constant function w ith no degree. The graphs of polynom ial functions share
certain characteristics.
Graphs of Polynomial Functions
Nonexamples
Example
J
V
0
Polynomial functions are defined and
continuous for all real numbers and have
smooth, rounded turns.
s
0
V
:A ,
X
Graphs of polynomial functions do not have breaks, holes, gaps, or
sharp corners.
97
Recall that the graphs of even-degree, non-constant monom ial functions resem ble the graph of
fix ) = x 2, while the graphs of odd-degree monomial functions resem ble the graph of f i x ) = x 3.
You can use the basic shapes and characteristics of even- and odd-degree m onomial functions and
what you learned in Lesson 1-5 about transform ations to transform graphs of monom ial functions.
liM m
l l l Graph Transformations of Monomial Functions
G raph each function,
a. f( x ) = i x -
b. gix) = - x4 + 1
2 )5
This is an odd-degree function, so its
graph is sim ilar to the graph of y = x 3.
The graph of/(x) = (x — 2 ) 5 is the graph
of y = x 5 translated 2 units to the right.
y
This is an even-degree function, so its graph
is similar to the graph of y = x 2. The graph
of gix) = —x 4 + 1 is the graph of y = x 4
reflected in the x-axis and translated 1 unit up.
J ! iy
gix) = - x 4 + 1
i
I1
(
I
J0
/
X
6
i
fix)
1
j
T
— X
I I
1
I I
\
X
1
►GuidedPractice
1A. f{ x ) = 4 - x 3
1B. gix ) = (x + 7)4
In Lesson 1-3, you learned that the end behavior of a function describes how the function behaves,
rising or falling, at either end of its graph. As x —* —oo and x —> oo, the end behavior of any
polynomial function is determ ined by its leading term. The lead ing term test uses the power
and coefficient of this term to determ ine polynom ial end behavior.
KeyConcept Leading Term Test for Polynomial End Behavior
The end behavior of any non-constant polynomial function f(x) = a„x" h—
+ a ,x + a0 can be described in one of
the following four ways, as determined by the degree n of the polynomial and its leading coefficient a„.
n odd, a„ positive
lim fix) =
—
*—
oo
n odd, a„ negative
- o o and lim
X—
*oo f(x) =
oo
lim fix) = <
X—
>oo
lim fix) =
X—
>-oo
lim f{x) = oo and lim fix) = - o o
♦—OO
lim fix) =
X—
*oc
-o o
lim f(x) =
X—
►
—
oo
oo and lim
lim f(x ) = o o i
X—
>oo fix) =
1
-o
n even, a„ negative
oo
lim fix) =
X—
►
—
oo
lim f(x ) = c
y=fix)\
lim fix) =
X—
►
—
oo
| Lesson 2-2 | P o ly n o m ia l F u n c tio n s
y
lim fix) =
X—
►
—
oo
n even, a„ positive
98
X—KX>
-
- o o and lim
X—
>oofix) =
-o o
M W n fy ^ Apply the Leading Term Test
Describe the end behavior of the graph of each polynom ial function using limits.
Explain your reasoning using the leading term test.
a. f( x ) — 3x4 — 5x2 — 1
yu
The degree is 4, and the leading coefficient is 3. Because
the degree is even and the leading coefficient is positive,
lim f(x ) = oo and lim f(x ) = oo.
vi r
v
V
—KVl
--------- t 0 1
A
\l
\k
■ (x ) = 3 x 4 — 5 x 2 - 1 -
WatchOut!
Standard Form The leading term
of a polynomial function is not
necessarily the first term of a
polynomial. However, the leading
term is always the first term of a
polynomial when the polynomial is
written in standard form. Recall
that a polynomial is written in
standard form if its terms are
written in descending order of
exponents.
X
>
b. g(x) = —3 x 2 — 2 x 7 + 4x 4
Write in standard form as g(x) = —2x 7 + 4x4 — 3x2. The
degree is 7, and the leading coefficient is —2. Because
the degree is odd and the leading coefficient is negative,
lim /(x) = oo and lim f( x ) = —oo.
1I
ty
g(x = - 2 x 7 + 4 x 4 — 3 x 2
X
f
c.
h(x) = x 3 — 2x2
The degree is 3, and the leading coefficient is 1. Because
the degree is odd and the leading coefficient is positive,
lim f(x) = —oo and lim f(x) = oo.
GuidedPractice
2A. g(x) = 4x 5 — 8 x 3 + 20
2B. h(x) = —2xb + l l x 4 + 2x 2
Consider the shapes of a few typical third-degree polynom ial or cubic functions and fourth-degree
polynom ial or quartic functions shown.
Typical Cubic Functions
Typical Quartic Functions
Observe the number of x-intercepts for each graph. Because an x-intercept corresponds to a real
zero of the function, you can see that cubic functions have at m ost 3 zeros and quartic functions
have at m ost 4 zeros.
Turning point‘dindicate where the graph of a function changes from increasing to decreasing, and
vice versa. M axim a and minim a are also located at turning points. N otice that cubic functions have
at m ost 2 turning points, and quartic functions have at m ost 3 turning points. These observations
can be generalized as follows and show n to be true for any polynom ial function.
&
connîD^grewjhHiLco^
99
KeyC oncept Zeros and Turning Points of Polynomial Functions
StudyTip
Look Back Recall from Lesson 1 -2
that the x-intercepts of the graph
of a function are also called the
zeros of a function. The solutions
of the corresponding equation are
called the roots of the equation.
A polynomial function f of degree n > 1 has at most n
distinct real zeros and at most n - 1 turning points.
E x a m p le
80
Let f{x) = 3 x 6 — 10 x 4 — 15 x 2. Then f has
at most 6 distinct real zeros and at most
5 turning points. The graph of f suggests that
the function has 3 real zeros and 3 turning points.
-4
—
/
y
Ix
A x
\
/
h)
fix) = 3 x 6 - 1 0 x 4 - 1 5 x 2
i i i
Recall that if/ is a polynom ial function and c is an x-intercept of the graph o f f , then it is equivalent
to say that:
• c is a zero of/,
• x = c is a solution of the equation f( x ) =
0
, and
• (x — c) is a factor of the polynomial/(x).
You can find the zeros of some polynom ial functions using the same factoring techniques you
used to solve quadratic equations.
Zeros of a Polynomial Function
State the num ber of possible real zeros and turning points of f( x ) = x 3 — 5x2 + 6x.
Then determ ine all of the real zeros by factoring.
The degree of the function is 3, so/ h as at most 3 distinct real zeros and at m ost 3 — 1 or
2 turning points. To find the real zeros, solve the related equation/(x) = 0 by factoring.
x3 - 5 x 2 + 6x = 0
x(x 2 — 5x +
6
)= 0
x(x - 2)(x - 3) = 0
Set f(x) equal to 0.
Factor the greatest common factor, x.
Factor completely.
So,/ has three distinct real zeros, 0, 2, and 3. This is consistent w ith a cubic function having at
most 3 distinct real zeros.
CHECK You can use a graphing calculator to graph
f( x ) = x 3 — 5x 2 + 6 x and confirm these zeros.
Additionally, you can see that the graph has
2 turning points, which is consistent w ith cubic
functions having at m ost 2 turning points.
f
GuidedPractice
State the num ber of possible real zeros and turning points of each function. Then determine
all of the real zeros by factoring.
StudyTip
3A. f( x ) = x — 6x — T lx
3B. f( x )
■8 x 2 + 15
Look Back To review techniques
for solving quadratic equations,
see Lesson 0-3.
In some cases, a polynom ial function can be factored using quadratic techniques if it has
quadratic form .
KeyConcept Quadratic Form
W o rd s
A polynomial expression in x is in quadratic form if it is written as au2 + bu+ c for any numbers a, b,
and c , a £ 0, where u is some expression in x.
S y m b o ls
x 4 — 5x 2 - 14 is in quadratic form because the expression can be written as (x 2 ) 2 - 5 (x 2) - 14.
If u = x 2, then the expression becomes u2 — 5u —14.
V
100
I Lesson 2 -2 | P o ly n o m ia l F u n c tio n s
J
P E S S S E E êros ° f a Polynomial Function in Quadratic Form
State the num ber of possible real zeros and turning points fo r# (x) = x 4 — 3x2 — 4.
Then determ ine all of the real zeros by factoring.
The degree of the function is 4 , so g has at m ost 4 distinct real zeros and at m ost 4 — 1 or
3 turning points. This function is in quadratic form because x 4 — 3x2 — 4 = (x 2) 2 — 3 (x 2) — 4 .
Let u = x 2.
(x 2) 2 — 3 ( x 2) — 4 = 0
« 2 — 3m — 4
= 0
(u + 1 )(w — 4 )
= 0
(x2 + l)(x 2 — 4 )
= 0
(x2 + l)(x + 2)(x — 2)
= 0
y
0
X
Set g(x) equal to 0.
x2 + l = 0
or
/A
V
Factor the quadratic expression.
Substitute x2 for u.
Factor completely.
x + 2 = 0
x = ± V —1
- X
Substitute u for x2.
or
x = —2
x —2 = 0
x = 2
Solve for x.
Because ± V —1 are not real zeros, g has two distinct real zeros, —2 and 2. This is consistent with
a quartic function. The graph of g(x) = x 4 — 3 x 2 — 4 in Figure 2.2.1 confirm s this. Notice that
there are 3 turning points, w hich is also consistent w ith a quartic function.
Figure 2.2.1
f
GuidedPractice
State the nu
numbe:
m ber o f p o ssible real zeros and turning points o f each fu nction. T h en determ ine
all of the real zei
zeros b y factoring.
4A. g(x) = x 4 — 9x 2 + 18
4B.h(x)= x 5 — 6 x 3 — 16x
If a factor (x — c) occurs more than once in the com pletely factored form of/(x), then its related
zero c is called a repeated zero. W hen the zero occurs an even num ber of times, the graph will be
tangent to the x-axis at that point. W hen the zero occurs an odd num ber of times, the graph will
cross the x-axis at that point. A graph is tangent to an axis w hen it touches the axis at that point,
but does not cross it.
■ E H jH S J P o ly n o m ia l Function with Repeated Zeros
State the n um ber o f p o ssible real zeros and turning points o f h(x) = —x 4 — x 3 + 2x2.
Th en determ ine all o f the real zeros b y factoring.
The degree of the function is 4, so h has at m ost 4 distinct real zeros and at m ost 4 — 1 or
3 turning points. Find the real zeros.
-x
k
4
- x3 +
2
—x 2 (x 2 + x —x 2(x - l)(x +
x2 =
0
Set h(x) equal to 0.
2
)=
0
Factor the greatest common factor, — x2.
2
)=
0
Factor completely.
The expression above has 4 factors, bu t solving for x yields only 3 distinct real zeros, 0 ,1 , and
—2. O f the zeros, 0 occurs twice.
Figure 2.2.2
The graph of h(x) = —x 4 — x 3 + 2x 2 show n in Figure 2.2.2 confirm s these zeros and shows that h
has three turning points. Notice that at x = 1 and x = —2, the graph crosses the x-axis, but at
x = 0 , the graph is tangent to the x-axis.
p Guided Practice
State the n um ber of p o ssib le real zeros and turning points of each function. T h en determ ine
all of the real zeros b y factoring.
5A. g(x) = —2x 3 — 4x 2 + 16x
5B./(x) = 3x 5 — 18x4 + 2 7x3
101
In h(x) = - x 2(x — l)(x + 2) from Exam ple 5, the zero x = 0
occurs 2 times. In k(x) = (x — l ) 3(x + 2)4, the zero x = 1
occurs 3 times, while x = —2 occurs 4 times. N otice that
in the graph of k shown, the curve crosses the x-axis at
x = 1 but not at x = —2. These observations can be
generalized as follows and shown to be true for all
KeyConcept Repeated Zeros of Polynomial Functions
StudyTip
v
If (x - c f is the highest power of (x - c) that is a factor of polynomial function f then c is a zero of m ultiplicity m of f,
where m is a natural number.
Nonrepeated Zeros A nonrepeated
zero can be thought of as having a
multiplicity of 1 or odd multiplicity.
A graph crosses the x-axis and has
a sign change at every nonrepeated
zero.
• If a zero c has odd multiplicity, then the graph of ^crosses the x-axis at x = c and the value of f{x) changes signs
at x = c.
• If a zero c has even multiplicity, then the graph of f is tangent to the x-axis at x = c and the value of f(x) does not
change signs at x = c.
____________
You now have several tests and tools to aid you in graphing polynom ial functions.
W s fffifflB P ] Graph a Polynomial Function
F o r /(x ) = x(2x + 3)(x — l ) 2, (a) apply the leading-term test, (b ) determ ine the zeros and state
the m ultiplicity of any repeated zeros, (c) find a few additional points, and then (d) graph
the function.
a. The product x(2x + 3)(x — l ) 2 has a leading term of x(2x)(x ) 2 or 2x4, so / has degree 4 and
leading coefficient 2. Because the degree is even and the leading coefficient is positive,
lim fix ) = oo and lim f(x ) = oo.
X—►—oo
x —>oo
b. The distinct real zeros are 0, —1.5, and 1. The zero at 1 has multiplicity 2.
C. Choose x-values that fall in the intervals determ ined by the zeros of the function.
Interval
x-value in Interval
( - 0 0 ,-1.5)
- 2
(-1.5, 0)
-1
f(x)
(x, f[x))
/(—2 ) = 18
(-2,18)
f ( - 1) = - 4
( - 1 ,- 4 )
(0 , 1 )
0.5
f(0.5) = 0.5
(0.5, 0.5)
(1 , 0 0 )
1.5
f(1.5) = 2.25
(1.5,2.25)
d. Plot the points you found (Figure 2.2.3).
The end behavior of the function tells you
that the graph eventually rises to the left
and to the right. You also know that the
graph crosses the x-axis at nonrepeated
zeros —1.5 and 0, but does not cross the
x-axis at repeated zero 1 , because its
m ultiplicity is even. Draw a continuous
curve through the points as show n in
Figure 2.2.4.
Figure 2.2.3
Figure 2.2.4
p GuidedPractice
For each function, (a) apply the leading-term test, (b) determ ine the zeros and state the
m ultiplicity of any repeated zeros, (c) find a few additional points, and then (d) graph the
function.
6A. f(x ) = —2x(x — 4)(3x — l ) 3
V
102
Lesson 2-2 | Polynom ial Functions
6B.h(x) = —x 3 + 2x 2 +
.......................
8
x
9 M odel D ata You can use a graphing calculator to model data that exhibit linear, quadratic,
mm cubic, and quartic behavior by first exam ining the num ber of turning points suggested by a
scatter plot of the data.
Real-World Example 7 Model Data Using Polynomial Functions
SAVINGS Refer to the beginning of the lesson. The average personal savings as a percent of
disposable income in the United States is given in the table.
Year
% Savings
1970
1980
1990
1995
2000
2001
2002
2003
2004
2005
9.4
10.0
7.0
4.6
2.3
1.8
2.4
2.1
2.0
-0 .4
Source: U.S. Department of Commerce
a. Create a scatter plot of the data and determ ine the type of polynom ial function that could
be used to represent the data.
Enter the data using the list feature of a graphing calculator.
Let L1 be the num ber of years since 1970. Then create a scatter
plot of the data. The curve of the scatter plot resem bles the
graph of a quadratic equation, so we will use a quadratic
regression.
A college graduate planning to
retire at 65 needs to save an
average of $10,000 per year
toward retirement.
Source: Monroe Bank
[ - 1 , 3 6 ] scl: 1 by [ - 1 , 1 1 ] scl: 1
b. Write a polynomial function to model the data set. Round each coefficient to the nearest
thousandth, and state the correlation coefficient.
Using the QuadReg tool on a graphing calculator and rounding each coefficient to the nearest
thousandth yields/(x) = —0.009x2 + 0.033* 4- 9.744. The correlation coefficient r1 for the data
is 0.96, w hich is close to 1, so the model is a good fit.
We can graph the com plete (unrounded) regression by sending
it to the |Y= 1menu. If you enter Lv L2, and Y 1 after QuadReg,
as show n in Figure 2.2.5, the regression equation will be entered
into Y 1. Graph this function and the scatter plot in the same viewing
window. The function appears to fit the data reasonably well.
Figure 2.2.5
C. Use the model to estimate the percent savings in 1993.
Because 1993 is 23 years after 1970, use the CALC feature on a calculator to find/(23).
The value of/(23) is 5.94, so the percent savings in 1993 w as about 5.94%.
d.
Use the model to determ ine the approximate year in which the percent savings
reached 6.5%.
Graph the line y = 6.5 for Y 2. Then use 5: intersect on the
CALC menu to find the point of intersection of y = 6.5 with/(.t).
The intersection occurs when x ~ 21, so the approxim ate year in
w hich the percent savings reached 6.5% was about 1970 + 21
or 1991.
- 1 , 3 6 ] scl: 1 by [ - 1 , 1 1 ] scl: 1
^ GuidedPractice
7.
POPULATION The m edian age of the U.S. population by year predicted through 2080 is shown.
Year
1900
1930
1960
1990
2020
2050
2080
M edian Age
22.9
26.5
29.5
33.0
40.2
42.7
43.9
Source: U.S. Census Bureau
a. Write a polynom ial function to model the data. Let L1 be the num ber of years since 1900.
b.
Estim ate the m edian age of the population in 2005.
C. According to your m odel, in what year did the m edian age of the population reach 30?
103
Exercises
= Step-by-Step Solutions begin on page R29.
43. RESERVOIRS The num ber of feet below the maxim um
w ater level in W isconsin's Rainbow Reservoir during
ten m onths in 2007 is shown. (Example 7)
Graph each function. (Example 1)
1. f i x ) = ix + 5 ) 2
2. f i x ) = i x -
3. f i x ) = x 4 - 6
4. fi x ) = x 5 + 7
5. f i x ) = (2x ) 4
6. f i x ) = i l x f - 16
7. f i x ) = ix - 3 ) 4 +
Month
10. f i x ) = ( i t ) 3 +
13.
f i x ) = 2x6 + 4x5 + 9x2
14. g(x) = 5x4 + 7x5 - 9
15.
g(x) = —7.x3 + 8x4 — 6x6
16. h{x) = 8 x 2 + 5 - 4.r3
17.
hix) = 4x 2 + 5x 3 — 2x5
11
September
16.5
April
9
November
11.5
May
7.5
December
8.5
a. Write a model that best relates the water level as a
function of the num ber of m onths since January.
b. Use the m odel to estim ate the water level in the
reservoir in October.
Use a graphing calculator to write a polynomial function to
model each set of data. (Example 7)
44.
18. f{ x ) = x(x + l)(x - 3)
19.
gix ) = x \ x 4- 4)(—2x + 1)
20. f i x ) = - x ( x - 4)(x + 5)
21.
g{x) = x 3{x + l) { x 2 - 4)
45.
46.
a. Graph the function using a graphing calculator.
b. Describe the end behavior of the graph of the function
using limits. Explain using the leading term test.
State the num ber of possible real zeros and turning points
of each function. Then determine all of the real zeros by
factoring. (Examples 3-5)
(2 3 ) f ix ) = x5 + 3x4 + 2x 3
, 24. fi x )
25. fi x ) = x4 + 4x2 - 21
26. fi x )
27. fi x ) = x 6 - 6x3 - 16
( g ) fi x )
29. fi x ) = 9x6 - 36x4
M b fix )
fi x ) = x(x + 4){x - l ) 2
W / M = x 2ix - 4)ix + 2)
35.
fi x ) = —x(x + 3)2(x - 5)
36.
37.
fi x ) = - x i x - 3)ix + 2 ) 3
M f ( x ) = - ( x + 2)2 ( x - 4
39.
fi x ) = 3x 3 - 3x2 - 36x
'40. f i x ) = —2x3 — 4x 2 + 6x
41.
f i x ) = x 4 + x 3 - 20x2
42>M x) = x 5 + 3x 4 — 10x 3
fi x ) = 2x(x + 5 )2{ x — 3)
P o ly n o m ia l F u n c tio n s
-1
0
1
2
3
fix)
8.75
7.5
6.25
5
3.75
2.5
1.25
5
7
8
10
11
2
5
6
4
-1
I
!
47.
12
15
16
-3
5
9
-2 .5 3
-2
-1 .5
-1
-0 .5
0
0.5
1
1.5
23
11
7
6
6
5
3
2
4
30
35
40
45
50
55
60
65
70
75
52
41
32
44
61
88
72
59
66
93
48. ELECTRICITY The average retail electricity prices in the
U.S. from 1970 to 2005 are shown. Projected prices for
2010 and 2020 are also shown. (Example 7)
Price
Year
id / kWh)
33.
| Lesson 2-2
-2
Year
For each function, (a) apply the leading-term test,
(b) determine the zeros and state the m ultiplicity of any
repeated zeros, (c) find a few additional points, and then
(d) graph the function. (Example
104
-3
32x
m
31. fi x ) = 4x4 - 4x3 - 3x 2
X
f(x)
22. ORGANIC FOOD The number of acres in the United States
used for organic apple production from 2000 to 2005 can
be modeled by a(x) = 43.77x 4 - 498.76x3 + 1310.2*2 +
1626.2* + 6821.5, where x = 0 is 2000. (Example 2)
5.5
9
Source: Wisconsin Valley Improvement Company
Describe the end behavior of the graph of each polynomial
function using limits. Explain your reasoning using the
leading term test. (Example 2)
8
July
August
March
WATER If it takes exactly one minute to drain a 10-gallon
12. fi x ) = - 5 x 7 + 6x4 +
4
Level
10
February
8
tank of water, the volume of water remaining in the tank
can be approximated by v(t) = 1 0 ( 1 — f)2, where f is time
in minutes, 0 < t < 1. Graph the function. (Example 1)
Month
Level
January
8. f i x ) = ix + 4 ) 3 - 3
6
9. f( x ) = \ ix - 9 f
11.
6 )3
)2
Price
(c / kWh)
1970
6.125
1995
7.5
1974
7
2000
6.625
1980
7.25
2005
6.25
1982
9.625
2010
6.25
1990
8
2020
6.375
Source: Energy Information Administration
a. W rite a model that relates the price as a function of the
num ber of years since 1970.
b. Use the model to predict the average price of
electricity in 2015.
c. According to the m odel, during w hich year w as the
price 7<t for the second time?
49. COMPUTERS The numbers of laptops sold each quarter
from 2005 to 2007 are shown. Let the first quarter of 2005
be 1, and the fourth quarter of 2007 be 12.
Quarters
Sale
(Thousands)
1
423
2
462
3
495
4
634
5
587
6
498
7
798
8
986
9
969
10
891
11
1130
12
1347
a. Predict the end behavior of a graph of the data as x
approaches infinity.
Determine w hether the degree n of the polynomial for each
graph is ev en or o d d and w hether its leading coefficient a„ is
p o s itiv e or n eg a tiv e.
68. MANUFACTURING A com pany manufactures
alum inum containers for energy drinks.
b. Use a graphing calculator to graph and model the
a. W rite an equation V that represents the
volum e of the container.
data. Is the model a good fit? Explain your reasoning.
b. Write a function A in terms of r that
C. Describe the end behavior of the graph using limits.
represents the surface area of a container
w ith a volum e of 15 cubic inches.
Was your prediction accurate? Explain your reasoning.
C. Use a graphing calculator to determine
Determine w hether each graph could show a polynomial
function. Write yes or no. If not, explain w hy not.
the m inim um possible surface area of
the can.
Determine a polynom ial function that has each set of zeros.
M ore than one answ er is possible.
69. 5 , - 3 , 6
70. 4 , - 8 , - 2
71. 3 , 0 , 4 , - 1 , 3
72. 1 , 1 , - 4 , 6 , 0
7 3
.2 .
4'
y
,,
4'
_ 2.
74. - 1 , - 1 , 5 , 0 ,|
3
6
Year
-i
—L
— 4
o
_ 4
POPULATION The percent of the United States population
living in m etropolitan areas has increased.
c V
J
_ 3
M
)'x
-
Percent of
Population
1950
56.1
1960
63
1970
6 8 .6
1980
7 4 .8
1990
7 4 .8
2000
7 9 .2
X
Find a polynomial function of degree n with only the
following real zeros. M ore than one answ er is possible.
54. —1; n = 3
55. 3 ;n = 3
56.
57. - 5 , 4; n = 4
a. W rite a model that relates the percent as a function of
the num ber of years since 1950.
58. 7 ;n = 4
59. 0, —4; n = 5
b. Use the model to predict the percent of the population
60. 2 ,1 ,4 ; n = 5
61. 0, 3, - 2 ; n = 5
6
, -3 ; n = 4
1 ,62. n o real zeros; n = 4-V
63. no real zeros; n =
that w ill be living in m etropolitan areas in 2015.
6
c. Use the model to predict the year in which 85% of the
population will live in m etropolitan areas.
105
Create a function with the following characteristics.
Graph the function.
76. degree = 5, 3 real zeros, lim = oo
°
X—>00
89. f£n MULTIPLE REPRESENTATIONS In this problem , you will
investigate the behavior of com binations of polynom ial
functions.
a. GRAPHICAL Graph/(x), gix), and h(x) in each row on
77. degree = 6 , 4 real zeros, lim = —oo
X—
>oo
°
the same graphing calculator screen. For each graph,
m odify the window to observe the behavior both on a
large scale and very close to the origin.
78. degree = 5, 2 distinct real zeros, 1 of which has a
multiplicity of 2 , lim = oo
1
y
X—
>00
1
X—>oo
J
9(x) =
ft(x) =
X2
X
X3
-X
X3 + X2
X3
X2
CO
f(x) =
X2 + X
I
, 3 distinct real zeros, 1 of which has a
multiplicity of 2 , lim = —oo
6
><
79. degree =
80. WEATHER The temperatures in degrees Celsius from
10 a.m. to 7 p.m. during one day for a city are shown
where x is the number of hours since 1 0 a.m.
Time
Temp.
Time
Temp.
0
1
4.1
5
10
5.7
6
7
2
7.2
7
4.6
3
7.3
8
2.3
4
9.4
9
- 0 .4
b. ANALYTICAL Describe the behavior of each graph of fix )
in terms of gix) or h{x) near the origin.
C.
ANALYTICAL D escribe the behavior of each graph of fix )
in terms of gix) or h(x) as x approaches oo and —oo.
d, VERBAL Predict the behavior of a function that is
a com bination of two functions a and b such that
f i x ) = a + b, where a is the term of higher degree.
a. Graph the data.
b. Use a graphing calculator to model the data using a
polynomial function with a degree of 3.
C.
Repeat part b using a function with a degree of 4.
d.
W hich function is a better model? Explain.
For each of the following graphs:
a. Determine the degree and end behavior.
H.O.T. Problem s
90. ERROR ANALYSIS Colleen and M artin are modeling
the data shown. Colleen thinks the model should be
fi x ) = 5.754x 3 + 2.912x2 — 7.516x + 0.349. M artin thinks it
should be fix ) = 3.697x2 + 11.734x — 2.476. Is either of
them correct? Explain your reasoning.
X
f(x )
X
the zeros are integral values.
-2
-1 9
0.5
-2
Use the given point to determine a function that fits
the graph.
-1
5
1
1.5
0.4
2
43
b. Locate the zeros and their multiplicity. Assume all of
C.
y
20
91. REASONING Can a polynom ial function have both an
absolute m axim um and an absolute minim um ? Explain
your reasoning.
60
- 8
Vi
X
—BO
I
-1 2 0
\
\k
8
92. REASONING Explain why the constant function fix ) = c,
_
-1 2 8
c / 0 , has degree 0 , but the zero function/(x) =
degree.
)
I ‘
OO
P
has no
zeros of f i x ) = x 3 + 5x 2 — x 2 — 5x — 12x — 60. Explain
each step.
94. REASONING How is it possible for more than one function
X
to be represented by the same degree, end behavior, and
distinct real zeros? Provide an exam ple to explain your
reasoning.
I4
.
0
93. CHALLENGE Use factoring by grouping to determ ine the
y
\
I \)
-3, -9)
0
V
-8
95. REASONING W hat is the m inim um degree of a polynomial
State the num ber of possible real zeros and turning points
of each function. Then find all of the real zeros by factoring.
85.
f{ x ) = 16.r4 + 72x2 + 80
86.
f( x ) = - 1 2 x 3 - 44x2 - 40x
(8 7 ) f( x ) = —24x4 + 24x3 — 6x2 88. f i x ) = x 3 + 6x2 — 4x — 24
106
Lesson 2-2
function that has an absolute maximum, a relative maximum,
and a relative minim um ? Explain your reasoning.
96. WRITING IN MATH Explain how you determ ine the best
polynom ial function to use when m odeling data.
Sp ira l R ev iew
Solve each equation. (Lesson 2-1)
97.
98. d +
Vz + 3 = 7
Vd2 —
8
99. V x - 8 = V l 3 + ;
= 4
100. REMODELING An installer is replacing the carpet in a 12-foot by 15-foot living room.
The new carpet costs $13.99 per square yard. The form ula/(x) = 9x converts square yards to
square feet. (Lesson 1-7)
a. Find the inverse/_ 1 (x). W hat is the significance of/_ 1 (x)?
b. How much will the new carpet cost?
Given fix ) = 2x2 — 5x + 3 and gix) = 6x + 4, find each function. (Lesson 1-6)
101. (/ + # )(*)
103. [g ° f ] { x )
m.[f°g]{x)
Describe how the graphs of fix ) = x 2 and gix) are related. Then w rite an equation for g Or). (Lesson 1-5)
105.
104.
12
t+ z
y
X
- 8
-Ug(x)\h-~
—L O
L
>4
- 8'
12
gix)
\
\
*
\
107. BUSINESS A com pany creates a new product that costs $25 per item to produce. They hire a
marketing analyst to help determ ine a selling price. After collecting and analyzing data
relating selling price s to yearly consumer dem and d, the analyst estim ates dem and for the
product using d = —200s + 15,000. (Lesson 1-4)
a. If yearly profit is the difference betw een total revenue and production costs,
determine a selling price s > 25, that will m axim ize the com pany's yearly profit P.
(Hint: P = sd — 25d)
b. W hat are the risks of determ ining a selling price using this method?
The scores for an exam given in physics class are given. (Lesson 0-8)
82, 77, 84, 98, 93, 71, 76, 64, 89, 95, 78, 89, 65, 88, 54,
96, 87, 92, 80, 85, 93, 89, 55, 62, 79, 90, 86, 75, 99, 62
108. Make a box-and-whisker plot of the test.
109. W hat is the standard deviation of the test scores?
S k ills R ev iew fo r S ta n d a rd iz e d T e sts
110. SAT/ACT The figure shows the
112. MULTIPLE CHOICE W hich of the follow ing equations
intersection of three lines. The
figure is not drawn to scale.
represents the result of shifting the parent function
y = x 3 up 4 units and right 5 units?
x =
A y + 4 = (x + 5 ) 3
C y + 4 = (x - 5 ) 3
B y - 4 = (x + 5 ) 3
D y — 4 = (x — 5 ) 3
A 16
D 60
B 20
E 90
113. REVIEW W hich of the follow ing describes the numbers
C 30
in the dom ain of h(x) =
111. Over the domain 2 < x < 3, w hich of the following
functions contains the greatest values of I/?
X+ 3
x -2
x —5
G y =
x+1
f y
„
______ 2 .
H y = x —3
F x + 5
G x > f
x ^?
x —5
H x>|,x#5
J
J V = 2x
co n n ectE D .m cg raw -h ill.co m l
107
oooo
oooo
oooo
Hidden Behavior of Graphs
Objective
Use Tl-Nspire technology
to explore the hidden
behavior of graphs.
CDOO
Using graphing technologies such as computers and calculators is an efficient way to be able to graph
and evaluate functions. It is important, however, to consider the limitations of graphing technology
when interpreting graphs.
Activity 1
Hidden Behavior of Graphs
Determine the zeros of f(x ) = x 3 — x2 — 60.7x + 204 graphically.
EflSfiWI
O pen a new Graphs and Geometry page, and
graph the function.
RAD A U T O REAL
3 2 60.7x+
(x)-x -x -
In the default window, it appears that the function
has two zeroes, one betw een — 1 0 and — 8 and
one betw een 4 and 6 .
2
..................... d
3 2
• IS fl(x )-x -x
StudyTip
ETfflW From the Window m enu, choose Window Settings.
Change the dim ensions of the window as shown.
Window Settings You can choose
values for the window based on
observation of your graph, or you
can use one of the zoom tools
such as the b o x z o o m that
allows you to zoom in on a certain
area of a graph.
The behavior of the graph is m uch clearer in the
larger window. It still appears that the function has
two zeros, one betw een — 8 and — 1 0 and one
betw een 4 and 6 .
PTTTTil From the Window menu, choose Window Settings.
Change the window to [ 2 , 8 ] by [—2, 2],
By enlarging the graph in the area where it appears
that the zero occurs, it is clear that there is no zero
betw een the values of 4 and 6 . Therefore, the
graph only has one zero.
|
I
0,5
2
V
2 60.7x+204
f (x,)«x/-x -
V
Analyze the Results
1. In addition to the limitation discovered in the previous steps, how can graphing calculators limit
your ability to interpret graphs?
2. W hat are some ways to avoid these limitations?
Exercises
Determine the zeros of each polynomial graphically. Watch for hidden behavior.
3. x 3 + 6.5x 2 - 46.5x + 60
5.
108
| Lesson 2-2
x 5 + 7x3 + 4 x 2 -
x
+ 10.9
4. x4 - 3x3 + 12x 2 + 6x - 7
6
. x4 - 19x3 + 107.2*2 - 162x + 73
You factored
quadratic expressions
to solve equations.
(Lesson 0 -3 )
NewVocabulary
synthetic division
depressed polynomial
synthetic substitution
Divide polynomials
using long division
and synthetic division.
The redwood trees of Redwood
National Park in California are the
oldest living species in the world.
The trees can grow up to 350 feet
and can live up to 2 0 0 0 years.
Synthetic division can be used to
determine the height of one of the
trees during a particular year.
I Use the Remainder
i and Factor Theorems.
4 Divide Polynomials
Consider the polynom ial function/(x) = 6 x 3 — 25x2 + 18x + 9. If you
I know th at/ h as a zero at x = 3, then you also know that (x — 3) is a factor of/(x). Because/(x)
is a third-degree polynom ial, you know that there exists a second-degree polynom ial q(x) such that
f( x ) = (x - 3) •q(x).
This implies that q{x) can be found by dividing 6 x 3 — 25x2 + 18x + 9 by (x — 3) because
q (x)
:
/(*)
-, if x =/= 3.
x —3
To divide polynom ials, we can use an algorithm sim ilar to that of long division w ith integers.
Long Division to Factor Polynomials
Factor 6x3 — 25x2 + 18x + 9 com pletely using long division if (x — 3) is a factor.
6x
■7x —3
I
x - 3 )6 x 3 - 25x2 + 18x + 9
( - ) 6x3 - 18x2
Multiply divisor by 6 x 2 because
= 6 x 2.
Subtract and bring down next term.
—7x 2 + 18x
ix 1
Multiply divisor by —Ix because •
( - ) - 7 x 2 + 21x
-Ix.
Subtract and bring down next term.
—3x + 9
Qv
Multiply divisor by - 3 because - j - = — 3.
( - ) —3x + 9
* ----------------------------
0
Subtract.Notice that
the rem ainder is 0.
From this division, you can write 6 x 3 — 25x2 + 18x + 9 = (x — 3)(6x 2 — 7x — 3).
Factoring the quadratic expression yields 6 x 3 — 25x2 + 18x + 9 = (x — 3)(2x — 3)(3x + 1).
So, the zeros of the polynom ial function
f( x ) =
6
x 3 — 25x2 + 18x + 9 are 3,
and
——. The x-intercepts of the graph of/(x)
show n support this conclusion.
’ GuidedPractice
Factor each polynomial com pletely using the
given factor and long division.
IA. x 3 + 7x 2 + 4x — 12; x +
IB.
6
x 3 - 2x 2 - 16x -
8
6
; 2x - 4
109
StudyTip
Proper vs. Improper A rational
expression is considered improper
if the degree of the numerator is
greater than or equal to the
degree of the denominator. So in
the division algorithm,
f(X )
Long division of polynomials can result in a zero remainder, as in Exam ple 1, or a nonzero
remainder, as in the example below. Notice that just as with integer long division, the result
of polynomial division is expressed using the quotient, remainder, and divisor.
V
Divisor
is an
- Quotient
- x + 2 jx z + 5x — 4
- Dividend
( - ) x 2 + 2x
3x — 4
/Vnproper rational expression,
while
x + 3
is a proper rational
Remainder
Dividend
Divisor -
( - ) 3x + 6
expression.
Quotient
x2 + 5x - 4
x+ 2
:x + 3 +
x + 2'
x +
^
_ 2
Divisor
Excluded value -
Remainder — ► - 1 0
Recall that a dividend can be expressed in terms of the divisor, quotient, and remainder,
divisor
.
(x + 2)
quotient
+
remainder
(x + 3)
+
(-1 0 )
=
dividend
- v2 . 5x — 4
=
This leads to a definition for polynom ial division.
K eyC oncept Polynomial Division
Let f(x) and d(x) be polynomials such that the degree of d(x) is less than or equal to the degree of f(x) and d(x) ± 0.
Then there exist unique polynomials q{x) and r(x) such that
W ) = q{x) + ^ )
or
f(x) = d{x) ’ qW + r{x)'
where r(x) = 0 or the degree of r(x) is less than the degree of d(x). If r(x) = 0, then d(x) divides evenly into f(x).
I
,
.
...................................................................
Before dividing, be sure that each polynom ial is w ritten in standard form and that placeholders
with zero coefficients are inserted where needed for m issing powers of the variable.
StudyTip
Graphical Cheek You can also
check the result in Example 2
using a graphing calculator. The
graphs of Yi = 9 x 3 - x - 3
and Y2 = (3x2 - 2x+ 1) •
( 3 / + 2) - 5 are identical.
Long Division with Nonzero Remainder
D ivide 9x3 — x — 3 by 3x + 2.
First rewrite 9x3 — x — 3 as 9x2
3x - 2 x + l
6
You can write this result as
x2
3
= 3x 2 — 2x + 1 — ^ - x + - ^
3x + 2
3'
—6 x 2 ■•x
( - ) —6 x 2 ■■4x
CHECK M ultiply to check this result.
3x — 3
-5 , 5] scl: 1 by [ - 8 , 2] scl: 1
3. Then divide.
9x3 - x - 3 _
■= 3x 2 - 2x + 1 +
3x + 2
3x + 2
3x + 2 j9 'xj + Ox2
( - ) 9x 3 +
Ox2
( - ) 3x + 2
-5
(3x + 2)(3x 2 - 2x + 1) + ( - 5 ) = 9x 3 - x - 3
9x 3 — 6 x 2 + 3x +
6
x 2 — 4x + 2 — 5 = 9x 3 — x — 3
9xJ
— 3 = 9x 3 — x — 3 v'
p GuidedPractice
D ivide using long division.
2A. (8 x 3 - 18x2 + 21x - 20) + (2x - 3)
2B. (—3x 3 + x 2 + 4x — 6 6 ) + (x — 5)
W hen dividing polynom ials, the divisor can have a degree higher than 1. This can sometimes result
in a quotient with m issing terms.
110
I Lesson 2-3 | T h e R e m a in d e r a n d F a c to r T h e o re m s
J
H 2 2 J J J 3 J 3 3 Division by Polynomial of Degree 2 or Higher
StudyTip
Divide 2 x 4 — 4 x 3 + 1 3 x 2 + 3 x — 1 1 by x 2 — 2x + 7.
Division by Zero In Example 3,
this division is not defined for
x2 - 2x + 1 = 0. From this point
forward in this lesson, you can
assume that x cannot take on
values for which the indicated
division is undefined.
2xz
4x 3 + 13x2 + 3x - 11
x 2 — 2x + 7 j l x *
( - ) 2x4 - 4 x3 + 14x2
—x 2 + 3x — 11
( - ) - x 2 + 2x - 7
You can write this result as
■=
x2 —2x + 7
2
x2 -
1
+ xz —2x + 7
y GuidedPractice
Divide using long division.
3A.
(2x 3 + 5x 2 - 7x +
6
3 B . (6 x 5 —x 4 + 12x 2 + 15x)
) + (x 2 + 3x — 4)
4
- (3x 3 — 2x 2 + x)
Synthetic division is a shortcut for dividing a polynom ial by a linear factor of the form x — c.
Consider the long division from Exam ple 1.
Notice the coefficients highlighted in
colored text.
6
6 - 7
x — 3)6x 3 — 25x2 + 18x + 9
(—) 6 x 3 — 18x2
-25-
(-)
-1 8
—7x 2 + 18x
( - ) —7x 2 +
2 1
-3
-3)6
x
— 3x + 9
6
Change the signs of the divisor and the
numbers on the second line.
Collapse the long division vertically,
eliminating duplications.
Suppress x and powers of x.
x 2 - 7x - 3
Synthetic Division
Collapse Vertically
Suppress Variables
Long Division
+ 18+ 9
-7
+ 18
(-) - 7
+ 21
—3j
6
6
-2 5
18
9
-1 8
21
9
- 7 - 3
j J
R - 3
0
-2 5
18
6
0
18 | 9
-2 1
- 7 - 3
-9
0
The number now representing the
divisor is the related zero of the
binomial x — c. Also, by changing the
signs on the second line, we are now
adding instead of subtracting.
- 3 + 9
( - ) —3x + 9
6
+9
0
We can use the synthetic division show n in the exam ple above to outline a procedure for synthetic
division of any polynom ial by a binomial.
KeyConcept Synthetic Division Algorithm
To divide a polynomial by the factor x — c, complete each step.
Write the coefficients of the dividend in standard form.
Write the related zero c of the divisor x - c in the box.
Bring down the first coefficient.
Example
Divide 6 x 3 - 25x2 + 18x + 9 by x -
Multiply the first coefficient by c. Write the product
under the second coefficient.
E T T fllfl Add the product and the second coefficient.
B S B
Repeat Steps 2 and 3 until you reach a sum in the last
column. The numbers along the bottom row are the
coefficients of the quotient. The power of the first term
is one less than the degree of the dividend. The final
number is the remainder.
of quotient
= Add terms.
1=
^ = Multiply by c, and write
the product.
.....................
.mjg g a n m
fl|c o n n e c tE D .m c g ra w -h ill.c o m |
111
As with division of polynomials by long division, remember to use zeros as placeholders for any
missing terms in the dividend. W hen a polynom ial is divided by one of its binom ial factors x — c,
the quotient is called a depressed polynomial.
Synthetic Division
Divide using synthetic division.
(2x4 - 5x2 + 5.v - 2) -f (x + 2)
a.
Because x + 2 = x — (—2), c = —2. Set up the synthetic division as follows, using zero as a
placeholder for the m issing x3-term in the dividend. Then follow the synthetic
division procedure.
^ 2J
2
0
-5
5
-2
= Add terms.
^
— Multiply by c, and
write the product.
coefficients of
depressed polynomial
TechnologyTip
The quotient has degree one less than that of the dividend, so
Using Graphs To check your
division, you can graph the
polynomial division expression
and the depressed polynomial
with the remainder. The graphs
should coincide.
2x4 - 5x2 + 5x - 2 _ 2X3 _
x+2
b.
+
4 * 2
Check ,hfs rgsu|t
(10x 3 - 13x2 + 5x - 14) - f (2x - 3)
Rewrite the division expression so that the divisor is of the form x — c.
10x3 - 13x2 + 5x - 14
(10x3 - 13x2 + 5x - 14) + 2
------------------------------ = --------------------------------------- or
2x - 3
(2x-3)~2
5* 3
~^
+ f x~ 7
— -.
x- |
So, c = —. Perform the synthetic division.
3
n
13
5
2
2
. 1
c
So,
_7
3
, r
10x3 — 13x2 + 5 x - 1 4 c
, a
2 ,
= 5xz + x + 4
2x — 3
2
1
T _
7
5
2
15
3
,
~2
2
6
~l
il
r 2
3
2
f
13
2
a
2
or 5xz + x + 4 — ----2x - 3
GuidedPractice
4A. (4x 3 + 3x 2 - x +
2
8
4B.(6 x 4 + l l x 3 - 15x2 - 12x + 7) -f- (3x + 1)
) -r (x - 3)
The R em ainder and Factor Theorem s
W hen d(x) is the divisor (x — c) w ith degree 1,
the remainder is the real number r. So, the division algorithm sim plifies to
f( x ) = (x - c) •q(x) + r.
Evaluating/(x) for x = c, we find that
/(c) = (c — c) •q(c) + r =
0
•q(c) + r or r.
So,/(c) = r, which is the remainder. This leads us to the follow ing theorem.
K eyC oncept Remainder Theorem
If a polynomial f(x) is divided by x —c, the remainder is r = f(c).
112
Check this result.
| Lesson 2-3 j T h e R e m a in d e r a n d F a c to r T h e o re m s
The Rem ainder Theorem indicates that to evaluate a polynom ial function/(x) for x = c, you can
divide/(x) by x — c using synthetic division. The rem ainder will be/(c). Using synthetic division
to evaluate a function is called synthetic substitution.
Real-World Example 5 Use the Remainder Theorem
FOOTBALL T h e num ber o f tickets sold during the N orthside H igh School fo otb all season can
b e m odeled by f (x) = x 3 — 12x 2 + 48x + 74, w here x is the num ber o f gam es played. Use the
R em ainder T heorem to fin d the n um ber o f tickets sold during the tw elfth game of the
N orthside H igh School fo otb all season.
To find the num ber of tickets sold during the tw elfth game, use synthetic substitution to
evaluate f(x) for x = 1 2 .
12J
High school football rules are
similar to most college and
professional football rules. Two
major differences are that the
quarters are 12 minutes as
opposed to 15 minutes and kickoffs take place at the 40-yard line
instead of the 30-yard line.
-12
48
The rem ainder is 650, so t(12) = 650.
Therefore, 650 tickets were sold during
the tw elfth game of the season.
74
12
0 576
0
48 I 650
CHECK You can check your answer using direct substitution.
Original function
t(x) = x 3 — 12x 2 + 48x + 74
t(12) = (12 ) 3 - 12(12 ) 2 + 48(12) + 74 or 650 ✓
Substitute 12 for x and simplify.
Source: National Federation of State
High School Associations
p GuidedPractice
5. FOOTBALL Use the Rem ainder Theorem to determ ine the num ber of tickets sold during
the thirteenth game of the season.
If you use the Rem ainder Theorem to evaluate/(x) at x = c and the result is/(c) = 0, then you know
that c is a zero of the function and (x — c) is a factor. This leads us to another useful theorem that
provides a test to determ ine whether (x — c) is a factor o ffix ) .
KeyConcept Factor Theorem
A polynomial f(x) has a factor (x — c) if and only if f(c) = 0.
You can use synthetic division to perform this test.
iu u iy i
'»
'' vV
"v
Use the Factor Theorem
'
. '".V'
Use the Factor Theorem to determ ine if the b in o m ials given are factors o f f(x ). Use the
b in o m ials that are factors to w rite a factored form o f fix ) .
a. f( x ) = 4 x 4 + 2 1 x 3 + 2 5 x 2 — 5 x + 3 ; (x — 1 ) , ix + 3 )
Use synthetic division to test each factor, (x — 1) and (x + 3).
u
21
25
-5
3
21
-1 2
4
25
50
45
25
50
45
48
Because the rem ainder when /(x) is
divided by (x — 1) is 4 8 ,/ (l) = 48 and
(x — 1 ) is not a factor.
25
-2 7
-5
6
3
-3
-2
0
Because the rem ainder w hen/(x) is
divided by (x + 3) is 0,/ (—3) = 0 and
(x + 3) is a factor.
Because (x + 3) is a factor of fix ) , we can use the quotient o f f i x ) -5- (x + 3) to write a
factored form of/(x).
f i x ) = (x + 3)(4x 3 + 9x 2 - 2x + 1)
113
CHECK If (x + 3) is a factor of f( x ) = 4x4 + 21 x 3 +
TechnologyTip
2
Zeros You can confirm the zeros
on the graph of a function by
using the zero feature on the
CALC menu of a graphing
calculator.
25,t — 5x + 3, then —3 is a zero of the function
and (—3 , 0 ) is an x-intercept of the graph.
G raph /(x) using a graphing calculator and
confirm that (—3, 0 ) is a point on the graph. ✓
>
[ - 1 0 , 1 0 ] scl: 1 by [ - 1 0 , 3 0 ] scl: 2
b. f(x )
= 2x3 - x 2 - 41x -
20; (x + 4 ) , I t - 5 )
I k
Use synthetic division to test the factor (x + 4).
-4 1 2
2
-1
-4 1
-2 0
36
20
-9
-5
0
Because the rem ainder w hen/(x) is divided by (x + 4) is 0,/ (—4) = 0 and (x + 4) is a
factor of f(x ).
Next, test the second factor, (x — 5), w ith the depressed polynom ial 2x 2 — 9x — 5.
5J
2 - 9 - 5
2
10
5
1
| 0
Because the rem ainder when the quotient of /(x) -j- (x + 4) is divided by (x — 5) is 0,
/(5) = 0 and (x — 5) is a factor of /(x).
Because (x + 4) and (x — 5) are factors of/(x), w e can use the final quotient to write a factored
form of/(x).
\
fi x ) = (x + 4)(x - 5)(2x + 1)
CHECK The graph of/(x) = 2x 3 — x 2 - 41x — 20
confirms that x = —4, x = 5, and x =
zeros of the function. ✓"
are
p GuidedPractice
Use the Factor Theorem to determ ine if the binom ials given are factors of fix ). Use the
binomials that are factors to w rite a factored form of fix ).
6A. fi x ) = 3x 3
22x + 24; (x - 2), (x + 5)
6B. fi x ) = 4x3 - 34x + 54x + 36; (x -
6
), (x - 3)
You can see that synthetic division is a useful tool for factoring and finding the zeros of polynomial
functions.
C o n cep tS u m m ary Synthetic Division and Remainders
If r is the remainder obtained after a synthetic division of f(x) by (x - c), then the following statements are true.
• ris the value of ^(c).
• If r = 0, then {x — c) is a factor of f(x).
• If r = 0, then c is an x-intercept of the graph of f.
1
114
| Lesson 2-3
•
If r = 0, then x = c is a solution of f(x) = 0.
The R em ainder and Factor Theorem s
J
\
[I-
Exercises
*
Factor each polynomial com pletely using the given factor
and long d ivision. (Example 1)
\;v
'b
1. x3 + 2x2 — 23x — 60; x + 4
AN
x3 +
2
x2 -
2 1
o
x + 18;x-3
\ L ‘6
0
31. f( x ) = 4x 5 — 3x 4 + x 3 — 6 x 2 +
4x 3 + 20x 2 — 8 x - 96; x + 3
6
7. x4 + 12x 3 + 38x2 + 12x - 63; x 2 +
A
6
33. f(x ) = 2x 6 + 5xs — 3x 4 +
34. f( x ) = 4x 6 +
x + 9
\ -
x + 240; x 2 - 4x - 12
6 8
8
35. f{ x ) = 10x 5 +
6
6
37. fix ) = —2x 8 +
x 2 — x + 12) -f (x — 4)
(fijL (x 6 - 2x 5 + x 4 - x 3 + 3x 2 - x + 24)
x4 -
8
8
6
x — 4; c =
8
x 5 - 4x 4 + 12x 3 -
6
x + 24; c = 4
Use the Factor Theorem to determ ine if the binomials given
are factors of fi x ) . Use the binom ials that are factors to write
a factored form of fix ) . (Example 6 )
38.
fi x ) = x 4 - 2x 3 - 9x 2 + x +
13.
(6 x 6 - 3x 5 +
39.
fi x ) = x 4 + 2x 3 - 5x 2 +
(R )
(108x5 - 36x4 + 75x 2 + 36x + 24) + (3x + 2)
40.
f( x ) = x 4 - 2x 3 + 24x2 + 18x + 135; (x -
15.
(x 4 + x 3 +
(@
(4x 4 - 14x3 - 14x2 + HOx - 84) + (2x 2 + x - 12)
17
6 x 5 - 1 2 x 4 + 1 0 x 3 - 2 x 2 - 8x + 8
6
6
) + (2x - 1)
x 2 + 18x - 216) + (x 3 - 3x 2 + 18x - 54)
3x3 + 2x + 3
1g
1 2 x 5 + 5 x 4 - 1 5 x 3 + 1 9 x 2 - 4x - 28
3x3 + 2x2 — x + 6
- 6
x 4 + 12x 3 - 15x2 - 9x + 64; c = 2 I ,jf
(2x 4 - l x 3 - 38x2 + 103x + 60) -h (x - 3)
x 4 - 15x3 + 2x 2 + lOx -
6
6
;c=
@
6
I
x 3 — 9x 2 + 3x — 4; c = 5
x 3 + 7x2 - 3x +
8
to
x - 3; c = 4
(4x 4 -
6
x + 12) + (2x + 4)
6
x — 15; c = 3
11.
8
x 3 + 12x 2 -
(x + 2)
6
x 5 — 6 x 3 — 5x 2 +
( g ) f(x ) = —6 x 7 + 4x 5 Divide using long division. (Examples 2 and 3)
9. (5x 4 — 3x 3 +
8
32. f( x ) = 3x6 - 2x 5 + 4x 4 - 2x 3 +
6
x 3 - 7x 2 - 29x - 12; 3x + 4
''(£) x4 - 3x 3 - 36x2 +
\l
Find each/(c) using synthetic substitution. (Example 5)
3. x3 + 3x2 — 18x — 40; x — 4
5. —3x 3 + 15x2 + 108x - 540; x -
30. SKIING The distance in meters that a person travels on
skis can be m odeled by d(t) = 0.2t2 + 3 f, where t is the
time in seconds. Use the Rem ainder Theorem to find the
distance traveled after 45 seconds. (Example 5)
(~~9?T
8
6
; (x + 2), (x - 1)
x + 12; (x - 1), (x + 3)
5), (x + 5)
41. fix )
= 3x 4 - 22x 3 +
42. fix )
= 4x 4 - x 3 - 36x2 - l l l x + 30; (4x - 1), (x -
43. f{ x )
= 3x 4 - 35x 3 +
38x2 + 56x + 64; (3x - 2), (x + 2)
44. f{ x )
= 5x 5 + 38x4 -
6 8
13x2 + 118x - 40; (3x - 1), (x - 5) )
6
)
x 2 + 59x + 30; (5x - 2), (x +
8
)
45. fix ) = 4x 5 - 9x 4 + 39x 3 + 24x2 + 75x + 63; (4x + 3), (x - 1)
Divide using synthetic division. (Example 4)
19.
(x 4 - x 3 + 3x 2 -
6
x-
. 20) (2x 4 + 4x 3 - 2x 2 +
8
6
46. TREES The height of a tree in feet at various ages in years
is given in the table.
) + (x - 2)
x - 4) + (x + 3)
Age
(21) (3x 4 - 9x 3 - 24x - 48) - (x - 4)
22. (x 5 - 3x 3 +
6
x 2 + 9x +
6
) -f (x + 2)
23. (12x 5 + 10x 4 - 18x3 - 12x 2 — 8 ) 4- (2x - 3)
24. (36x4 -
6
x 3 + 12x 2 - 30x - 12) -I- (3x + 1)
25. (45x 5 +
6
x 4 + 3x 3 +
26. (48x5 + 28x4 +
6 8
8
x + 12)
Height
Age
2
3. 3
24
7 3 .8
6
1 3.8
26
8 2 .0
10
2 3 .0
28
9 1 .9
14
4 2 .7
30
1 0 1 .7
20
6 0 .7
36
1 1 1 .5
(3x - 2)
x 3 + l l x + 6 ) -h (4x + 1)
a. Use a graphing calculator to write a quadratic
equation to model the growth of the tree.
27. (60x6 + 78x5 + 9x 4 - 12x 3 - 25x - 20)
28.
4
- (5x + 4)
1 6 x 6 - 5 6 x 5 - 2 4 x 4 + 9 6 x 3 - 4 2 x 2 - 3 0 x + 105
2x
b. Use synthetic division to evaluate the height of the tree
at 15 years.
-7
29. EDUCATION The number of U.S. students, in thousands,
that graduated w ith a bachelor's degree from 1970 to 2006
can be modeled by g(x) = 0.0002x 5 — 0.016x4 + 0.512x3 —
7.15x2 + 47.52x + 800.27, where x is the number of years
since 1970. Use synthetic substitution to find the number
of students that graduated in 2005. Round to the nearest
thousand. (Example 5)
47. BICYCLING Patrick is cycling at an initial speed v0 of
4 m eters per second. W hen he rides downhill, the bike
accelerates at a rate a of 0.4 m eter per second squared.
The vertical distance from the top of the hill to the bottom
1
?
of the hill is 25 meters. Use d(t) = v 0t + —a r to find how
long it will take Patrick to ride dow n the hill, where d{t) is
distance traveled and t is given in seconds.
L,
- .............. -.........................................
fl[c o n n e c tE D .m c g r a w -h ill.c o m |
1 1 5
U
Factor each polynomial using the given factor and long
division. Assume n > 0.
60. t^f MULTIPLE REPRESENTATIONS In this problem , you will
explore the upper and lower bounds of a function.
a. GRAPHICAL Graph each related polynom ial function,
48.
x 3n + x 2n - 14x” - 24; x n + 2
49.
x3n + x ln - \2xn + 10; x n - 1
50.
Ax3” + 2xln - W xn + 4; 2 x n+ 4
51.
9x3n + 2Ax2n - 171x” + 54; 3x" - 1
and determ ine the greatest and least zeros. Then copy
and com plete the table.
Greatest
Zero
Polynomial
Least
Zero
x 3 — 2 x 2 — 11x + 12
52. MANUFACTURING An 18-inch by 20-inch sheet of cardboard
x 4 + 6 x 3 + 3 x 2 — 10x
I
x
CO
CN
J
x
I
is cut and folded into a bakery box.
b. NUMERICAL Use synthetic division to evaluate each
function in part a for three integer values greater than
the greatest zero.
18 in.
c. VERBAL M ake a conjecture about the characteristics of
the last row when synthetic division is used to
evaluate a function for an integer greater than its
greatest zero.
a. Write a polynomial function to model the volume
of the box.
d. NUMERICAL Use synthetic division to evaluate each
C.
d.
function in part a for three integer values less than the
least zero.
The com pany wants the box to have a volume of
196 cubic inches. Write an equation to model this
situation.
e. VERBAL M ake a conjecture about the characteristics
of the last row when synthetic division is used to
evaluate a function for a num ber less than its least
Find a positive integer for x that satisfies the equation
found in part c.
Find the value of k so that each rem ainder is zero.
53.
x 3 —far2 + 2x - 4
x -2
54.
x3 + 18x2 + kx + 4
x+2
55.
x3 + 4x2 - kx + 1
x + l
56.
2x3 —x2 + x + k
x -l
H.O.T. Problems
(61) CHALLENGE Is (x - 1) a factor of 18x165 - 15x135 +
15x55 + 4? Explain your reasoning.
8
x 105 -
57. SCULPTING Esteban will use a block of clay that is 3 feet
by 4 feet by 5 feet to m ake a sculpture. He wants to
reduce the volume of the clay by removing the same
amount from the length, the width, and the height.
a. Write a polynomial function to model the situation.
62. WRITING IN MATH Explain how you can use a graphing
calculator, synthetic division, and factoring to com pletely
factor a fifth-degree polynom ial with rational coefficients,
three integral zeros, and two non-integral, rational zeros.
C. He wants to reduce the volume of the clay to — of
the original volume. Write an equation to model
the situation.
63. REASONING Determ ine whether the statem ent below is
true o r false. Explain.
If h(y) = (y + 2)(3y 2 + 11 y — 4) — 1, then the remainder
d. How much should he take from each dim ension?
Use the graphs and synthetic division to completely factor
each polynomial.
58. /(x) = 8 x 4 + 26x3 - 103x2 - 156x + 45 (Figure 2.3.1)
o (f
CHALLENGE Find k so that the quotient has a 0 remainder.
64.
x 3 + kx2 - 34x + 56
x+ 7
65.
x6 + fcc4 - 8x3 + 173x2 - 16x - 120
x -l
66 .
kx3 + 2x2 - 22x - 4
x -2
59. /(x) = 6 x 5 + 13x4 - 153x3 + 54x 2 + 724x - 840
(Figure 2.3.2)
- i■
s —11 .
y+2
67. CHALLENGE If 2x 2 — dx + (31 — d 2)x + 5 has a factor x — d,
w hat is the value of d if d is an integer?
Figure 2.3.1
116
Lesson 2-3 | The R em ainder and Factor Theorem s
68. WRITING IN MATH Compare and contrast polynomial
division using long division and using synthetic division.
Spiral Review
Determine whether the degree n of the polynomial for each graph is e ve n or o d d and whether
its leading coefficient a n is p o s itiv e or n e g a tiv e . (Lesson 2-2)
)
- V
- >
X
V
72. SKYDIVING The approximate time t in seconds that it takes an object to fall a distance of
d feet is given by t = \J~^- Suppose a skydiver falls 11 seconds before the parachute opens.
How far does the skydiver fall during this time period? (Lesson 2 -1)
73. FIRE FIGHTING The velocity v and maximum height h of water being pumped into the air are
related by v = \j2gh, where g is the acceleration due to gravity (32 feet/second2). (Lesson 1-7)
a. Determine an equation that w ill give the m axim um height of the water as a function of
its velocity.
b.
The Mayfield Fire D epartm ent m ust purchase a pump that is powerful enough to propel
water 80 feet into the air. Will a pump that is advertised to project water w ith a velocity
of 75 feet/second m eet the fire departm ent's needs? Explain.
Solve each system of equations algebraically. (Lesson 0-5)
74.
5x — y = 16
2x + 3y = 3
75.
76.
77.
2x + 5y = 4
3x + 6y = 5
78.
79.
3x — 5y = —8
x + 2y = 1
y =
x - 4.5 + y
7x + 12y = 16
5 y - 4 x = -2 1
6
—x
4x + 5y =
3 x - 7 y = 10
-8
Skills Review for Standardized Tests
80. SAT/ACT In the figure, an equilateral triangle is drawn
w ith an altitude that is also the diam eter of the circle.
If the perim eter of the triangle is 36, w hat is the
circumference of the circle?
82. REVIEW The first term in a sequence is x. Each
subsequent term is three less than twice the preceding
term. W hat is the 5th term in the sequence?
A
8
x — 21
C \6x — 39
B
8
x — 15
D 16x-45
E 32*-43
83. Use the graph of the polynom ial function. W hich is
not a factor
A 6 V 2 tt
C 12V 2tt
B 6 a/ 3 tt
D 12V3-JV
E 36-77
81. REVIEW If (3, - 7 ) is the center of a circle and (8 ,5 ) is
on the circle, w hat is the circum ference of the circle?
F 13-77
G 15-77
H
18tt
J 25-77
K 26-tt
F (x -
2
)
G (x +
2
)
H (X -
1
)
1
)
J (X +
y
'V£
—
4
I
\
—12 ~v
'
|
V
fix) = x 5 + x4 - 3x3 - 3x 2 - 4x—4
conneHEDj7cgraîlTH!'otT^
117
Mid-Chapter Quiz
Lessons 2-1 through 2-3
Graph and analyze each function. Describe its domain, range,
intercepts, end behavior, continuity, and where the function is
increasing or decreasing. (Lesson 2-1)
1
.
f(x) = 2 x 3
2
. f(x) = - | x 4
2
3. f (x ) = 3 x - 8
4. f(x) = 4 x 5
Describe the end behavior of the graph of each polynomial function
using limits. Explain your reasoning using the leading term
test. (Lesson 2-2)
14. f(x) = —7 x 4 - 3 x 3 - 8 x 2 + 2 3 x + 7
15. f(x) - - 5 x 5 + 4 x 4 + 12 x 2 - 8
16. ENERGY C rystal’s electricity consum ption m easured in kilow att
5.
TREES The heights of several fir trees and the areas under their
hours (kW h) fo r th e past 12 m onths is show n below. (Lesson 2-2)
branches are show n in the table. (Lesson 2-1)
Month
Consumption
Month
Consumption
(kWh)
(kWh)
Height (m)
Area (m 2)
4 .2
3 7 .9 5
J a n u a ry
240
J u ly
300
2.1
7.44
February
135
August
335
3.4
23.54
March
98
September
390
1.7
4.75
April
110
October
345
4.6
46.48
May
160
November
230
June
230
December
100
b. Determine a power function to model the data.
c. Predict the area under the branches of a fir tree that is 7.6
meters high.
a. Determine a model for the number of kilowatt hours Crystal used
as a function of the number of months since January.
b.
Use the model to predict how many kilowatt hours Crystal will
use the following January. Does this answer make sense?
Explain your reasoning.
Solve each equation. {Lesson 2-1)
6
Divide using synthetic division. (Lesson 2-3)
. \ / 5 x + 7 = 13
17. (5x 3 — 7 x 2 +
7. V 2 x - 2 + 1 = x
8
8
x — 13) + (x — 1)
18. (x 4 — x 3 — 9x + 18) + (x — 2)
. \ / 3 x + 10 + 1 = V * + 11
19. (2x 3 - 11x 2 + 9 x — 6 ) -f ( 2 x — 1)
9. - 5 = V ( 6 * + 3 ) 3 - 3 2
Determine each f(c) using synthetic substitution. (Lesson 2-3)
State the number of possible real zeros and turning points of each
function. Then find all of the real zeros by factoring. (Lesson 2-2)
20. f(x) = 9 x 5 + 4 x 4 - 3 x 3 + 18x2 - 16x +
21. f(x) =
10 . f(x) = x 2 — 1 1 x — 26
6
x 6 - 3x5 +
8
x 4 + 12x 2 -
6
22. f(x) = - 2 x 6 + 8 x 5 - 12x 4 + 9 x 3 -
8
;c= 2
x + 4; c = - 3
8
x2 +
6
x - 3; c = - 2
11. f(x) = 3 x 5 + 2 x 4 - x 3
12. f(x) = x 4 + 9x 2 - 10
Use the Factor Theorem to determine if the binomials given are factors
of f(x). Use the binomials that are factors to w rite a factored form of
f(x). (Lesson 2-3)
13. MULTIPLE CHOICE Which of the following describes the possible
end behavior of a polynomial of odd degree? (Lesson 2-2)
23. f(x) = x 3 + 2 x 2 - 25x - 50; (x + 5)
24. f(x) = x 4 -
118
A
X—
KX)
B
X—
>oo
6
x 3 + 7x2 +
6
x-
8
; (x - 1), (x - 2)
lim f(x) = 5; lim f(x) = 5
X—
>—O
O
lim f(x) =
-o o ;
lim f(x) -
lim f(x) =
C
X—>oo
oo;
D
lim f(x) =
X—
>oo
-o o ;
|
lim fix) =
-o o
X—
*—oo
X—
►
—oo
C h a p te r 2
oo
lim fi x) =
X—
>—
oo
|
oo
M id -C h a p te r Q u iz
25. MULTIPLE CHOICE Find the remainder when
f(x) = x 3 - 4x + 5 is divided by x + 3. (Lesson 2-3)
F
-1 0
G
8
H 20
J 26
■■:
I
. ...wc___________
!
Zeros of Polynomial Functions
: Why?
Now
You learned that a
polynomial function • 1
of degree n can have
at most n real zeros.
2
(Lesson 2-1)
Find real zeros of
Find complex zeros of
t/ {)
S I"
m NewVocabulary 1
Rational Zero Theorem
lower bound
upper bound
Descartes’ Rule of Signs
Fundamental Theorem of
Algebra
Linear Factorization
Theorem
Conjugate Root Theorem
complex conjugates
irreducible over the reals
A company estimates that the profit P in thousands of dollars
from a certain model of video game controller is given by
P(x) = - 0 . 0 0 0 7 / 2 + 2.45x, where x is the num ber of
thousands of dollars spent marketing the controller. To find the
number of advertising dollars the company should spend to
make a profit of $1,500,000, you can use techniques presented
in this lesson to solve the polynomial equation P(x) = 1500.
>,
Real Zeros Recall that a polynom ial function of degree n can have at m ost n real zeros.
These real zeros are either rational or irrational.
Rational Zeros
Irrational Zeros
f(x{= 3 x 2 + Ix - 6 or fix) = (x+ 3)(3x - 2)
gix) = x2 - 5 or g(x) = (x+ V 5 ) (x - V 5 )
There are two rational zeros, - 3 or
There are two irrational zeros, ± V 5 .
The Rational Zero Theorem describes how the leading coefficient and constant term of a
polynom ial function w ith integer coefficients can be used to determ ine a list of all possible
rational zeros.
KeyConcept Rational Zero Theorem
If f is a polynomial function of the form fix) = anxn + a „ _ 1x',_ 1 + . . . + a2x 2 + a^x+ a0, with degree n > 1, integer
p
coefficients, and a0 ± 0, then every rational zero of f has the form —, where
• p and q have no common factors other than + 1 ,
• p is an integer factor of the constant term a0, and
• q is an integer factor of the leading coefficient a„.
C o ro lla ry If the leading coefficient a„ is 1, then any rational zeros of f are integer factors of the constant term a0.
Once you know all of the possible rational zeros of a polynom ial function, you can then use direct or
synthetic substitution to determ ine which, if any, are actual zeros of the polynomial.
Leading Coefficient Equal to 1
List all possible rational zeros of each function. Then determ ine which, if any, are zeros.
a.
f{x ) = x 3 + 2x + l
ETffiTI Identify possible rational zeros.
Because the leading coefficient is 1, the possible rational zeros are the integer factors
of the constant term 1. Therefore, the possible rational zeros of/ are 1 and —1.
ETTffW Use direct substitution to test each possible zero.
/(I) = ( l ) 3 + 2(1) + l o r 4
/(—l) = ( - 1
)3
+
2
(—1 ) +
1
or
- 2
B ecau se/ (l)
0 and/(—1) =/=0, you can conclude that
/ has no rational zeros. From the graph of/ y ou can
see th at/h as one real zero. A pplying the Rational
Zeros Theorem shows that this zero is irrational.
mm
- 5 , 5] scl: 1 by [ - 4 , 6] scl: 1
-
—mm —
mmmmmm
lTlconnectED.mcgraw-hill.com |
119
b.
g (*) = x 4 + 4 x 3 - 1 2 * - 9
Step 1 Because the leading coefficient is 1, the possible rational zeros are the integer factors
of the constant term -9 . Therefore, the possible rational zeros of g are ±1, ±3, and ±9.
Step 2 Begin by testing 1 and —1 using synthetic substitution.
lj
1
4
0 - 1 2 - 9
1
1
5
5
5
Zl l
5 -7
- 7 |—16
1
4
0 - 1 2 - 9
- 1 - 3
3 —3
1
Because g (—1) = 0, you can conclude that
—1 is a zero of g . Testing —3 on the
depressed polynom ial shows that —3
is another rational zero.
1
3
—9~I
9
0
3 - 3 - 9
-3
i
0
9
o
o
Thus, g(x) = (x + l)(x + 3)(x 2 — 3). Because the factor (x2 — 3) yields no rational zeros,
w e can conclude that g has only two rational zeros, — 1 and —3.
CHECK The
and
two
and
[-5 , 5] scl: 1 by [-2 0 ,1 0 ] scl: 3
Figure 2.4.1
graph of g(x) = * 4 + 4x 3 — 12* — 9 in Figure 2.4.1 has x-intercepts at —1 and —3,
close to (2, 0) and (—2, 0). By the Rational Zeros Theorem, we know that these last
zeros must be irrational. In fact, the factor (x2 — 3) yields two irrational zeros, V 3
—\p3. ✓
^ GuidedPractice
List all possible rational zeros of each function. Then determ ine w hich, if any, are zeros.
1A. f( x ) =
* 3
+ 5*
2
1B. h{x) = x 4 + 3 *
- 4* - 2
3
- 7x2 + 9x - 30
W hen the leading coefficient of a polynom ial function is not 1, the list of possible rational zeros can
increase significantly.
Leading Coefficient not Equal to 1
List all possible rational zeros of h(x) = 3x3 — 7x2 — 22x + 8. Then determ ine which, if any,
are zeros.
EflSBn The leading coefficient is 3 and the constant term is 8 .
g
Possible rational zeros:
g
± 8
or ± 1 , ± 2 , ± 4 ,
±8, ±1, ± f, ± f, ± f
RTTTO By synthetic substitution, you can determ ine that —2 is a rational zero.
—2J
3
—7
- 6
3 - 1 3
-2 2
8
26
- 8
4 |
0
By the division algorithm, h(x) = [x + 2)(3x2 - 13x -I- 4). Once 3x 2 — 13x + 4 is factored,
the polynom ial becom es h{x) = (x + 2)(3x - l)(x — 4), and you can conclude that the
rational zeros of h are —2,
y
and 4. Check this result by graphing.
GuidedPractice
List all possible rational zeros of each function. Then determ ine which, if any, are zeros.
2A. g(x) = 2 x 3 - 4x2 + 18* - 3 ^ ' \
120
| Lesson 2 -4 | Zeros o f Polynom ial Functions
2B./ (*) = 3 *
4
- 1 8 *3 + 2 * - 21
Real-World Example 3 Solve a Polynomial Equation
BUSINESS A fter the first half-hour, the num ber of video games that were sold by a company
on their release date can be m odeled by g(x) — 2x3 + 4xz — 2x, where g(x) is the num ber of
games sold in hundreds and x is the num ber of hours after the release. How long did it take
to sell 400 games?
Because g (x ) represents the num ber of games sold in hundreds, you need to solve g(x) = 4 to
determ ine how long it will take to sell 400 games.
g (x ) = 4
2x3 + 4x2 - 2x = 4
2x 3 + 4 x 2 — 2x — 4 = 0
Write the equation.
Substitute 2x3 + 4x2 - 2 xfo r g(x).
Subtract 4 from each side.
Apply the Rational Zeros Theorem to this new polynom ial function,/(x) = 2x + 4x
Possible rational zeros:
•2x — 4.
Factors o f 4 _ ±1/ ± 2 , ± 4
F actors o f 2
±1, ±2
= ± 1, +2, ± 4,
± 7
1
By synthetic substitution, you can determ ine that 1 is a rational zero.
R eal-W orld Link
1J
A recent study showed that
2
4 - 2 - 4
almost a third of frequent video
game players are between 6 and
0
17 years old.
Source: NPD Group Inc
Because 1 is a zero of /, x = 1 is a solution of/(x) = 0. The depressed polynomial
2x 2 + 6x + 4 can be written as 2(x + 2)(x + 1). The zeros of this polynomial are —2 and —1.
Because time cannot be negative, the solution is x = 1. So, it took 1 hour to sell 400 games.
p GuidedPractice
3. VOLLEYBALL A volleyball that is returned after a serve w ith an initial speed of 40 feet per
second at a height of 4 feet is given by f( t ) = 4 + 4 0 1 — 16f2, where /(£) is the height the ball
reaches in feet and t is time in seconds. At w hat time(s) will the ball reach a height of 20 feet?
One way to narrow the search for real zeros is to determ ine an
interval within w hich all real zeros of a function are located. A
real num ber a is a low er bound for the real zeros o f/ if f( x ) =/=0
for x < a. Similarly, b is an upper bound for the real zeros o f/ if
f( x ) ^ 0 for x > b .
The real zeros of fare
in the interval [a, b].
You can test whether a given interval contains all real zeros of a function by using the following
upper and lower bound tests.
Reading Math
KeyConcept Upper and Lower Bound Tests
__________ _______
_________
Nonnegative and Nonpositive
Remember that a nonnegative
Let f be a polynomial function of degree n > 1, real coefficients, and a positive leading coefficient. Suppose f(x) is divided
value is one that is either
by x — c using synthetic division.
positive or zero, and a
• If
nonpositive value is one that is
either negative or zero.
c < 0 and every number in the last line of the division is alternately nonnegative and nonpositive, then c is a lower
bound tor the real zeros of f.
• If c > 0 and every number in the last line of the division is nonnegative, then c is an upper bound for the real zeros of f.
121
To m ake use of the upper and lower bound tests, follow these steps.
Graph the function to determ ine an interval in w hich the zeros lie.
ETHTW Using synthetic substitution, confirm that the upper and lower bounds of your interval are in
fact upper and lower bounds of the function by applying the upper and lower bound tests.
n f f li n Use the Rational Zero Theorem to help find all the real zeros.
Use the Upper and Lower Bound Tests
StudyTip
Upper and Lower Bounds
Upper and lower bounds of a
Determine an interval in which all real zeros of h(x) = 2xi — l l x 3 + 2x2 — 44x — 24 m ust lie.
Explain your reasoning using the upper and low er bound tests. Then find all the real zeros.
function are not necessarily
unique.
Graph h(x) using a graphing calculator. From this
graph, it appears that the real zeros of this function
lie in the interval [—1, 7].
lower bound of c = —1 and
c = 7.
2
Z^J
- 1 1
2
-2 4
- 2
13
-1 5
59
2
-1 3
15
-5 9
35
2
- 1 1
2
-4 4
-2 4
2 1
161
819
23
117
795
14
2
E S E
-4 4
3
Values alternate signs in the
last line, so —1 is a lower bound.
Values are all nonnegative in
last line, so 7 is an upper bound.
Js e the Rational Zero Theorem.
TD
■i i
.•
i
Factors o f 24
± I, + 2 , + 3 , + 4 , + 6 , ± 8 , ± 1 2 , + 2 4
= — :----- ----- '
' —— —
Factors o f 2
±1, +2
Possible rational zeros: - -
= ± 1, ± 2 , + 4, + 6 , ± 8 , +12, +24, ± ± , ± |
Because the real zeros are in the interval [—1, 7], you can narrow this list to ju st ± 1 , ± j ,
± ~ , 2, 4, or 6 . From the graph, it appears that only
Begin by testing 6 .
6
J
2
- 1 1
12
2
~1
6
and ~
are reasonable.
N ow test ——in the depressed polynomial.
2
6
8
-4 4
-2 4
48
24
4~|
—U
2
1
2
By the division algorithm, h(x) = 2(x -
6
4
O
- 1
0
8
o
n
-
4
0
)|x + -|j(x 2 + 4). Notice that the factor (x2 + 4)
has no real zeros associated w ith it because x 2 + 4 = 0 has no real solutions. So,/ has two
real solutions that are both rational,
44x — 24 supports this conclusion.
6
and —j . The graph of h(x) = 2x4 — l l x 3 + 2x2 —
k GuidedPractice
Determine an interval in which all real zeros of the given function m ust lie. Explain your
reasoning using the upper and lower bound tests. Then find all the real zeros.
4A. g (x ) = 6x4 + 70x3 - 21x 2 + 35x - 12
122
| Lesson 2 -4 | Zeros o f Polynom ial Functions
4B. f( x ) = 10x5 - 50x4 - 3x 3 + 22x 2 - 41x + 30
ReadinsMath
Variation in Sign a variation in
Another w ay to narrow the search for real zeros is to use D escartes' Rule of Signs. This rule gives
us inform ation about the num ber of positive and negative real zeros of a polynom ial function by
looking at a polynom ial's variations in sign.
s/gn occurs in a polynomial
written in standard form when
consecutive coefficients have
opposite signs.
K eyC oncept Descartes’ Rule of Signs
If f(x) = a „ x n + a n _ 1x n _ l + . .. + a^x + a 0 is a polynomial function with real coefficients, then
• the number of positive real zeros of f is equal to the number of variations in sign of f(x) or less than that number by some
even number and
• the number of negative real zeros of f \s the same as the number of variations in sign of f(—x) or less than that number
by some even number.
Describe the possible real zeros of g(x) = —3x3 + Zx2 — x — 1.
Exam ine the variations in sign for g(x) and for g (—x).
+ to —
g (x) = —3x 3 + 2x2 ~ x ~ 1
g ( - x ) = - 3 ( - x ) 3 + 2 (—x )2 - ( - * ) - 1
-to +
= 3x3 + 2 x 2 + x j l
'— r
-Hto —
The original function g(x) has two variations in sign, while g (—x) has one variation in sign. By
D escartes' Rule of Signs, you know that g(x) has either 2 or 0 positive real zeros and 1 negative
real zero.
From the graph of g(x) show n, you can see that the function has
one negative real zero close to x = —0.5 and no positive real zeros.
► GuidedPractice
Describe the possible real zeros of each function.
5A. h(x) = 6x5 + 8x2 - lOx - 15
5B.f(x ) = - l l x 4 + 20x 3 + 3x2 - x + 18
W hen using D escartes' Rule of Signs, the number of real zeros indicated includes any repeated
zeros. Therefore, a zero w ith m ultiplicity m should be counted as m zeros.
9 Complex Zeros
A French mathematician, scientist,
Just as quadratic functions can have real or im aginary zeros, polynomials
mm of higher degree can also have zeros in the com plex num ber system. This fact, combined with
the Fundam ental Theorem of Algebra, allows us to im prove our statem ent about the number of
zeros for any wth-degree polynomial.
and philosopher, Descartes wrote
many philosophical works such
as Discourse on Method anti
mathematical works such as
Geometry.
K eyC oncept Fundamental Theorem of Algebra
A polynomial function of degree n, where n > 0, has at least one zero (real or imaginary) in the complex number system.
C o r o lla r y
A polynomial function of degree n has exactly n zeros, including repeated zeros, in the complex number
system.
123
By extending the Factor Theorem to include both real and imaginary zeros and applying the
Fundam ental Theorem of Algebra, we obtain the Linear Factorization Theorem.
KeyConcept Linear Factorization Theorem
If f(x) is a polynomial function of degree n > 0, then f has exactly n linear factors and
f(x) = a n( x - c 1) { x - c 2) . . . ( x - c n)
where a„ is some nonzero real number and cv c 2
c„ are the complex zeros (including repeated zeros) of f.
According to the Conjugate Root Theorem, when a polynom ial equation in one variable w ith real
coefficients has a root of the form a + bi, where b =/=0 , then its com plex conjugate, a — bi, is also a
root. You can use this theorem to write a polynom ial function given its com plex zeros.
m
m
Find a Polynomial Function Given Its Zeros
m
Write a polynomial function of least degree with real coefficients in standard form that has
—2, 4, and 3 — i as zeros.
Because 3 — i is a zero and the polynom ial is to have real coefficients, you know that 3 + i must
also be a zero. Using the Linear Factorization Theorem and the zeros —2 ,4 , 3 — i, and 3 + i, you
can write/(x) as follows.
StudyTip
Infinite Polynomials Because a
can be any nonzero real number,
there are an infinite number of
polynomial functions that can be
written for a given set of zeros.
f{x ) = a[x - (- 2 )](x - 4)[x - (3 - i)][x - (3 + i)]
W hile a can be any nonzero real number, it is sim plest to let a = 1. Then write the function in
standard form.
f(x ) = (l)(x + 2)(x - 4)[x - (3 - /)][* - (3 + t')]
Let a = 1.
= (x 2 — 2x — 8)(x2 — 6x + 10)
Multiply.
x 3 + 14x2 + 28x - 80
Multiply.
= x4 -
8
Therefore, a function of least degree that has —2 , 4 , 3 — i, and 3 + i as zeros is f(x ) = x 4 — 8 x 3 +
14x2 + 28x — 80 or any nonzero m ultiple of f(x).
p GuidedPractice
Write a polynomial function of least degree with real coefficients in standard form with the
given zeros.
6A. —3 ,1 (multiplicity: 2), 4i
StudyTip
Prime Polynomials Note the
difference between expressions
6B. 2\/3, —2\/3,1 + t
In Exam ple 6 , you wrote a function w ith real and com plex zeros. A function has com plex zeros
w hen its factored form contains a quadratic factor w hich is irreducible over the reals. A quadratic
E xp ression is irreducible over the reals w hen it has real coefficients but no real zeros associated
with it. This exam ple illustrates the follow ing theorem.
which are irreducible over the
reals and expressions which are
prime. The expression x 2 — 8 is
prime because it cannot be
factored into expressions with
integral coefficients. However,
x 2 - 8 is not irreducible over the
K eyC oncept Factoring Polynomial Functions Over the Reals
Every polynomial function of degree n > 0 with real coefficients can be written as the product of linear factors and
irreducible quadratic factors, each with real coefficients.
reals because there are real zeros
associated with it,V 8 and - V 8 -
As indicated by the Linear Factorization Theorem, when factoring a polynom ial function over the
complex num ber system , we can write the function as the product of only linear factors.
124
| Lesson 2 -4 j Zeros of Polynomial Functions
Factor and Find the Zeros of a Polynomial Function
StudyTip
Using Multiplicity Sometimes a
Consider k(x) = x 5 — 18x3 + 30x2 — 19x + 30.
rational zero will be a repeated
a.
zero of a function. Use the graph
of the function to determine
Write k(x) as the product of linear and irreducible quadratic factors.
The possible rational zeros are ± 1 , ± 2 , + 3 , + 5 , ± 6 , ± 1 0 , ± 1 5 , ± 3 0 . The original polynomial
has 4 sign variations.
whether a rational zero should be
tested using synthetic substitution
in succession.
k ( - x ) = ( - x ) 5 - 18(—x)3 + 30(—x)2 - 19(—x) + 30
= - x 5 + 18x3 + 30x2 +. 19x + 30
fc(—x) has 1 sign variation, so k(x) has 4, 2, or 0 positive real zeros and 1 negative real zero.
The graph shown suggests —5 as one real zero of k(x).
Use synthetic substitution to test this possibility.
1
1
0
-1 8
30
-5
25
-35
-5
7
-5
-1 9
30
25 - 3 0
0
B, 8] scl: 1 by [- 1 0 0 , 800] scl: 50
Because k(x) has only 1 negative real zero, you do not need
to test any other possible negative rational zeros. Zoom ing
in on the positive real zeros in the graph suggests 2 and 3 as
other rational zeros. Test these possibilities successively in
the depressed quartic and then cubic polynomials.
1
StudyTip
-5
7
2
-6
1
-3
1
-3
1
-3
1
-3
Quadratic Formula You could
also use the Quadratic Formula to
-5
6
2
Begin by
testing 2.
[ - 8 , 8] scl: 1 by [- 2 0 , 20] scl: 4
find the zeros of x2 + 1 in order’
to factor the expression.
- 0 ± V o 2 - 4(1 )(1)
2 (1)
3
0
0
2
/
= ± f or ± /
So, / and —/ are zeros and
and
(x + i) are factors.
(x - i)
>
Now test 3 on the
depressed polynomial.
3
0
The remaining quadratic factor (x 2 + 1 ) yields no real zeros and is therefore irreducible
over the reals. So, k(x) written as a product of linear and irreducible quadratic factors is
k{x) = (x + 5)(x - 2)(x - 3)(x 2 + 1).
Write k(x) as the product of linear factors.
You can factor x 2 + 1 by writing the expression first as a difference of squares x 2 — (V —l ) 2 or
x 2 — i 2. Then factor this difference of squares as (x — i)(x + »). So, k(x) written as a product of
linear factors is as follows.
fc(x) = (x + 5)(x - 2)(x - 3)(x - i)(x + i )
C. List all the zeros o f k(x).
Because the function has degree 5, by the corollary to the Fundam ental Theorem of Algebra
k(x) has exactly five zeros, including any that m ay be repeated. The linear factorization gives
us these five zeros: —5, 2 , 3 , i, and —i.
» GuidedPractice
Write each function as (a) the product of linear and irreducible quadratic factors and (b) the
product of linear factors. Then (c) list all of its zeros.
7A. f(x ) = x 4 + x 3 - 26x2 + 4x - 120
7B. f(x ) = x 5 - 2x 4 - 2x 3 -
$
6
x 2 - 99x + 108
connectED.mcgraw-hill.com I
125
You can use synthetic substitution w ith com plex num bers in the same w ay you use it w ith real
numbers. Doing so can help you factor a polynom ial in order to find all of its zeros.
WatchOut!
Complex Numbers Recal1 from
Lesson 0-2 that all real numbers
are also complex numbers.
Find tfie Zeros ° * a Polynomial When One is Known
>
Find all complex zeros of p(x) = x4 — 6x3 + 20x2 — l l x — 13 given that 2 — 3 i is a zero of p.
Then write the linear factorization of p(x).
Use synthetic substitution to verify that 2 — 3i is a zero of p(x).
2 — 3i |
1
- 6
2 -3 i
1
-1 3
(2 - 3/)(—4 - 3/') = - 8 + 6/4- 9i2
— —8 _j_ 6 / 9(—
i
6
= -1 7 4 - 6 /
2 0
- 6
1
-1 7 +
- 6
1
1
2 -3 i
1
6
i
- 2 2
-1 3
- 2 2
—17 + 6i
24 + 3 i
3 +
2 + 3i
i
6
/) = 6 -f 3/— 18/2
= 6 + 3 / - 18(—1)
= 24 + 3/
0
2 0
6
(2 - 3/)(3 +
24 + 3 i
3 4- 6 i
1
1
2 - 3i
1
2 - 3j |
-1 7 +
- 2 2
- 4 - 3i
1
2 — 3i |
2 0
-1 3
(2 - 3/)(2 4- 3/) = 4 - 9/2
= 4 —9(—1)
= 4 4- 9 or 13
13
0
Because 2 — 3i is a zero of p, you know that 2 + 3i is also a zero of p. Divide the depressed
polynomial by 2 4- 3 i.
2 + 3i\
1
3i
3 +
3i
-4 -
-2
6
i
6
i
2 + 3i
- 2
- 3i
- 1
0
Using these two zeros and the depressed polynom ial from this last division, you can write
p(x) = [x - (2 - 3i)][x - (2 + 3 i)](x2 - 2x - 1).
Because p(x) is a quartic polynom ial, you know that it has exactly 4 zeros. Having found 2, you
know that 2 more remain. Find the zeros of x 2 — 2x — 1 by using the Q uadratic Formula.
—b ± \]b2 - 4ac
2a
Quadratic Formula
-(-2 ) ± V(-2)2 -
4 (1 )(—1)
a= 1 , b -
2 (1)
2 + V8
1
±V
— 1
Simplify.
2
=
- 2 , and c =
Simplify.
2
Therefore, the four zeros of p(x) are 2 — 3i, 2 + 3i, 1 + V 2 , and
1 — \/2. The linear factorization of p(x) is [x — (2 — 3t)] •
StudyTip
[x -
Dividing Out Common Factors
Before applying any of the
Using the graph of p(x), you can verify that the function has two
methods in this lesson, remember
real zeros at 1 + \[2 or about 2.41 and 1 — \ Jl or about —0.41.
(2
+ 3 i)][x -
(1
+ V 2 )][x -
(1
- \/2 )].
to factor out any common
monomial factors.
For example, g(x) = - 2 x 4 +
6 / 3 - 4 * 2 - 8x should first be
factored as g(x) = - 2 x ( x 3 -
3x2 + 2 x + 4), which implies that
0 is a zero of g.
^ Guided Practice
For each function, use the given zero to find all the com plex zeros of the function. Then write
the linear factorization of the function.
8A. §{x) = x 4 - 10x3 + 35x2 - 46x + 10; 2 + V 3
8B. h(x) = x 4 - 8x3 + 26x2 - 8x - 95; 1 - V 6
126
| Lesson 2 -4 | Zeros o f P olynom ial Functions
Exercises
Step-by-Step Solutions begin on page R29.
List all possible rational zeros of each function. Then
determine which, if any, are zeros. Examples 1 and 2 )
0
gix) = x 4 - 6x3 - 31x2 + 216x - 180
2.
f( x ) = 4x 3 - 24j
2
-x + 6
5. h(x) = 6x4 + 13x3 -
7xz + 56x + 20
7x2 - 156x - 60
27. fi x ) =
10x 4 - 3x 3 +
8
x 2 - 4x -
8
6
29. fi x ) =
12x 4 +
30. g(x) =
4x 5 + 3x 4 + 9x 3 — 8 x 2 + 16x —
24
31. hix) =
—4x 5 + x 4 -
- 124
6
x 3 + 3x 2 - 2x + 12
8
x 3 - 24x2 + 64x
. f( x ) = 18x4 + l l x 3 + 56x2 + 48x - 64
9. MANUFACTURING The specifications for the dim ensions
of a new cardboard container are shown. If the volume
of the container is modeled by V(h) = I h 3 —9h 2 + 4h and
it will hold 45 cubic inches of merchandise, what are the
container's dim ensions? (Example 3)
(Example 6)
32.
33. - 2 , - 4 , - 3 , 5
I
x 5 + 18x4 - 5x 3 - 71x2 - 162x + 45
I—1
8
On
. g(x) =
Write a polynomial function of least degree with real
coefficients in standard form that has the given zeros.
I
7. h(x) = x 5 - l l x 4 + 49x 3 - 147x2 + 360x - 432
8
- I x 3 - 3x 2 + 4x + 7
f( x ) =
CO
6
6
8
26.
28. f( x ) = —3x 4 - 5x 3 + 4x 2 + 2x -
g(x) = x4 — x 3 — 3 lx 2 + x + 30
4. gix) = -A x 4 + 35x 3 -
Describe the possible real zeros of each function. (Example 5)
34. - 5 , 3 , 4 + i
35. - 1 , 8 ,
36. 2 V 5 , -2\/5, - 3 , 7
37. - 5 , 2 ,4 - V 3 , 4 + V 3
38. V 7 , - V 7 , 4 i
39. V 6 , - V
6
, 3 - 4i
40. 2 + V 3 , 2 - \/3,4 + 5x
41.
6
+V 5,
6
- i
6
- V 5,
8
-3 i
Write each function as (a) the product of linear and
irreducible quadratic factors and (b) the product of
linear factors. Then (c) list all of its zeros. (Example 7)
/
2 1 - 1
Solve each equation. (Example 3)
42.
gix) =
x 4 - 3x 3 -
12x 2 + 20x
(fa x 4 + 2x 3 — 7x 2 — 20x — 12 = 0
+
48
-
216
(4 3 ) gix) =
x 4 - 3x 3 -
12x 2 +
11. x 4 + 9x 3 + 23x2 + 3x - 36 = 0
44.
hix) =
x4 + Ix 3 -
15x2 + 18x
12. x 4 — 2x 3 — 7x 2 +
45.
fix )=
4x 4 - 35x 3
+ 140x2 - 295x + 156
8
x + 12 = 0
13. x 4 - 3x 3 - 20x 2 + 84x - 80 = 0
46. f i x ) = 4x 4 - 15x3 + 43x 2 + 577x + 615
14. x4 + 34x =
47. hix) = x 4 - 2x 3 - 17x2 + 4x + 30
15.
6
6
x 3 + 21x 2 - 48
x 4 + 41x3 + 42x2 - 96x +
6
= -2 6
8
48. g(x) = x 4 + 31x2 - 180
—12x 4 + 77x 3 = 136x2 - 33x - 18
17. SALES The sales S(x) in thousands of dollars that a store
makes during one m onth can be approximated by
S(x) = 2x 3 — 2x 2 + 4x, where x is the num ber of days after
the first day of the month. How many days will it take the
store to make $16,000? (Example 3)
Use the given zero to find all complex zeros of each function.
Then write the linear factorization of the function. (Example 8)
49. hix) = I x 5 + x 4 - 7x 3 + 21x 2 - 225x + 108; 3i
50. hix) = 3x5 - 5x4 - 13x3 - 65x2 - 2200x + 1500; - 5 i
51. g(x) = x 5 - 2x 4 - 13x3 + 28x2 + 46x - 60; 3 - i
Determine an interval in which all real zeros of each
function must lie. Explain your reasoning using the upper
and lower bound tests. Then find all the real zeros. (Example 4)
52. ^(x) = 4x 5 - 57x 4 + 287x3 - 547x 2 + 83x + 510; 4 + i
18. f( x )
53. fi x ) = x 5 - 3x 4 - 4x 3 + 12x 2 - 32x + 96; - 2 i
=
x4 - 9x 3 +
12x 2 + 44x - 48
54. gix) = x 4 - 10x 3 + 35x 2 - 46x + 10; 3 + i
19. /(x) =
2x 4 - x 3 -
29x2 + 34x + 24
55. ARCHITECTURE An architect is constructing a scale model
of a building that is in the shape of a pyramid.
20. g(x) = I x 4 + 4x 3 - 18x2 - 4x + 16
2 1
. g(x) =
22. f( x )
=
6
x 4 - 33x 3
2x 4 - 17x3
-
6
is
x 2 + 123x - 90
length and its base is a square, write a polynomial
function that describes the volume of the model in
terms of its length.
+ 39x 2 - 16x - 20
23. f( x ) = 2x 4 - 13x3 + 21x 2 + 9x - 27
24. h{x) — x 5
a. If the height of the scale model is 9 inches less than its
■9x 3 + 5x 2 + 16x - 12
25. h(x) = 4x5 - 20x4 + 5x3 + 80x2 - 75x + 18
b.
If the volum e of the model is 6300 cubic inches, write
an equation describing the situation.
C. W hat are the dim ensions of the scale model?
flJconnectEamcgrawT^
127
56. CONSTRUCTION The height of a tunnel that is under
construction is 1 foot more than half its width and its
length is 32 feet more than 324 times its width. If the
volume of the tunnel is 62,231,040 cubic feet and it is a
rectangular prism, find the length, width, and height.
72.
$ MULTIPLE REPRESENTATIONS In this problem , you will
explore even- and odd-degree polynom ial functions.
a.
ANALYTICAL Identify the degree and num ber of zeros of
each polynom ial function.
i. m
Write a polynomial function of least degree with integer
coefficients that has the given num ber as a zero.
3/
57. ^ 6
58.y
5
- \ fl
59.
= x 3 — x 2 + 9x — 9
ii. g(x) = 2x 5 + x 4 - 32x - 16
iii. h(x) = 5x 3 + 2x 2 — 13x + 6
iv. fi x ) = x 4 + 25x2 + 144
V. h(x) = 3x 6 + 5x 5 + 46x4 + 80x3 — 32x
60.-S / 7
vi. g t o = 4x 4 — l l x 3 + 10x 2 — l l x +
Use each graph to write g as the product of linear factors.
Then list all of its zeros.
g(x) = 3x 4 - 15x3 + 87x2 - 375x + 300
61.
b. NUMERICAL Find the zeros of each function.
c. VERBAL Does an odd-degree function have to have a
minim um num ber of real zeros? Explain.
H.O.T. Problems
62. g(x) = 2x5 + 2x4 + 28x3 + 32x2 - 64x
6
73. ERROR ANALYSIS Angie and Julius are using the Rational
Zeros Theorem to find all the possible rational zeros of
f(x ) = 7x 2 + 2x 3 — 5x — 3. Angie thinks the possible zeros
are ± y , ± y , ± 1 , ± 3 . Julius thinks they are + 1 , ± ~ , ± 1 ,
± 3 . Is either of them correct? Explain your reasoning.
74. REASONING Explain why g(x) = x 9 - x 8 + x 5 + x 3 x 2 + 2 m ust have a root betw een x = —I and x = 0 .
[ - 4 , 4] scl: 1 by [- 4 0 , 80] scl: 12
Determine all rational zeros of the function.
63. h{x) =
6
x 3 - 6x2 + 12
64. f( y ) = \ yA + |y 3 - y 2 +
2
y-
a. -/ (x )
8
65. w(z) = z4 - 10z 3 + 30z2 - lOz + 29
6 6
. b(a) = a5 - f a 4 + |fl3 - \ a 2 - \a 4-
O
O
J
o
\o
(6 ?) ENGINEERING A steel beam is supported by two pilings
200 feet apart. If a weight is placed x feet from the
piling on the left, a vertical deflection represented by
d = 0.0000008x2(200 — x) occurs. How far is the weight
from the piling if the vertical deflection is 0 . 8 feet?
Write each polynomial as the product of linear and
irreducible quadratic factors.
68.
70.
8
128
75. CHALLENGE Use/(x) = x 2 + x - 6 ,/(x) = x 3 + 8 x 2 +
19x + 12, and/(x) = x 4 — 2x 3 — 21x 2 + 22x + 40 to make a
conjecture about the relationship betw een the graphs
and zeros of/(x) and the graphs and zeros of each of the
following.
x3 —3
69. z 3 + 16
x3 + 9
71. 27x6 + 4
| Lesson 2 -4 | Zeros of Polynomial Functions
b. / ( - x )
76. OPEN ENDED Write a function of 4 th degree w ith an
imaginary zero and an irrational zero.
77. REASONING Determ ine whether the statem ent is true or
false. If false, provide a counterexample.
A third-degree polynom ial with real coefficients has at least one
nonreal zero.
CHALLENGE Find the zeros of each function if h(x) has zeros
at xv x2, and x3.
78. c(x) = 7h(x)
79. k(x) = h(3x)
80. g(x) = h(x — 2 )
81. /(x) = h (—x)
82. REASONING I f x - c is a factor of/(x) = a^x5 — a 2x 4 + ...,
w hat value m ust c be greater than or equal to in order to
be an upper bound for the zeros of/(x)? Assum e a =/=0.
83. WRITING IN MATH Explain why a polynom ial w ith real
coefficients and one imaginary zero must have at least two
imaginary zeros.
Spiral Review
D ivide using synthetic d ivision. (Lesson 2-3)
84.
8 6
(x 3 - 9x2 + 27x - 28) + (x - 3)
. (3x 4 - 2 x
85. (x 4 + x 3 - 1) + (x - 2)
+ 5x 2 - 4 x - 2 ) + (x + l )
3
87. (2x 3 - 2x - 3) + (x - 1)
D escribe the end b ehav ior o f the graph of each polynom ial fu nction using lim its. Explain
your reasoning using the lead ing term test. (Lesson 2-2)
8 8
. f( x ) = —4x 7 + 3x 4 +
89. /(x) = 4x 6 + 2x 5 + 7x 2
6
90. #(x) = 3x 4 + 5x 5 - 11
Estim ate to the nearest 0.5 un it and classify the extrem a fo r the graph o f each fu nction.
Support the answ ers num erically. (Lesson 1-4)
nJ
—I
I
-4
k
i
8*
4 n
/7 1 7
f (x) = (x - 1)(x + 1)(x + 3)
94. FINANCE Investors choose different stocks to com prise a balanced portfolio. The matrices
show the prices of one share of each of several stocks on the first business day of July,
August, and September. (Lesson 0-6)
July
August
September
Stock A
[33.81
30.94
27.25]
Stock B
[15.06
13.25
8.75]
Stock C
[54
54
46.44]
[52.06
44.69
34.38]
Stock
D
a. Mrs. Rivera owns 42 shares of stock A, 59 shares of stock B, 21 shares of stock C, and
18 shares of stock D. Write a row matrix to represent Mrs. Rivera's portfolio.
b.
Use m atrix multiplication to find the total value of Mrs. Rivera's portfolio for each
month to the nearest cent.
95. SAT/ACT A circle is inscribed in a
square and intersects the square at
points A, B, C, and D. If AC = 12,
what is the total area of the shaded
regions?
A 18
D 24 tt
B 36
E 72
C 18 tt
97. Find all of the zeros of p(x) = x 3 + 2x 2 — 3x + 20.
A - 4 , 1 + 2i, 1 - 2 i
C - 1 , 1 , 4 + t, 4 - i
B 1, 4 + t, 4 - i
D 4 ,1 + i, 1 - i
98. REVIEW W hich expression is equivalent to
it2 + 31 - 9)(5 - f)-1 ?
F
f+ 8 —
31
5
- t
-t ■
96. REVIEW f{ x ) = x 1 — 4x + 3 has a relative m inimum
located at which of the follow ing x-values?
F -2
G 2
H 3
H - t -
8
+
J —f — 8 —
31
5
- t
31
5 —f
J 4
129
• You identified points • 1 Analyze and graph
rational functions.
of discontinuity and
end behavior of
Solve rational
graphs of functions
equations.
using limits.
2
(Lesson 1-3)
NewVocabulary
rational function
asymptote
vertical asymptote
horizontal asymptote
oblique asymptote
holes
• Water desalination, or
removing the salt from sea
water, is currently in use in
areas of the world with
limited water availability
and on many ships and
submarines. It is also being
considered as an alternative
for providing water in the
future. The cost for various
extents of desalination can
be modeled using rational
functions.
5. Freshwater storage
4. Post-treatment
3 .Reverse
osmosis
process
Concentrated sea
water disposal
Treated
2. Pretreatment
system
Mesh
spacer
1. Seawater supply
1 R ational Functions
A rational fu n ction f(x ) is the quotient of two polynom ial functions
a(x) and b(x), where b is nonzero.
The dom ain of a rational function is the set of all real num bers excluding those values for which
b(x) = 0 or the zeros of b(x).
One of the sim plest rational functions is the reciprocal function
f( x ) = —. The graph of the reciprocal function, like m any rational
functions, has branches that approach specific x- and y-values.
The lines representing these values are called asym ptotes.
The reciprocal function is undefined when x = 0, so its dom ain is
(—oo, 0) or (0, oo). The behavior o ff( x ) = j to the left (0 “ ) and right
(0 +) of x =
0
can be described using limits.
lim f(x) = oo
lim f( x ) = —oo
From Lesson 1-3, you should recognize 0 as a point of infinite discontinuity in the domain of/. The
line x = 0 in Figure 2.5.1 is called a vertical asym ptote of the graph o f f . The end behavior of/ can be
also be described using limits.
lim f( x ) =
lim f(x )
X—
»+OCT
0
=
0
The line y = 0 in Figure 2.5.2 is called a horizontal asym ptote of the graph o f f
V
x) —
*
’■'S 0
X
\
\
\
1
Figure 2.5.1
./
y
n
Figure 2.5.2
These definitions of vertical and horizontal asym ptotes can be generalized.
130
Lesson 2-5
You can use your know ledge of limits, discontinuity, and end behavior to determ ine the vertical
and horizontal asym ptotes, if any, of a rational function.
KeyConcept Vertical and Horizontal Asymptotes
ReadingM ath
Limit Notation The expression
W ords
the graph of f if lim_ f(x) =
X—
*C
left and the expression
lim
X—►£+
The line y = c is a horizontal asymptote of the graph
±00 or
lim fix) =
X—
►
—
00
o f f if
lim fix) = ± o o C
lim fix) is read as the limit o f f
o f x as x approaches c from the
W ords
The line x = c is a vertical asymptote of
co r lim f(x) = c.
X—
►
oo
X-tC+
E xam ple
E xam ple
f(x) is read as the limit o f f o f x
f(x) =
as x approaches c from the right.
(x +
2)2
vertical
asymptote:
x = —2
H ^ S E E U D Fincl Vertical and Horizontal Asymptotes
Find the domain of each function and the equations of the vertical or horizontal asymptotes,
if any.
a. f i x ) :
X ' 3 =0
*"3
3
Find the domain.
The function is undefined at the real zero of the denom inator b(x) = x — 3. The real
zero of b(x) is 3. Therefore, the dom ain o f/ is all real num bers except x = 3.
CTHTF3 Find the asymptotes, if any.
Check for vertical asymptotes.
Determ ine whether x = 3 is a point of infinite discontinuity. Find the lim it as x
approaches 3 from the left and the right.
X
f(x)
2.9
-6 9
2.99
3
2.999
-6 9 9
3.01
3.1
7001
701
71
undefined
-6 9 9 9
Because lim_ f(x ) = —0 0 and lim fi x ) =
x >3
3.001
00
x—* 3 ^
, you know that x = 3 is a vertical
asymptote of/.
Check for horizontal asymptotes.
Use a table to exam ine the end behavior o ff(x ).
-1 0 ,0 0 0
-1 0 0 0
-1 0 0
0
0.9993
0.9930
0.9320
-1 .3 3
.....100
_
1.0722
1000
10,000
1.0070
1.0007
The table suggests that lim fix ) = lim fix ) = 1. Therefore, you know that y = 1 is a
X—* —ocr
x —>oc
horizontal asymptote of/.
x + 4
CHECK The graph of fix ) =
x
these findings. S
— 3
show n supports each of
...................................................
co n n ec tE D .m cg ra w -h ill.c o m |
i
131
.
, .
8x2 + 5
b- S (x ) = —
2-----4xz + l
|The zeros of the denom inator b(x) = 4 x 2 + 1 are imaginary, so the domain of g is all
real numbers.
Because the dom ain of g is all real num bers, the function has no vertical asymptotes.
Using division, you can determ ine that
g (x) ■
8 x2 + 5
4xz + 1
2
+
ix 2 + i
As the value of \x\ increases, 4,t 2 + 1 becom es an increasing large positive number
and — — - decreases, approaching 0. Therefore,
lim g(x) = lim g(x) =
X —►
—OO
X —>oo
2
+
0
or 2 .
CHECK You can use a table of values to support this
gx 2 I c
reasoning. The graph of g (x ) = — — —shown
also supports each of these findings. ✓
p GuidedPractice
if any.
1A. m(x) = f £ >
1B. h{x) = ! ^ f f *
The analysis in Exam ple 1 suggests a connection betw een the end behavior of a function and
its horizontal asymptote. This relationship, along w ith other features of the graphs of rational
functions, is sum marized below.
KeyConcept Graphs of Rational Functions
If f is the rational function given by
f(Y )_ a M _
b(x)
where b(x)
a„x" + a „ _ 1xn- 1 + . . . + a 1x + a 0
bmxm +
+ ... + * , x + b0'
0 and a(x) and b(x) have no common factors other than ± 1 , then the graph of f has the following
characteristics.
StudyTip
V ertical A sym p to tes Vertical asymptotes may occur at the real zeros of b(x).
Poles A vertical asymptote in the
H orizontal A s ym p to te The graph has either one or no horizontal asymptotes as determined by comparing the degree
graph of a rational function is also
\
called a pole of the function.
,
n of a(x) to the degree m of b(x).
>
• If n < m, the horizontal asymptote is y = 0.
• If n = m, the horizontal asymptote is y =
• If n > m, there is no horizontal asymptote.
In tercep ts The x-intercepts, if any, occur at the real zeros of a(x). The y-intercept, if it exists, is the value of f
when x = 0.
1
132
| Lesson 2-5 | R ational Functions
/
\
To graph a rational function, simplify/, if possible, and then follow these steps.
StudyTip
R f f l n Find the domain.
Test Intervals A rational function
ETEflFJ
can change sign at its zeros and
its undefined values, so when
these x-values are ordered, they
Find and sketch the asym ptotes, if any.
E3SH Find and plot the x-intercepts and y-intercept, if any.
>E9
3
divide the domain of the function
E] Fin d and plot at least one point in the test intervals determ ined by any x-intercepts and
vertical asymptotes.
into test intervals that can help
you determine if the graph lies
above or below the x-axis.
P E S S H U l GraPh Rational Functions: n < m and n > m
For each function, determ ine any vertical and horizontal asym ptotes and intercepts.
Then graph the function, and state its domain.
a.
g(x) =
6
x
3
CTTffn The function is undefined at b(x) = 0, so the dom ain is {x |x =/= —3, x g R }.
WM'.H There is a vertical asym ptote at x = —3.
The degree of the polynom ial in the num erator is 0, and the degree of the polynomial
in the denom inator is 1. Because 0 < 1, the graph of g has a horizontal asymptote at
V = °'
k W .H The polynom ial in the num erator has no real zeros, so g has no x-intercepts. Because
g( 0 ) = 2 , the y-intercept is 2 .
ETHT1 G raph the asym ptotes and intercepts. Then choose x-values that fall in the test
intervals determ ined by the vertical asym ptote to find additional points to plot
on the graph. Use sm ooth curves to com plete the graph.
StudyTip
Hyperbola The graphs of the
reciprocal functions f(x) = - and
R
gix) = —2— are called
Interval
^
x+3
hyperbolas. You will learn more
(-oo, -3)
about hyperbolas in-Chapter 7.
(-3, oo)
b.
k(x) =
X
(*, ffM )
-8
(-8,-1.2)
-6
(-6, -2)
-4
(-4, -6)
-2
(-2,6)
2
(2,1.2)
x2 - 7x + 10
x —3
Factoring the num erator yields k(x) ■
(:x - 2)(.v - 5)
. N otice that the num erator and
x —3
denom inator have no com m on factors, so the expression is in sim plest form.
The function is undefined at b(x) = 0, so the dom ain is {x |x # 3, x G R}.
There is a vertical asym ptote at x = 3.
Compare the degrees of the num erator and denominator. Because 2 > 1, there is no
horizontal asymptote.
ETfTTil The num erator has zeros at x = 2 and x = 5, so the x-intercepts are 2 and 5.
A:(0) = —^-y-, so the y-intercept is at about — 3.3.
F I T m G raph the asym ptotes and intercepts.
Then find and plot points in the test intervals
determ ined by the intercepts and vertical
asymptotes: (— oo, 0), (0, 3), (3, oo). Use sm ooth
curves to com plete the graph.
p GuidedPractice
2A. h(x) = —r— —
2B. n(x) =
xz + x - 2
c o n n e c tE D jT ic g ra ^
133
In Exam ple 3, the degree of the num erator is equal to the degree of the denominator.
Graph a Rational Function: n = m
3x2 —3
Determine any vertical and horizontal asym ptotes and intercepts for fix ) = —
x
Factoring both numerator and denom inator yields/(x) =
.
9
+ ^ w ith no com m on factors.
The function is undefined at b(x) = 0, so the dom ain is {x |x ^ —3, 3, x e R }.
There are vertical asymptotes at x = 3 and x = —3.
There is a horizontal asym ptote at y = —or y = 3, the ratio of the leading coefficients of
the num erator and denominator, because the degrees of the polynom ials are equal.
HTBin The x-intercepts are 1 and —1, the zeros of the numerator. The y-intercept is
because /( 0 ) =
F f T m Graph the asymptotes and intercepts. Then find and plot points in the test intervals
(—oo, - 3 ) , ( - 3 , - 1 ) , ( - 1 , 1 ) , (1 ,3 ), and (3, oo).
/te a
V
■
L
f(x) =
>■
y = 3
3x2 - 3
x2- 9
^
ir
'
f GuidedPractice
For each function, determ ine any vertical and horizontal asymptotes and intercepts.
Then graph the function and state its domain.
3A. h{x) -
3B. h{x) =
x + 2
x
— 4
5x2 — 5
StudyTip
Nonlinear Asymptotes
Horizontal, vertical, and oblique
asymptotes are all linear.
A rational function can also have a
>When the degree of the num erator is exactly one more than the degree of the denominator, the graph
has a slant or oblique asymptote.
nonlinear asymptote. For example,
V3
the graph of f(x) = -^ —r has a
x- 1
quadratic asymptote.
KeyConcept Oblique Asymptotes
E xam ple
if f is the rational function given by
f(x) =
a(x)
b(x)
a
/ + a „ _ /-
bmxm+
+ ... + b ,x + b0
where b(x) has a degree greater than 0 and a(x ) and b(x) have no
common factors other than 1, then the graph of f has an oblique
asymptote if n = m + 1. The function for the oblique asymptote is the
quotient polynomial q(x) resulting from the division of a(x) by b(x).
W ~ b(x) ~ q(x) + b(x)
function for oblique asymptote
134
Lesson 2-5
I
R ational Functions
p F T E f f f f l n Graph a Rational Function: n = m + 1
2x
Determine any asymptotes and intercepts for fix ) =
. Then graph the function, and
x 1 + .V - 12
state its domain.
Factoring the denom inator yields/(x)
E SH
2
x3
(x + 4)(x-3 y
The function is undefined at b(x) = 0, so the domain is {x |x =£ —4, 3, x £ R }.
ETffiW There are vertical asym ptotes at x = —4 and x = 3.
The degree of the num erator is greater than the degree of the denominator, so there is
no horizontal asymptote.
Because the degree of the num erator is exactly one more than the degree of the
denom inator,/has a slant asymptote. Using polynom ial division, you can write
the following.
/ (*) =
2 x3
x + x —1 2
: Zx -
26x - 24
2 +
x
2 + x - 1 2
Therefore, the equation of the slant asym ptote is y = 2x — 2.
EflCTfl The x- and i/-intercepts are 0 because 0 is the zero of the num erator and
StudyTip
m
End-Behavior Asymptote
In Example 4, the graph
of f approaches the slant
asymptote y = 2 x - 2 as
x —►± 00. Between the vertical
asymptotes x = - 4 and x = 3,
>
= o.
ET7?m G raph the asym ptotes and intercepts. Then find and plot points in the test intervals
(— oo, — 4), (—4,0), (0,3), and (3, oo).
however, the graph crosses the
line y = 2.x - 2. For this reason, a
slant or horizontal asymptote is
sometimes referred to as an
end-behavior asymptote.
► GuidedPractice
For each function, determ ine any asymptotes and intercepts. Then graph the function and
state its domain.
4A. h(x)
StudyTip
x 2 + 3x —3
x+ 4
4B. p(x) =
x 2 - 4x + 1
2x —3
W hen the num erator and denom inator of a rational function have com m on factors, the graph of the
function has removable discontinuities called holes, at the zeros of the com m on factors. Be sure to
indicate these points of discontinuity when you graph the function.
Removable and Nonremovable
Discontinuities If the function is
not continuous at x = a, but could
be made continuous at that point
by simplifying, then the function
has a removable discontinuity at
x = a. Otherwise, it has a
nonremovable discontinuity x = a.
>
(x-— r^T(x - b)
/ (*) = ix—-vr>
r (x - c)
Divide out the common factor in
the numerator and denominator.
The zero of x — a is a.
connectED.m cgraw-hill.com I
135
B 2 S S 3 3 S 0 GraPh a Rati° nal Function with Common Factors
Determine any vertical and horizontal asym ptotes, holes, and intercepts for h(x) = —p
2x-8
x
-
x-
(x -
x-
(x
Factoring both the num erator and denom inator yields h(x) =
PTTim The function is undefined at b(x) = 0, so the domain is {x |x
2
;X + -2 .
4
—2, 4, x G R }.
ETBTEI There is a vertical asym ptote at x = 4, the real zero of the simplified denominator.
There is a horizontal asym ptote at y = y or 1, the ratio of the leading coefficients of the
num erator and denominator, because the degrees of the polynom ials are equal.
StudyTip
Hole For Example 5 , / + 2 was
The x-intercept is 2, the zero of the sim plified numerator. The y-intercept is
divided out of the original
—because h( 0 ) = —.
1
expression. Substitute - 2 into the
new expression.
( - 2) -
*(—2 ) = ■:-2> - 4
1
ETHT1 Graph the asymptotes and intercepts.
Then find and plot points in the test
intervals (— oo, 2), (2,4), and (4, oo).
2
There is a hole at
There is a hole at
•2 x- i
(—2, -|-j because
the original function is undefined
when x = —2 .
f GuidedPractice
For each function, determine any vertical and horizontal asymptotes, holes, and intercepts.
Then graph the function and state its domain.
5A. g(x) =
x2 +
lO x + 2 4
5B. c(x) =
x 2 + x — 12
x —2x —3
x 2 — 4x — 5
Rational Equations Rational equations involving fractions can be solved by m ultiplying
mm each term in the equation by the least com m on denom inator (LCD) of all the terms of the
equation.
B E 2 I 0 3 Q Solve a Rational Equation
StudyTip
Solve x + ■
Check for Reasonableness
■— 0 .
You can also check the result in
Example 6 by using a graphing
calculator to graph y=x-t—
x + -
=
0
)=
0
X- I
~
X—o
x(x — 8 ) + ■
Use the CALC menu to locate the
- (x -
8
Original equation
(x -
8
)
Multiply by the LCD, x — i
zeros. Because the zeros of the
graph appear to be at about
x
— 8x + 6 = 0
x = 7.16 and x = 0.84, the
8 ± V ( — 8 )2 — 4 (1 )(6 )
solution is reasonable.
Quadratic Formula
2 ( 1)
:=
...................
Y=0
.
rrrr
or 4 ± V 10
Simplify.
6A.
Lesson 2-5
+ 2VT(3
p GuidedPractice
[- 2 0 , 20] scl: 2 by
[ - 2 0 , 20] scl: 2
136
8
I R ational
20
x + 3
Functions
-4 = 0
6
B.
9x
=
x-2
6
StudyTip
'• Solving a rational equation can produce extraneous solutions. Always check your answers in the
original equation.
Intersection You can use the
Solve a Rational Equation with Extraneous Solutions
intersection feature of your
graphing calculator to solve a
rational equation by graphing
each side of the equation and
finding all of the intersections of
the two graphs.
>
Solve —
- -------- =
x2 —6x +
8
x—
2
+ —— .
x—
4
The LCD of the expressions is (x — 2)(x — 4), which are the factors of x 2 — 6x
4
x —6x + f
(x — 2)(x - 4)
3x .
2
x —2
x —4
Original equation
------—j
= (x — 2)(x — 4 ) (—52— |
\Ux -—2 2
x -4 )
4 = 3x(x - 4) + 2(x - 2)
4 = 3 x 2 — lOx - 4
0 = 3 x 2 — lOx — 8
0 = (3x + 2)(x - 4)
Factor.
Multiply by the LCD.
x 2 — 6x + 8
Solve.
Because the original equation is not defined w hen x = 4, you can elim inate this extraneous
solution. So, the only solution is ■
3'
p GuidedPractice
7A. - ^ - +
x+3
27
3 x —6
x 2 —3x — 18
7B. -
12
2
x 2 + 6x
x+6
x -2
Real-World Example 8 Solve a Rational Equation
ELECTRICITY The diagram of an electric circuit shows three parallel
resistors. If R is the equivalent resistance of the three resistors,
1
1
1
1
then — = — + — + — . In this circuit,
is tw ice the resistance
Iv
1
2
3
of R 2, and R 3 equals 20 ohms. Suppose the equivalent resistance
is equal to 10 ohms. Find R 2 and R 2-
i
= i
+ i
+ i
^ = 2 fe + i
Electrician
+ i j
Original equation
ff = 10, /?, = 2R2, and ff3 = 20
Electricians install
and maintain various components
i o = 2T2 + T 2
of electricity, such as wiring and
Subtract ~ from each side.
fuses. They must maintain
compliance with national, state,
(20jR2) ^ = (20R 2) | ^ - +
Multiply each side by the LCD, 2 0 %
and local codes. Most electricians
complete an apprenticeship
R2 =
program that includes both
1 0
+
2 0
or 30
Simplify.
classroom instruction and
on-the-job training.
R 2 is 30 ohms and R j = 2 R 2 or 60 ohms.
GuidedPractice
8.
ELECTRONICS Suppose the current I, in amps, in an electric circuit is given by the formula
I = t+
•*_ , where t is time in seconds. A t what time is the current 1 amp?
coTi^ÊiDjniii^raw^
K
137
W h ''
*
\1 ( V "
4
Exercises
Find the domain of each function and the equations of the
vertical or horizontal asymptotes, if any. (Example 1)
'Jo
x 2 - 2
o
_ x3 —_ 8
2 . hix) =
1 . f(x ) =
x+4
x 2 - 4
3. f{x ) =
x(x - l)(x + 2 ) 2
ix + 3)(x - 4)
5. hix) =
2x2 - 4x + 1
x2 + 2 x
4. g(x) =
6
. fix ) =
8
. gix) =
(x - l)(x + 1 )
7. hix) =
(x - 2)2(x + 4);
30. STATISTICS A num ber x is said to be the harmonic mean of y
1
1
(Example 7)
a. Write an equation for w hich the solution is the
harm onic m ean of 30 and 45.
b. Find the harm onic m ean of 30 and 45.
x —6
(x + 3)(x + 5)
x 2 + 9x + 20
x —4
1
and z if — is the average of —and
31. OPTICS The lens equation is \
+ -f-, where f is the
/
d0
focal length, d; is the distance from the lens to the image,
and d 0 is the distance from the lens to the object. Suppose
the object is 32 centimeters from the lens and the focal
length is 8 centim eters. (Example 7)
(x - 4)(x + 2)
(x + l)(x - 3)
^ tiov-£‘5'Kyl-<5)
For each function, determine any asymptotes and intercepts.
Then graph the function, and state its domain. (Examples 2-5)
9. f(x ) =
1 1
. f( x ) =
ix + 2)(x - 3)
(x + 4)(x - 5)
8
(x -
2
1 2
)(x + 2 )
(x + 2)(x + 5)
13. gix) =
(x + 5) 2(x -
6
)
x2(x —2)(x + 5)
15. hix) =
x 2 + 4x + 3
. /(x) =
14. /i(x) =
16. f ix ) =
(2x + 3)(x - 6 )
(x + 2 )(x - 1 )
x+2
x(x —6 )
b. Find the distance from the lens to the image.
(x + 6 )(x + 4)
x(x - 5)(x + 2)
x(x + 6)2(x —4)
Solve each equation. (Examples 6 -8
x 2 - 5x - 24
J
3 4
. A z iJ_ + £ ± i
2x - 4
3x
= 1
3 5
.
19. SALES The business plan for a new car w ash projects that
profits in thousands of dollars will be modeled by the
3z
function p(z) =
•, where z is the w eek of
2z + 7z + 5
operation and z = 0 represents opening. (Example 4)
intercepts for p iz).
C.
37.
X — 1 _ J_
x
20
39.
x — 1 . 3x + 6
•= 3
x -2
2x + 1
41.
X
38.
a. State the domain of the function.
b. Determine any vertical and horizontal asymptotes and
23
36. — 1
40.
2
X+ l
X
_
x+ l
-
X2 + X
4
<N
6
3:
y
1
32. y + | = 5
II
N
-4
18. £(x) =
x +
x 2 + 4x + 5
V 'V A 'f
a. Write a rational equation to model the situation.
G
O|Nt
,7 . /(*)■:
10. £(x) =
y+2
2
4
x —2
2
x
2
y2 + 4
y 2 —4
4
x+
6
x- 3
u+ 3
y
—y
14
x 2 —2 x
2
3
4 —a
12
x 2 —x —6
2a —2
a2 —a — 12
Graph the function.
42. WATER The cost per day to remove x percent of the salt
For each function, determine any asymptotes, holes, and
intercepts. Then graph the function and state its domain.
(Examples 2-5)
. fix ) =
24. h{x) =
1& . fi x ) =
28. fix )
138
994x
—,
a. Graph the function using a graphing calculator.
b. G raph the line y = 8000 and find the intersection with
3x —4
2 0 . h{x) =
x3
2 2
from seaw ater at a desalination plant is c(x) =
w here 0 < x < 1 0 0 .
2 1
x 2 + 2x - 15
x 2 + 4x + 3
„3
x -t- 3
x 2 - 4x - 21
x 3 + 2x2 - 5x - (
(x + 4)(x - 1)
(x - l)(x + 3)
Lesson 2-5
. hix)
4x 2 - 2x + 1
3x 3 + 4
the graph of c(x) to determ ine w hat percent of salt can
be rem oved for $8000 per day.
C.
According to the m odel, is it feasible for the plant to
remove 100% of the salt? Explain your reasoning.
23. g ix ) :
x+ 7
x —4
25. g ix )
x 3 + 3x 2 + 2x
x —4
| SM
x 2 —4
x 3 + x 2 - 4x - 4
43. x-intercepts at x = 0 and x = 4, vertical asym ptotes at
(2x + l)(x - 5)
44. x-intercepts at x = 2 and x = —3, vertical asym ptote at
7
29. gix)
Rational Functions
(x - 5)(x + 4) 2
Write a rational function for each set of characteristics.
x =
1
and x =
6
, and a horizontal asym ptote at y =
x = 4, and point discontinuity at (—5, 0)
0
TRAVEL W hen distance and time are held constant, the
average rates, in miles per hour, during a round trip can
30r,
be modeled by r 2 =
where r 1 represents the
average rate during the first leg of the trip and r 2
represents the average rate during the return trip.
53. #
MULTIPLE REPRESENTATIONS In this problem , you will
investigate asym ptotes of rational functions.
a. TABULAR Copy and com plete the table. Determine the
horizontal asym ptote of each function algebraically.
function, if any. Verify your answer graphically.
b. Copy and com plete the table shown.
1
45
50
55
60
65
Horizontal
Asymptote
Function
a. Find the vertical and horizontal asymptotes of the
w
= *2- 5x+4
x3 + 2
n \x ) —
x 3 — 3x2 + 4x — 12
4
x4 - 4
70
s w = 4x 5 +^ 3
c. Is a dom ain of r1 > 30 reasonable for this situation?
b. GRAPHICAL Graph each function and its horizontal
asym ptote from part a.
c. TABULAR Copy and com plete the table below. Use the
Use your know ledge o f asym ptotes and the provided points
to express the fu nction represented by each graph.
Rational Zero Theorem to help you find the real zeros
of the num erator of each function.
Real Zeros ot
Numerator
Function
x 3 — 3x2 + 4 x — 12
x4 — 4
9(*) =
4x5 +^ 3
VERBAL M ake a conjecture about the behavior of
Use the intersection feature of a graphing calculator to solve
each equation.
48.
x4 - 2x3 +
x
50j
1
0
'
3 + 6
3x -4 xz + )
4x + 2 x —1
51
. 2x4 - 5x2 + 3 _
x* + 3x2 —4
1
2x5 —3x3 + 5x
,
= 6
4x3 + 5x - 12
—
the graph of a rational function when the degree
of the denom inator is greater than the degree of
the num erator and the num erator has at least
one real zero.
H.O.T. Problem s
, yvill f( x ) sometimes,
dx3 + ex2 + /
always, or never have a horizontal asym ptote at y = 1 if t
b, c, d, e, an d / are constants w ith a =j= 0 and d =jfc 0. Explain.
54. REASONING Given/(x)
52. CHEMISTRY W hen a 60% acetic acid solution is added to
1 0 liters of a 2 0 % acetic acid solution in a 1 0 0 -liter tank,
the concentration of the total solution changes.
a L, 60% acetic
1/= 100 L<
10 L, 20% acetic acid
a. Show that the concentration of the solution is f(d ) =
+
where a is the volum e of the 60% solution.
5a + 50
b. Find the relevant domain of f( a ) and the vertical or
horizontal asymptotes, if any.
C.
Explain the significance of any dom ain restrictions or
asymptotes.
d. Disregarding dom ain restrictions, are there any
additional asymptotes of the function? Explain.
55. PREWRITE Design a lesson plan to teach the graphing
rational functions topics in this lesson. M ake a plan that
addresses purpose, audience, a controlling idea, logical
sequence, and time frame for completion.
56. CHALLENGE Write a rational function that has vertical
asym ptotes at x = —2 and x = 3 and an oblique
asym ptote y = 3x.
57. WRITING IN MATH Use w ords, graphs, tables, and equations
to show how to graph a rational function.
58. CHALLENGE Solve for k so that the rational equation has
exactly one extraneous solution and one real solution.
2
x2 —4 x + k
2x
x —1
1
x —3
59. WRITING IN MATH Explain why all of the test in terv
be used in order to get an accurate graph of a rat;
function.
&
connectED.mcgraw-hill.c
£
"~
Spiral Review
List all the possible rational zeros of each function. Then determ ine which, if any, are
zeros. (Lesson 2-4)
60. f( x ) = x + 2x — 5x — 6
61. /(x) = x 3 -
2
x 2 + x + 18
62. f( x ) = x 4 - 5 x 3 + 9 x 2 - 7 x + 2
Use the Factor Theorem to determine if the binomials given are factors of fi x ) . Use the
binomials that are factors to write a factored form of fi x ) . (Lesson 2-3)
63. fix ) = x 4 -- 2 x 3 - 13x2 + U x + 24; x - 3, x - 2
64. f i x ) = 2x4 — 5x 3 -- l l x 2 — 4x; x - 4, 2x — 1
II
66. fi x )
*
-v
II
+ 59x 3 + 138x2 - 45x - 50; 3x - 2, x - 5
— 3 x 3 -- 12x 2 + 17x -
6
; 4x - 3; x - 1
+ 15x4 + 12x 3 - 4 x 2; x + 2 , 4x + 1
1
*
OO
LO
II
*
68. f i x )
II
XO l
67. fi x )
*
65. fi x )
- 5x 3 + 10x 2 + x - 2 ; x + l , x - l
Graph each function. (Lesson 2-2)
69. fi x ) = (x + 7 ) 2
70. f i x ) = (x - 4 ) 3
71. f{ x ) = x 4 - 5
72. RETAIL Sara is shopping at a store that offers $10 cash back for every $50 spent. Let
fix ) =
and hix) = lOx, where x is the amount of m oney Sara spends. (Lesson 1-6)
a. If Sara spends money at the store, is the cash back bonus represented by f[h ix )] or h [/(*)]? Explain your reasoning.
b. Determine the cash back bonus if Sara spends $312.68 at the store.
73. INTERIOR DESIGN Adrienne Herr is an interior designer. She has been asked to locate an
oriental rug for a new corporate office. The rug should cover half of the total floor area
w ith a uniform w idth surrounding the rug. (Lesson 0-3)
a. If the dim ensions of the room are 12 feet by 16 feet, write an equation to m odel the
area of the rug in terms of x.
b. Graph the related function.
C.
W hat are the dim ensions of the rug?
Simplify. (Lesson 0-2)
74.
i 10- H 2
75.
(2 + 3«) +
( - 6
+ i)
76. (2.3 + 4.1*) - ( - 1 .2 - 6.3i)
77. SAT/ACT A company sells ground coffee in two sizes
of cylindrical containers. The sm aller container holds
10 ounces of coffee. If the larger container has twice
the radius of the smaller container and 1.5 times the
height, how many ounces of coffee does the larger
container hold? (The volume of a cylinder is given by
the formula V = Ttr2h.)
A 30
C
60
B 45
D 75
F x = l,x = -2
G x = -2 ,x = l
Lesson 2 -5
x2
x
H x = l + V 3 ,x = l - V 3
J*=
R atio n al Functions
6
T + 12 = 7T
C
|
°
T
e - T
~ j
+
1 2
= |
12
80. Diana can put a puzzle together in three hours. Ella
can put the same puzzle together in five hours. How
long will it take them if they work together?
\ ?
1 + V3
-x
A
B X
T _ T - 12
7
E 90
78. What are the solutions of 1 =
79. REVIEW Alex wanted to determ ine the average of his
6 test scores. He added the scores correctly to get T
but divided by 7 instead of 6 . The result w as 12 less
than his actual average. W hich equation could be
used to determ ine the value of T?
F 1—hours
1 - V3
G 1—hours
H
1
— hours
4
J 1 -j hours
• You solved
polynomial and
rational equations.
•
(Lessons 2-3 and 2-4)
NewVocabulary
polynomial inequality
sign chart
rational inequality
•
■§ Solve polynomial
1 inequalities.
Solve rational
2 inequalities.
Many factors are involved when starting a
new business, including the amount of the
initial investment, maintenance and labor
costs, the cost of manufacturing the product
being sold, and the actual selling price of the
product. Nonlinear inequalities can be used to
determine the price at which to sell a product
in order to make a specific profit,
Polynomial Inequalities If/(x) is a polynom ial function, then a polynomial inequality
has the general form /(x) < 0,f{ x ) < 0,/(x)
0,/(x) > 0, or f(x ) > 0. The inequality /(x) < 0 is
true w hen/(x) is negative, while/(x) > 0 is true when/(x) is positive.
1
In Lesson 1-2, you learned that the x-intercepts of a polynom ial function are the real zeros of the
function. W hen ordered, these zeros divide the x-axis into intervals for w hich the value of/(x) is
either entirely positive (above the x-axis) or entirely negative (below the x-axis).
By finding the sign of f(x ) for just one x-value in each interval,
you can determ ine on w hich intervals the function is positive
or negative. From the test intervals represented by the sign chart
at the right, you know that:
• /(x) < 0 on (—4, —2) U (2, 5) U (5, oo),
• /(x) < 0 on [—4, —2] U [2, oo),
• /(x) = 0 at x = —4, —2 ,2 , 5,
• f( x ) > 0 on (—oo, —4) U (—2, 2), and
• /(x) > 0 on (—oo, —4] U [—2, 2] U [5, 5].
m
s m
(+) ( - )
(+)
(-) (-)
-— I
1---------------1------ I— ►
- 4 - 2
2
5
Solve a Polynomial Inequality
Solve x 2 — 6x — 30 > —3.
Adding 3 to each side, you get x 2 — 6 x — 27 > 0. Let/(x) = x 2 — 6 x — 27. Factoring yields
/(x) = (x -I- 3)(x — 9), so/(x) has real zeros at —3 and 9. Create a sign chart using these zeros.
Then substitute an x-value in each test interval into the factored form of the polynom ial to
determ ine if/(x) is positive or negative at that point.
/U) = U + 3 ) U - 9 )
Think: (x + 3)
and (x - 9) are
both negative
when x = —4.
f(x) = (x+3)(x-9)
T e s t x = - 4 . T e s t x = 0 . Test x = 10.
-►
(-)(-)
(+ )(-)
(+ )(+ )
(+)
(-)
+
-3
Because/(x) is positive on the first and last intervals, thesolution set of x 2 — 6 x — 30 > —3 is (—oo, —3) U (9, oo). The
graph of /(x) supports this conclusion, because /(x) is
above the x-axis on these same intervals.
p GuidedPractice
1A.
x 2 + 5x +
6
< 20
1B. (x — 4 )2 > 4
connectED.m cgraw -hi'll'co'm l
141
If you know the real zeros of a function, including their multiplicity, and the function's end
behavior, you can create a sign chart w ithout testing values.
H 2 3 2 2 3 3 So,ve a Polynomial Inequality Using End Behavior
StudyTip
Polynomial Inequalities You
can check the solution of a
polynomial inequality by graphing
the function and evaluating the
Solve 3x 3 — 4x2 — 13% — 6 < 0.
ETEfln Let f(x ) = 3x 3 — 4 x 2 — 13x — 6 . Use the techniques from Lesson 2-4 to determ ine that
o
/ has real zeros w ith m ultiplicity 1 at —1, ——, and 3. Set up a sign chart.
truth value of the inequality for
+
each interval of the solution.
-1
EflSfiW Determ ine the end behavior o f / ( x ) . Because the degree o f/ is odd and its leading
coefficient is positive, you know lim / ( x ) = —o o and lim f( x ) = o o . This m eans that
the function starts off negative at the left and ends positive at the right.
(-)
(+)
1
1
-
1
-1
E S E
Because each zero listed is the location of a sign change, you can com plete the sign chart.
(-)
(+)
(-)
(+)
-1
The solutions of 3x3 — 4 x 2 — 13x — 6 < 0 are x-values such that/(x) is negative or equal
to 0. From the sign chart, you can see that the solution set is (—o o , —1] U
CHECK The graph of /(x) = 3x 3 — 4 x 2 — 13x — 6 is on or below
the x-axis on (—o o , —1 ] U
3
> GuidedPractice
2A. 2 x 2 — lOx < 2x — 16
2B. 2x + 7x —12x — 45 > 0
When a polynom ial function does not intersect the x-axis, the related inequalities have
unusual solutions.
^ ^ 5 5 3 2 0 0 Polynomial Inequalities with Unusual Solution Sets
Solve each inequality,
a. x 1 + 5x + 8 < 0
The related function/(x) = x 2 + 5x + 8 has no real zeros, so
there are no sign changes. This function is positive for all real
values of x. Therefore, x 2 + 5x + 8 < 0 has no solution. The
graph of/(x) supports this conclusion, because the graph is
never on or below the x-axis. The solution set is 0 .
b.
x 2 + 5x +
8 > 0
[- 1 2 , 8] scl: 1 by [ - 5 , 1 0 ] scl: 1
Because the related function/(x) = x 2 + 5x + 8 is positive for all
real values of x, the solution set of x 2 + 5x + 8 > 0 is all real numbers
or (—o o , o o ).
142
| Lesson 2 -6
I Nonlinear Inequalities
x2 — lOx + 25 > 0
The related function/(x) = x2 — lOx + 25 has one real zero,
5, w ith m ultiplicity 2, so the value of /(x) does not change
signs. This function is positive for all real values of x except
x = 5. Therefore, the solution set of x2 — 10% + 25 > 0 is
(—oo, 5) U (5, oo). The graph o f f(x) supports this conclusion.
f(x) = X2 -
\
Ox + 2 5 1\
[ - 2 , 8 ] scl:
1
J
/
by [ - 2 , 8 ] scl:
1
d. x2 - lO.r + 25 < 0
The related function /(x) = x 2 - lOx + 25 has a zero at 5. For all other values of x, the function
lOx + 25 < 0 is {51.
is positive. Therefore, the solution set of x 2
P GuidedPractice
StudyTip
Rational Inequalities Remember
to include all zeros and undefined
points of a rational function when
creating a sign chart.
V_________ ________________
3A. x2 + 2x + 5 > 0
3B.
3C. x 2 - 2x - 15 < - 1 6
3D. x 2 - 2x - 15 > - 1 6
2x + 5 < 0
R ational In eq u alities Consider the rational function at the right.
Notice the intervals on which/(x) is positive and negative. W hile
a polynom ial function can change signs only at its real zeros, a
rational function can change signs at its real zeros or at its points of
discontinuity. For this reason, when solving a rational inequality, you
m ust include the zeros of both the num erator and the denom inator in
your sign chart.
2
You can begin solving a rational inequality by first writing the inequality in general form with a
single rational expression on the left and a zero on the right.
I S
H
E
x —6
Solve a Rational Inequality
>0
■+ ■
x+ l
Original inequality
4x + 4 + 2 x - 1 2
>0
(x - 6)(x + 1 )
6x — 8
(x - 6)(x + 1 )
Let/(x)
>
Use the LCD, ( x — 6)(x + 1 ) , to rewrite each fraction. Then add.
Simplify.
0
6x — 8
(x - 6)(x + 1 )
numerator,
The zeros and undefined points of the inequality are the zeros of the
and denominator,
6
and —1. Create a sign chart using these numbers. Then choose
and test x-values in each interval to determ ine if/(x) is positive or negative.
f(x) = ;
6
x—
8
/(*) =
U - 6 ) U + 1)
T e s tx = -2 .
(-)
(-)(-)
undef.
T e s tx = 0 .
-)(+)
T e s tx = 2 .
(+)
(-)(+)
6x — 8
U —6 )U + 1 )
T e s tx = 7 .
undef.
(+)
(+)(+)
(-)
und e f.
(+ )
------------
1
(-)
und e f.
(+ )
+
-1
Figure 2.6.1
The solution set of the original inequality is the union of those intervals for which/(x) is positive,
1, -|-j U (6, oo). The graph of/(x) =
2
x —6
+ j - j - y in Figure 2.6.1 supports this conclusion.
&
143
^ GuidedPractice
4A.
>i
4x — 3
4B.
x —2
<3
4C. — > ■
x
x + 5
You can use nonlinear inequalities to solve real-world problems.
Real-World Example 5 Solve a Rational Inequality
AMUSEMENT PARKS A group of high school students is renting a bus for $600 to take to an
amusement park the day after prom. The tickets to the am usem ent park are $60 less an extra
$0.50 group discount per person in the group. Write and solve an inequality that can be used
to determine how m any students m ust go on the trip for the total cost to be less than $40 per
student.
Let x represent the number of students.
+
bus cost per student
+
600
Ticket cost per student
60 — 0.5x
60 - 0.5x + ^ 0 < 40
60 - 0.5x + ^ 0 _
4 0
6 0 * - 0 .5 x2 + 600 - 40x
-0 .5 x 2 + 20x + 600
■40x - 1200
The Kingda Ka roller coaster at Six
must be less than
<
40
Write the inequality.
< o
<0
Use the LCD, x, to rewrite each fraction. Then add.
<
Simplify.
0
>0
Multiply each side by —2. Reverse the inequality sign.
>
Factor.
Flags Great Adventure in New
Jersey is the tallest and fastest
(x + 20) (x - 60)
roller coaster in the world. The
0
ride reaches a maximum height of
456 feet in the air and then
Let/(x) =
— — •The zeros of this inequality are —20, 60, and 0. Use these num bers to
create and com plete a sign chart for this function.
plunges vertically into a 270°
spiral, while reaching speeds of
up to 128 miles per hour.
Source: Six Flags
(jr +
f(x) = '
Test
2 0
)(jr —60)
f(x) =
(x + 2 0 ) U - 60)
x = - 3 0 . Test x = - 1 0 . Test x = 10. Test x = 70.
(-)
(-)
undef.
W
zero
T fT
(_ )
zero
1
-2 0
So, the solution set of 60 — 0.5x +
60
-2 0
(+ )
und e f.
1
0
(_ )
zero
1------------60
< 40 is (—2 0 ,0 ) U (60, oo).
Because there cannot be a negative num ber of students, m ore than 60 students must go to the
amusement park for the total cost to be less than $40 per student.
GuidedPractice
\
5. LANDSCAPING A landscape architect is designing a fence that will enclose a rectangular garden
that has a perimeter of 250 feet. If the area of the garden is to be at least 1000 square feet,
write and solve an inequality to find the possible lengths of the fence.
\..
1 4 4 | Lesson 2 -6 | N o n lin e a r In e q u a litie s
Exercises
Find the solution set of fix ) — gix) > 0.
Solve each inequality. (Examples 1-3)
I ( x - - 6 )(x +
1. (x + 4)(x - 2) < 0
3. (3x + l)(x -
8
1
)>
0
( * - - 4 )(—2x + 5) < 0
)> 0
6. ) 2 x 3 - 9x 2 - 20x + 12 < 0
7. —8 x 3 - 30x2 - 18x < 0
8. 5x 3 — 43x 2 + 72x + 36 > 0
2
2
x2
< 5 — 2x
12.
2
x2
b 2 + \ 6 < b 2 + 8b
14. c 2 + 12 < 3 -
1
00
8
1
13.
10.
1
11. 4x 2 +
— 1 0
x >
w
6
VI
9. x 2 +
+
OO
X
IV
5. (4 - 6y)(2y + 1) < 0
6
c
16. 3 d 2 + 16 > —d 2 + 16rf
15. - a2 > 4« + 4
17. BUSINESS A new com pany projects that its first-year
revenue will be r(x) = 120x — 0.0004x2 and the start-up
cost will be c(x) = 40x + 1,000,000, where x is the number
of products sold. The net profit p that they will m ake the
i
first year is equal to p = r — c. W rite and solve an
inequality to determ ine how m any products the com pany
must sell to make a profit of at least $2,000,000. (Example 1)
38. SALES A vendor sells hot dogs at each school sporting
event. The cost of each hot dog is $0.38 and the cost of
each bun is $0.12. The vendor rents the hot dog cart that
he uses for $1000. If he w ants his costs to be less than his
profits after selling 400 hot dogs, what should the vendor
charge for each hot dog?
39. PARKS AND RECREATION A rectangular playing field for a
com m unity park is to have a perimeter of 1 1 2 feet and an
area of at least 588 square feet.
a. Write an inequality that could be used to find the
possible lengths to w hich the field can be constructed.
Solve each inequality. (Example 4)
18.
x —3
>3
x+4
20.
2x + 1
>4
x —6
22.
3 —2x
<5
5x + 2
24.
(x + 2 )(2 x ■3)
<
(x - 3)(x + 1)
26.
12x + 65
■> 5
(x + 4)
19.
6
x+ 6
<1
x —5
b. Solve the inequality you wrote in part a and interpret
the solution.
3x —2
x+3
C.
<6
23.
4x + 1
> -3
3x —5
25.
(4x + l)(x - 2)
<4
(x + 3)(x - 1)
27.
2x + 4
29. PROM A group of friends decides to share a limo for
prom. The cost of rental is $750 plus a $25 fee for each
occupant. There is a m inim um of two passengers, and the
limo can hold up to 14 people. Write and solve an
inequality to determ ine how many people can share the
limo for less than $120 per person. (Example 5)
Find the domain of each expression.
V*2+ 5x + 6
Solve each inequality, iffin t: Test every possible solution
interval that lies within the domain using the original
inequality.)
<12
(x ~ 3)
28. CHARITY The Key Club at a high school is having a dinner
as a fundraiser for charity. A dining hall that can
accommodate 80 people will cost $1000 to rent. If each
ticket costs $ 2 0 in advance or $ 2 2 the day of the dinner,
and the same number of people bought tickets in advance
as bought the day of the dinner, write and solve an
inequality to determine the m inim um number of people
that must attend for the club to m ake a profit of at least
$500. (Example 5)
30.
How does the inequality and solution change if the
area of the field is to be no more than 588 square feet?
Interpret the solution in the context of the situation.
31. V x 2 - 3 x - 4 0
40. ^ 9 y + 19 - >/6y - 5 > 3
41. V 4x + 4 — V x — 4 < 4
42. a/12 y + 72 - yj6y - 11 > 7
43. \/25 - 12x - \ /l6 - 4x < 5
Determine the inequality shown in each graph
44.
jif
L / 'x
y
7 y
V
\
0
12 \
2
(2,
/
/
/
(4, - 6)
/
2)
y
45.
—4 -2
-2
sr
X
0
i t ), - -8
46. 2y4 - 9y3 - 29y2 + 60y + 36 > 0
47. 3«4 + 7a3 - 56a2 - 80a < 0
32. V l(T
33. \ fx 2
48. c 5 + 6c4 - 12c3 - 56c2 + 96c > 0
34.
< x/ P—~
25
</■3 6 - x 2
49. 3x5 + 13x4 - 137x3 - 353x2 + 330x + 144 < 0
50. PACKAGING A com pany sells cylindrical oil containers like
the one shown.
a. Use the volume of the container to express its surface
area as a function of its radius in centimeters. (Hint:
I liter = 1 0 0 0 cubic centimeters)
If k is nonnegative, find the interval for .v fo r w hich each
ineq u ality is true.
58.
x 2 + kx + c > c
59.(x + k)(x — k) < 0
60.
x 3 — kx2 — k2x + k 3 >
62.
ig i MULTIPLE REPRESENTATIONS In this problem , you will
investigate absolute value nonlinear inequalities,
61. x 4 —8k2x 2 + 16k4 >
0
0
a. TABULAR Copy and com plete the table below.
Function
b. The company wants the surface area of the container
to be less than 2400 square centimeters. Write an
inequality that could be used to find the possible radii
to meet this requirement.
Zeros
Undefined
Points
W - I ^ 2 l
p w
C. Use a graphing calculator to solve the inequality you
wrote in part b and interpret the solution.
-
I2; _ - 35'
51.
(x
+ 3 ) 2( x — 4) 3 (2x + l ) 2 < 0
52.
(y
- 5)2(y + l)(4y - 3 ) 4 > 0
53. (a - 3)3(a + 2)3(a -
> 0
6 )2
54. c2(c + 6)3 (3c - 4 )5(c - 3) < 0
b. GRAPHICAL Graph each function in part a.
C.
SYMBOLIC Create a sign chart for each inequality.
Include zeros and undefined points and evaluate the
sign of the num erators and denom inators separately.
= x -l
\x
55. STUDY TIME Jarrick determ ines that with the information
that he currently knows, he can achieve a score of a 75%
on his test. Jarrick believes that for every 5 com plete
minutes he spends studying, he will raise his score by 1 %.
a. If Jarrick wants to obtain a score of at least 89.5%, write
an inequality that could be used to find the time t that
he will have to spend studying.
b. Solve the inequality that you wrote in part a and
interpret the solution.
56. GAMES A skee ball machine pays out 3 tickets each time
a person plays and then 2 additional tickets for
every 80 points the player scores.
a. Write a nonlinear function to model the amount of
tickets received for an x-point score.
b. Write an inequality that could be used to find the score
a player would need in order to receive at least
I I tickets.
C. Solve the inequality in part b and interpret your
solution.
57. The area of a region bounded by a parabola and a
2
horizontal line is A = —bh, where b represents the base of
the region along the horizontal line and h represents the
height of the region. Find the area bounded b y / an d g.
+
2|
<0
\2x - 5[
>0
x —3
III.
\x + 4|
>0
|3x — 1|
d. NUMERICAL W rite the solution for each inequality in
part c.
H.O.T. Problems
63. ERROR ANALYSIS Aiay and Mae are solving —
. > 0.
' J
(3 - x) 2
Ajay thinks that the solution is (—oo, 0] or [0, oo), and Mae
thinks that the solution is ( —oo, oo). Is either of them
correct? Explain your reasoning.
64. REASONING If the solution set of a polynom ial inequality
is (—3, 3), what will be the solution set if the inequality
symbol is reversed? Explain your reasoning.
65. CHALLENGE Determ ine the values for which (a + b)2 >
(c + d)2 if a < b < c < d.
66. REASONING If 0 < c < d, find the interval on which
(x — c)(x — d) < 0 is true. Explain your reasoning.
(67) CHALLENGE W hat is the solution set of (x — a)2n > 0 if n is
a natural number?
68. REASONING W hat happens to the solution set of
(x + a)(x — b) < 0 if the expression is changed to
—(x + a)(x — b) < 0, where a and b > 0? Explain your
reasoning.
69. WRITING IN MATH Explain w hy you cannot solve ^
by multiplying each side by x — 2 .
146
| Lesson 2-6
Nonlinear Inequalities
< 6
Spiral Review
if any. (Lesson 2-5)
70. f(x ) =
73.
2x
x + 4
71. h(x)
72. f(x ) =
x + 6
x -1
(2x + l ) ( x - 5 )
GEOMETRY A cone is inscribed in a sphere with a radius of 15 centimeters. If the volum e of
the cone is 115277 cubic centim eters, find the length represented by x. (Lesson 2-4)
Divide using long division. (Lesson 2-3)
74.
(x2 - lOx - 24) + (x + 2)
75. (3a 4 -
76.
(z5 - 3z 2 - 20) -r (z - 2)
77. (x 3 + y 3) + (x + y)
78.
6
a3 - l a 2 + a -
6
) -f- (a + 1)
FINANCE The closing prices in dollars for a share of stock during a one-m onth period
are shown. (Lesson 2-2)
a. Graph the data.
b. Use a graphing calculator to model the data using a polynom ial function w ith a
degree of 3.
C.
79.
Use the model to estimate the closing price of the stock on day 25.
Day
Price(s)
Day
Price(s)
1
30.15
15
15.64
5
27.91
20
10.38
7
26.10
21
9.56
10
22.37
28
9.95
12
19.61
30
12.25
HOME SECURITY A com pany offers a hom e-security system that uses the num bers 0
through 9, inclusive, for a 5-digit security code. (Lesson 0-7)
a. How many different security codes are possible?
b. If no digits can be repeated, how many security codes are available?
C.
Suppose the hom eowner does not w ant to use 0 or 9 as the first digit and w ants the last
digit to be 1. H ow many codes can be formed if the digits can be repeated? If no repetitions
are allowed, how m any codes are available?
80.
SAT/ACT Two circles, A and B, lie in the same plane. If
the center of circle B lies on circle A, then in how
many points could circle A and circle B intersect?
I.
0
II.1
A I only
C I and III only
B III only
D II and III only
81 . A rectangle is
III.2
E I, II, and III
6 centim eters longer than it is wide.
Find the possible w idths if the area of the rectangle is
more than 216 square centimeters.
F a; > 1 2
H w > 18
G w < 12
J w < 18
82. FREE RESPONSE The amount of drinking water reserves in m illions of gallons available for a town
is modeled by/(f) = 80 + lOf — 4 12. The m inim um am ount of water needed by the residents is
modeled by g(t) = ( 2 f) 3, where f is the time in years.
a. Identify the types of functions represented by / ( f ) and g(t).
b.
W hat is the relevant dom ain and range fo r/ ( f ) and g(t)7 Explain.
C.
W hat is the end behavior o f/ ( f ) and g(t)7
d. Sketch/ ( f ) and g(t) for 0 < f < 6 on the same graph.
e. Explain why there must be a value c for [0, 6 ] such that/(c) = 50.
f. For what value in the relevant domain does/have a zero? W hat is the significance of the zero in this situation?
g. If this were a true situation and these projections were accurate, when would the residents be
expected to need more water than they have in reserves?
147
3*18
Study Guide and Review
Chapter Summary
KeyConcepts
KeyVocabulary
Power and Radical Functions (Lesson 2 - 1)
complex conjugates (p. 124)
power function (p. 86)
•
A power function is any function of the form f(x) = ax", where
a and n are nonzero real numbers.
extraneous solution 'p. 91)
quartic function (p. 99)
horizontal asymptote (p. 131)
rational function (p. 130)
•
A monomial function is any function that can be written as f(x) = a or
f(x) = ax", where a and n a re nonzero constant real numbers.
irreducible over the reals (p. 124)
repeated zero (p. 101)
•
A radical function is a function that can be written as f(x) = \ f r f ,
where n and p are positive integers greater than 1 that have no
common factors.
leading coefficient (p. 97)
sign chart (p. 141)
leading-term test
synthetic division (p. 111 )
Polynom ial Functions (Lesson 2 - 2)
•
A polynomial function is any function of the form f(x) = anx n + a „ _ 1
x n_1 + . . . + a ^ x + a0, where a „ =/= O.The degree is n.
p.
98)
lower bound (p. 121)
synthetic substitution (p. 1 13)
multiplicity (p. 102)
turning point (p. 99)
oblique asymptote (p. 134)
upper bound p. 121)
polynomial function (p. 97)
vertical asymptote (p. 1 31)
• The graph of a polynomial function has at most n distinct real zeros
and at most n - 1 turning points.
•
The behavior of a polynomial graph at its zero c depends on the
multiplicity of the factor (x - c).
VocabularyCheck
Identify the word or phrase that best completes each sentence.
The R em ainder and Factor Theorem s (Lesson 2-3)
•
Synthetic division is a shortcut for dividing a polynomial by a linear
factor of the form x - c.
• If a polynomial f is divided by x - c, the remainder is equal to f(c).
• x - c is a factor of a polynomial f if and only if f(c) = 0.
Zeros of Polynomial Functions (Lesson 2 -4 )
•
If f(x) = anx n + . . . + a^x + a0 w ith integer coefficients, then any
rational zero of f(x) is of the form
where p and q have no common
factors, p is a factor of a0, and q is a factor of an.
1. The coefficient of the term with the greatest exponent of the variable
is the (leading coefficient, degree) of the polynomial.
2. A (polynomial function, power function) is a function of the form
f(x) = anx n + an_ 1x n_1 + ... + a:x + a0, where av a2
an
are real numbers and n is a natural number.
3. A function that has multiple factors of (x - c) has (repeated zeros,
turning points).
4. (Polynomial division, Synthetic division) is a short way to divide
polynomials by linear factors.
• A polynomial of degree n has n zeros, including repeated zeros, in the
complex system. It also has n factors:
f(x) = an( x - c 1) ( x - c 2) . . .
(x — cn).
Rational Functions (Lesson 2 - 5)
•
•
The graph of f has a vertical asymptote x = c if
lim f( x ) = ± 0 0 or lim f(x) = + 00 .
x—>c
x—>c+
The graph of f has a horizontal asymptote y = c if
lim ft x ) = c o r lim f(x) = c.
X—►
O
O
•
X— O
O
A rational function f(x) -
six)
may have vertical asymptotes,
horizontal asymptotes, or oblique asymptotes, x-intercepts, and
/-intercepts. They can all be determined algebraically.
•
(Lesson 2- 6)
The sign chart for a rational inequality must include zeros and
undefined points.
148
C h a p te r 2
5. The (Remainder Theorem, Factor Theorem) relates the linear factors
of a polynomial with the zeros of its related function.
6. Some of the possible zeros for a polynomial function can be listed
using the (Factor, Rational Zeros) Theorem.
7. (Vertical, Horizontal) asymptotes are determined by the zeros of the
denominator of a rational function.
8. The zeros of the (denominator, numerator) determine the x-intercepts
of the graph of a rational function.
9. (Horizontal, Oblique) asymptotes occur when a rational function has a
denominator with a degree greater than 0 and a numerator with
degree one greater than its denominator.
10. A (quartic function, power function) is a function of the form f(x) = axn,
where a and n are nonzero constant real numbers.
Lesson-by-Lesson Review
Power and Radical Functions (pp. 86-95)
Graph and analyze each function. Describe the domain, range,
intercepts, end behavior, continuity, and where the function is increasing
or decreasing.
11. f(x) = 5 x 6
Example 1
Graph and analyze f(x) - - 4 x ~ 5. Describe the domain, range,
. i ■.
12. f(x) = - 8 x 3
13. f(x):
( u ) f(x) = j x \15?) f(x) = V 5 x - 6
11
16. f(x) = - W < 6 x 2 - 1 + 2
4
17. 2x = 4 + \ / 7 x — 12
2
0
undefined
1
-4
2
-0 .1 2 5
3
-0 .0 1 6
f
Range: (-o o , 0) u (0, oo)
lim
X—>—oc
fix) = 0 and lim fix) = 0
X—>00
Increasing: (0 , 00)
(pp. 9 7-107)
21. f{x) = - 4 x 4 + 7 x 3 - 8 x 2 + 1 2 x - 6
22.) f(x) = - 3 x 5 + 7 x 4 + 3 x 3 - 11x - 5
Example 2
Describe the end behavior of the graph of
f(x) = - 2 x 5 + 3 x 3 - 8 x 2 - 6 using limits. Explain your
reasoning using the leading term test.
The degree is 5 and the leading coefficient is - 2 . Because the
degree is odd and the leading coefficient is negative,
lim f(x) = oo and lim f(x) = -o o .
23. f(x) = | x 2 - 8 x - 3
24. f(x) = x 3( x - 5 ) ( x + 7 )
X—>—oo
function. Then determine all of the real zeros by factoring.
f(x) = x 4 — 1 0 x 2 - f 9
4
Increasing: ( - 00, 0)
using limits. Explain your reasoning using the leading term test.
27.
-1
f(x) = x 3 — 7 x 2 + 12x
0.125
End behavior:
+ 31 - 1 = 3
25.
-2
Intercepts: none
19. 4 = V 6 x + 1 — V 17 — 4x
. V*
0.016
Domain: ( - 00, 0) u ( 0 , 00)
! 1Ei. V 4 x + 5 + 1 = 4 x
2 0
-3
26. f(x) = x 5 + 8 x 4 - 2 0 x 3
28.
f(x) = x 4 — 25
For each function, (a) apply the leading term test, (b) find the zeros and
state the multiplicity of any repeated zeros, (c) find a few additional
points, and then (d) graph the function.
29. f(x) = x 3(x - 3)(x + 4)2 T 30J f(x) = (x - 5)*(x - 1)
X—>oo
Example 3
State the number of possible real zeros and turning points for
f(x) = x 3 + 6 x 2 + 9x. Then find all the real zeros by factoring.
The degree of f is 3, so f has at most 3 distinct real zeros and at most
3 - 1 or 2 turning points. To find the real zeros, solve the related
equation f(x) = 0 by factoring.
x 3 + 6 x 2 + 9 x = x (x 2 + 6 x + 9)
= x (x + 3)(x + 3) or x (x + 3)2
The expression has 3 factors but only 2 distinct real zeros, 0 and - 3 .
C r
149
150
C h a p te r 2 : Study G uide and Review
Rational Functions
(pp.
130 - 140)
Find the domain of each function and the equations of the vertical or
horizontal asymptotes, if any.
50. f(x) =
52. f(x) =
x '-1
51. f(x) =
x+ 4
x (x -3 )
53. f(x) =
(x — 5)2(x + 3)2
v2
x 2 — 25
Example 6
Find the domain of f(x) ■
asymptotes.
EflSflTl
(x — 5)(x — 2)
x+l
and any vertical or horizontal
x+1
Find the domain.
The fun ction is undefined a t the zero of the denom inator
(x+3)(x+9)
h(x) = x + 1, w hich is - 1 . The dom ain of 1 is all real
num bers except x = - 1 .
For each function, determine any asymptotes and intercepts. Then
graph the function, and state its domain.
54. f(x) =
56. f(x) =
X
(x + 3)(x —4)
x+4
The zero of the denom inator is - 1 , so there is a vertical
x (x + 7 )
57.
(x + 5)(x - 6)
Find the asymptotes, if any.
Check fo r vertical asym ptotes.
x -2
55.
x - 5
E T T fln
asym ptote a t x = - 1 .
(x + 6 )(x -3 )
Check fo r horizontal asym ptotes.
58. f(x) =
x+2
..2—
x
x 2 - 16
59.
H
1
x 3 - 6 x 2 + 5x
The degree of the num erator is equal to the degree of the
denom inator. The ratio of the leading coe fficient is j
60.
12 . + * — 8
61.
2
x+2
62.
63.
f
= 1
3 = _
x
x
x+2
1
2
d+ 4
d2 + 3 d - 4
1
=
/7 — 2
= 1.
Therefore, y = 1 is a horizontal asym ptote.
1
2 /7 + 1
n 2 + 2n — 8
1- d
2
n+ 4
(pp. 141 -
1 47)
Example 7
64.
(x + 5)(x — 3) < 0
65.
x 2 — 6 x — 16 > 0
Solve x 3 + 5x2 - 36x < 0.
66.
x 3 + 5x2 < 0
67.
2 x 2 + 1 3 x + 15 < 0
Factoring the polynom ial f(x) = x 3 + 5 x 2 -
3 6 x yields
f{x) = x ( x + 9)(x - 4), so f(x ) has real zeros a t 0, - 9 , and 4.
68.
x 2 + 12 x + 36 < 0
69. x 2 + 4 < 0
70.
x 2 + 4x + 4 > 0
71.
Create a sign cha rt using these zeros. Then substitute an x-value
72.
x+ 1
(12x + 6)(3x + 4)
x - 5
<0
from each te s t interval into the fun ction to determ ine w hether f(x) is
positive or negative at th a t point.
> 0
73.
x —3
x —4
>0
(-)
(+)
(-)
- — I----------------- 1
(
-9
(+)
--
0
Because f (x) is negative on the firs t and third intervals, the solution of
x 3 + 5x2 -
3 6 x < 0 is ( - o o , - 9 ] U [0 ,4 ],
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&
151
Continued
Applications and Problem Solving
74. PHYSICS Kepler’s Third Law of Planetary Motion implies that the
time f it takes for a planet to complete one revolution in its orbit
about the Sun is given by T = R 2, where R is the planet’s mean
distance from the Sun. Time is measured in Earth years, and
distance is measured in astronomical units. (Lesson 2-1)
a. State the relevant domain and range of the function.
78. BUSINESS A used bookstore sells an average of 1000 books each
month at an average price of $10 per book. Due to rising costs the
owner wants to raise the price of all books. She figures she will sell
50 fewer books for every $1 she raises the prices. (Lesson 2-4)
a. Write a function for her total sales after raising the price of her
books x dollars.
b. How many dollars does she need to raise the price of her books
c. The time for Mars to orbit the Sun is observed to be 1 .88 Earth
years. Determine Mars’ average distance from the Sun in miles,
given that one astronomical unit equals 93 million miles.
so that the total amount of sales is $11,250?
c. What is the maximum amount that she can increase prices and
still achieve $10,000 in total sales? Explain.
79. AGRICULTURE A farmer wants to make a rectangular enclosure
75. PUMPKIN LAUNCH Mr. Roberts’ technology class constructed a
catapult to compete in the county’s annual pumpkin launch. The
speed v \n miles per hour of a launched pumpkin after t seconds is
given. (Lesson 2-1)
0.5
1.0
1.5
2.0
2.5
3.0
85
50
30
20
15
12
using one side of her barn and 80 meters of fence material. Determine
the dimensions of the enclosure. Assume that the width of the
enclosure w will not be greater than the side of the barn. (Lesson 2-4)
b. Determine a power function to model the data.
c. Use the function to predict the speed at which a pumpkin is
traveling after 1.2 seconds.
d. Use the function to predict the time at which the pumpkin’s
speed is 47 miles per hour.
80. ENVIRONMENT A pond is known to contain 0.40% acid. The pond
contains 50,000 gallons of water. (Lesson 2-5)
76. AMUSEMENT PARKS The elevation above the ground for a rider on
the Big Monster roller coaster is given in the table.
Time (seconds)
Elevation (feet)
I 5
| 85
(Lesson 2-2)
10
15
20
25
62
22
4
17
a. Create a scatter plot of the data and determine the type of
polynomial function that could be used to represent the data.
a. How many gallons of acid are in the pond?
b. Suppose x gallons of pure water was added to the pond. Write
p(x), the percentage of acid in the pond after / gallons of pure
water are added.
c. Find the horizontal asymptote of p(x).
d. Does the function have any vertical asymptotes?
Explain.
b. Write a polynomial function to model the data set. Round each
coefficient to the nearest thousandth and state the correlation
coefficient.
c. Use the model to estimate a rider’s elevation at 17 seconds.
d. Use the model to determine approximately the first time a rider
81. BUSINESS For selling x cakes, a baker will make b{x) = x 2 5 x - 150 hundreds of dollars in revenue. Determine the minimum
number of cakes the baker needs to sell in order to make a
profit. (Lesson 2-6)
is 50 feet above the ground.
77. GARDENING Mark’s parents seeded their new lawn in 2001. From
2001 until 2011, the amount of crab grass increased following the
model f(x) = 0.021 x 3 - 0.336*2 + 1,945x - 0.720, where x is
the number of years since 2001 and f(x) is the number of square
feet each year. Use synthetic division to find the number of square
feet of crab grass in the lawn in 2011. Round to the nearest
thousandth. (Lesson 2-3)
152
| C h a p te r 2 | Study G uide and Review
82. DANCE The junior class would like to organize a school dance as a
fundraiser. A hall that the class wants to rent costs $3000 plus an
additional charge of $5 per person. (Lesson 2-6)
a. Write and solve an inequality to determine how many people
need to attend the dance if the junior class would like to keep
the cost per person under $ 1 0 .
b. The hall will provide a DJ for an extra $1000. How many people
would have to attend the dance to keep the cost under $ 1 0 per
person?
&&ziskJ
Practice Test
19. WEATHER The table shows the average high temperature in Bay
Town each month.
4
1. f(x) = 0.25x-3
2. f(x) = 8x3
3.
x= V4- x- 8
4.
V 5 x + 4 = V 9 - x -I- 7
5.
-2 + V 3x+ 2 = x
6.
56 - ^ 7 x 2 + 4 = 54
7.
x 4 - 5x3 — 14x2 = 0
8.
x 3 — 3x2 — 10x = —24
using limits. Explain your reasoning using the leading term test.
9. f(x) = 5x4 - 3x3 — 7x2 + 1 1 x - 8
Jan
Feb
M ar
Apr
May
Jun
62.3°
66.5°
73.3°
79.1°
85.5°
90.7°
Nov
Dec
72.0°
64.6°
Jul
Aug
Sep
Oct
93.6°
93.5°
89.3”
82.0°
a. Make a scatter plot for the data.
b.
Use a graphing calculator to model the data using a polynomial
function with a degree of 3. Use x = 1 for January and round
each coefficient to the nearest thousandth.
c. Use the model to predict the average high temperature for the
following January. L e tx = 13.
Write a polynomial function of least degree with real coefficients in
standard form that has the given zeros.
21. 5 , - 5 , 1 - i
20. - 1 , 4 , - V 3
10. f(x) = —3x5 - 8x4 + 7x2 + 5
function. Then find all of the real zeros by factoring.
22. MULTIPLE CHOICE Which function graphed below must have
imaginary zeros?
I
l
H
y
11. f(x) = 4x3 + 8x2 - 60x
y
12. f(x) = x 5 — 16x
—! —
A f(x) = x 4 - 4
B
f(x) = x 4 — 11 x 3
C f(x) =
Df(x)
=
8x
\
\
13. MULTIPLE CHOICE Which function has 3 turning points?
x 3+
9x2+ 20x
x4-
5x2+ 4
0
-8 -4
i
ix
-8
T
16
14. BASEBALL The height h in feet of a baseball after being struck by a
batter is given by h(t) = - 3 2 12 + 1 2 8 f+ 4, where t is the time in
seconds after the ball is hit. Describe the end behavior of the graph
of the function using limits. Explain using the leading term test.
For each function, (a) apply the leading term test, (b) find the zeros and
state the multiplicity of any repeated zeros, (c) find a few additional
points, and then (d) graph the function.
15. f(x) = x (x — 1 )(x + 3)
Divide using synthetic division.
23. f(x) = (x3 - 7x2 + 13) -r- (x - 2)
24. f(x) = (x4 + x 3 - 2x2 + 3 x + 8) -f ( x + 3)
16. f(x) = x 4 - 9x2
Determine any asymptotes and intercepts. Then graph the function and
state its domain.
x2 + x - 6
X- 4
2x —6
x+ 5
Use the Factor Theorem to determine if the binomials given are factors of
f(x). Use the binomials that are factors to write a factored form of f(x).
25. f(x) -
17. f(x) = x 3 - 3x2 — 1 3 x + 15; ( x + 3)
18. f(x) = x 4 - x 3 - 34x2 + 4 x + 120; (x + 5), ( x - 2 )
27. x 2 — 5x — 14 < 0
26. f(x) = -
28.
x —6
> 0
connectED.m cgraw-hiii.com | |
E
153
Connect to AP Calculus
Area Under a Curve
:• Objective
• Approximate the area
between a curve and the
x-axis.
Integral calculus is a branch of calculus that focuses on the processes
of finding areas, volumes, and lengths. In geometry, you learned
how to calculate the perimeters, areas, and volumes of polygons,
polyhedrons, and composite figures by using your knowledge of
basic shapes, such as triangles, pyramids, and cones. The
perimeters, areas, and volumes of irregular shapes, objects that are
not a combination of basic shapes, can be found in a sim ilar manner.
Calculating the area between the curve and the x-axis, as shown to
the right, is an application of integral calculus.
Activity 1 Approximate Area Under a Curve
Approxim ate the area between the curve fix ) — V —x2 + 8x and the x-axis using rectangles.
Draw 4 rectangles with a w idth of 2 units betw een
f{ x ) and the x-axis. The height of the rectangle should
be determ ined w hen the left endpoint of the rectangle
intersects/(x), as show n in the figure. N otice that the
first rectangle will have a height o f/( 0 ) or 0 .
KTH!W
Calculate the area of each rectangle.
Approxim ate the area of the region by taking the sum
of the areas of the rectangles.
V A nalyze the Results
1. W hat is the approxim ation for the area?
2. How does the area of a rectangle that lies outside the graph affect the approximation?
3. Calculate the actual area of the sem icircle. How does the approxim ation com pare to the
actual area?
4. How can rectangles be used to find a more accurate approximation? Explain your reasoning.
Using relatively large rectangles to estim ate the area under a
curve may not produce an approximation that is as accurate as 3
desired. Significant sections of area under the curve may go
unaccounted for. Similarly, if the rectangles extend beyond the
curve, substantial amounts of areas that lie above a curve may
be included in the approximation.
In addition, regions are also not always bound by a curve
intersecting the x-axis. You have studied many functions
w ith graphs that have different end behaviors. These graphs
do not necessarily have tw o x-intercepts that allow for
obvious start and finish points. In those cases, we often
estimate the area under the curve for an x-axis interval.
154
C h a p te r 2
Desired Area
Undesired
Area
Activity 2 Approximate Area Under a Curve
Approxim ate the area betw een the curve f i x ) = x 2 + 2 and the x-axis on the interval [1, 5]
using rectangles.
StudyTip
PflSIWI
Endpoints Any point within a
subinterval may be used to
determine the height of the
rectangles used to approximate
Draw 4 rectangles w ith a w idth of 1 unit betw een f i x ) and
the x-axis on the interval [1, 5], as show n in the figure. Use
the left endpoint of each sub interval to determ ine the height
of each rectangle.
^
1
6
Calculate the area of each rectangle.
area. The most commonly used
are left endpoints, right endpoints,
Approxim ate the area of the region by determ ining the sum
of the areas of the rectangles.
and midpoints.
P ftS T !
w
Repeat Steps 1 -3 using 8 rectangles, each w ith a w idth of 0.5 unit,
and 16 rectangles, each with a w idth of 0.25 unit.
Analyze the Results
5. W hat value for total area are the approxim ations approaching?
6
. Using left endpoints, all of the rectangles com pletely lie under the curve. How does this affect
the approxim ation for the area of the region?
7. Would the approxim ations differ if each rectangle's height w as determ ined by its right
endpoint? Is this always true? Explain your reasoning.
8
. W hat would happen to the approxim ations if w e continued to increase the num ber of
rectangles being used? Explain your reasoning.
9. M ake a conjecture about the relationship betw een the area under a curve and the number of
rectangles used to find the approximation. Explain your answer.
Model and Apply
10.
In this problem , you will approxim ate the area
betw een the curve/(x) = —x 2 + 1 2 x and the x-axis.
a.
Approxim ate the area by using 6 rectangles,
12 rectangles, and 24 rectangles. Determ ine the
height of each rectangle using the left endpoints.
b. W hat value for total area are the approximations
approaching?
C. Does using right endpoints opposed to left endpoints
for the rectangles' heights produce a different
approximation? Explain your reasoning.
11.
In this problem , you will approxim ate the area betw een
the curve/(x) = -^x3 — 3x 2 + 3x +
interval [1, 5],
a.
6
and the x-axis on the
Approxim ate the area by first using 4 rectangles and then using
8 rectangles. Determ ine the height of each rectangle using left
endpoints.
b. Does estim ating the area by using 4 or 8 rectangles give
sufficient approxim ations? Explain your reasoning.
C. Does using right endpoints opposed to left endpoints for the
rectangles' heights produce a different approximation?
M
.r o n n e c t^
155
C In Chapter 2, you
graphed and
analyzed power,
polynomial, and
rational functions.
C In Chapter 3, you will:
■ Evaluate, analyze, and graph
exponential and logarithmic
functions.
C ENDANGERED SPECIES Exponential functions are often used to
model the growth and decline of populations of endangered
species. For example, an exponential function can be used to model
the population of the Galapagos Green Turtle since it became an
endangered species.
■ Apply properties of logarithms.
* Solve exponential and
logarithmic equations.
PREREAD Use the Concept Summary Boxes in the chapter to
predict the organization of Chapter 3,
■ Model data using exponential,
logarithmic, and logistic
functions.
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algebraic functions
transcendental functions
QuickCheck
Simplify.
1.
natural base
(Lesson 0-4)
(3x2)4 • 2x3
, c4cf
5- I d
7.
exponential functions
2.
(3b3)(2b*)
4.
^
6.
2a/
8.
f(x) = V 4 - x 2
10.
f(x )= ^ f*-
12.
f(x )= -J =
V1 - x
1 1
p. 172 logaritmo
common logarithm
p. 173 logaritmo comun
natural logarithm
p. 174 logaritmo natural
logistic
growth function
linearize
funcion de
p. 202 crecimiento logistica
p. 204 linearize
13. g(x) =
Vx+7
4x3
one-to-one p. 66 d e u n o a u n o a function that passes the horizontal
line test, and no y-value is matched with more than one x-value
inverse functions p. 65 funciones inversas Two functions, f and
M , are inverse functions if and only if f [f _1(x)] = xand M [ f ( x ) ] = x.
Domain of f
C
N
Range of f
<
16. g(x) =
ReviewVocabulary
J
\
17. h(x) = —3x3
funcion logaritmica
p. 172 con base b
. g(x) = ± z ±
f(x) = 2x2
p. 160 base natural
9. f(x) = \ [ x + 2
14. STAMPS The function v(t) = 200(1,6)f can be used to predict the
value /o f a rare stamp after fyears. Graph the function, and
determine whether the inverse of the function is a function.
15.
p. 158 funciones exponenciales
logarithmic function
with base b
logarithm
Use a graphing calculator to graph each function. Determine whether
the inverse of the function is a function. (Lesson 1-7)
p. 158 funciones transcendentales
interes compuesto
p. 163 continuo
4/7
feet and the width by 5a3 feet. Determine the area of the carpet.
p. 158 funciones algebraicas
continuous
compound interest
(2 n Y
CARPET The length of a bedroom carpet can be represented by 2a2
Espanol
f
N
X
fix)
Range of f~
Domain of f_1
18. f(x) = - x 5
x — — y
Online Option Take an online self-check Chapter
Readiness Quiz at connectED.mcaraw-hill.com.
end behavior p. 159 comportamiento de final describes the
behavior of f(x) as x increases or decreases without bound— becoming
greater and greater or more and more negative
continuous function p. 159 funcion continua a function with a
graph that has no breaks, holes or gaps
157
•
You identified,
graphed, and
described several
parent functions.
•
(Lesson 1-5)
■# Evaluate, analyze,
•
1 and graph
exponential functions.
Worldwide water consumption has increased rapidly
over the last several decades. Most of the world’s
water is used for agriculture, and increasing
population has resulted in an increasing agricultural
demand. The increase in water consumption can be
modeled using an exponential function.
Solve problems
2 involving exponential
growth and decay.
NewVocabulary
algebraic function
transcendental function
exponential function
natural base
continuous compound
interest
Exponential Functions In Chapter 2, you studied power, radical, polynom ial, and
rational functions. These are exam ples of algebraic functions— functions with values that are
obtained by adding, subtracting, multiplying, or dividing constants and the independent variable
or raising the independent variable to a rational power. In this chapter, we will explore
exponential and logarithmic functions. These are considered to be transcendental functions
because they cannot be expressed in terms of algebraic operations. In effect, they transcend algebra.
1
Consider functions/(x) = x 3 and g(x) = 3 X. Both involve a base raised to a power; however, in/(x), a
power function, the base is a variable and the exponent is a constant. In g (x ), the base is a constant
and the exponent is a variable. Functions of a form similar to g(x) are called exponential functions.
KeyConcept Exponential Function
An exponential function with base b has the form f(x) = ab\ where x is any real number and a and b are real number
constants such that a j= 0, b is positive, and b ± 1.
Examples
f(x ) = 4* f(x) =
Nonexamples
f(x) = 7~x
f(x) = 2x~3, f(x) = 511, f(x) = 1 *
W hen the inputs are rational num bers, exponential functions can be evaluated using the properties
of exponents. For example, if/(x) = 4 X, then
/(f)
/(2) = 4 2
= 4^/
= 16
/ (-3 )
=4
- 3
43
_1_
64
Since exponential functions are defined for all real num bers, you must also be able to evaluate an
exponential function for irrational values of x, such as \ fl. But what does the expression 4 ^ mean?
The value of this expression can be approxim ated using successively closer rational approximations
of \ fl as shown below.
X
1
1.4
1.41
1.414
1.4142
1.41421
f(x) = 4 *
4
7.0
7.06
7.101
7.1029
7.10296
From this table, we can conclude that 4 ^ is a real number
approximately equal to 7.10. Since/(x) = 4 Xhas real num ber values
for every x-value in its domain, this function is continuous and can
be graphed as a sm ooth curve as shown.
158
Lesson 3-1
Example 1 Sketch and Analyze Graphs of Exponential Functions
Sketch and analyze the graph of each function. Describe its domain, range, intercepts,
asymptotes, end behavior, and where the function is increasing or decreasing.
a.
fix ) = 3*
Evaluate the function for several x-values in its domain. Then use a smooth curve to connect
each of these ordered pairs.
-2
-1
0
2
4
6
16
0.11
0.33
1
9
81
729
12
- 4
0.01
O
Domain: ( —00 , 00 )
Range ( 0 ,00 )
i/-Intercept: 1
Asym ptote: x-axis
Tfi,
End behavior: lim fix) = 0 and lim f(x ) = 00
'
x—
»oo
X—
*—oo
y
—L
o\
4
n\
8x
Increasing: ( — 00 , 00 )
StudyTip
b.
£(X) = 2 ~ x
Negative Exponents Notice that
y
f(x) = (-^)*and g(x) = b~x are
-6
equivalent because - L = ( 6 ~ V
64
b
-4
-2
0
2
4
6
16
4
1
0.25
0.06
0.02
2 \
8
or b~x.
Domain: (—0 0 ,
0 0
)
Range (0,
y-Intercept: 1
0 0
)
Asym ptote: x-axis
-4
End behavior: lim g ix ) =
X—>—OO°
Decreasing: (—0 0 ,
0 0
00
v _^
3
?(x) = 2 - x ... ....
r
4
and lim ?(x) = 0
x— >00 °
TTT
I
)
f GuidedPractice
1
A. f{ x ) = 6~x
1B. g ix ) = 5 X
1
C
•
= ( l) ‘
+
1
The increasing and decreasing graphs in Exam ple 1 are typical of the two basic types of exponential
functions: exponential growth and exponential decay.
K eyC on cept Properties of Exponential Functions
Exponential Growth
D o m a in : ( -
StudyTip
Coefficient a An exponential
00, 00)
R a n g e : (0,
Exponential Decay
00)
D o m a in : ( -
00, 00)
R a n g e : (0 , 00)
y - ln t e r c e p t: 1
x - ln t e r c e p t: none
y - ln t e r c e p t: 1
x - ln t e r c e p t: none
E x tre m a : none
A s y m p to te : x-axis
E x tre m a : none
A s y m p to te : x-axis
function of the form f(x) = a b x
has a y-intercept at (0, a).
E n d B e h a v io r:
lim f(x) = 0
X—
>—00
E n d B e h a v io r:
00 , 00)
00
X— >00 fix) =
and lim fix) = oo
X—
>00
C o n tin u ity : continuous on ( -
lim f(x) =
X—
>—
CO
and lim
C o n tin u ity : continuous on ( -
0
00 , 00)
connectED.m cgraw -hill.com Tj
8
159
The same techniques that you used to transform graphs of algebraic functions can be applied to
graphs of exponential functions.
Graph Transformations of Exponential Functions
Use the graph of f(x ) = 2Xto describe the transformation that results in each function.
Then sketch the graphs of the functions.
a.
g(x) = 2 V + 1
This function is of the form g(x) = f ( x + 1). Therefore, the graph of g(x) is the graph of
/(x) = 2 Xtranslated 1 unit to the left (Figure 3.1.1).
b.
h(x) = 2~x
This function is of the form h{x) = f ( —x). Therefore, the graph of h(x) is the graph of/(x) = 2X
reflected in the y-axis (Figure 3.1.2).
C. ; ( * ) = - 3 ( 2 * )
This function is of the form/(x) = —3f(x ). Therefore, the graph o fj( x ) is the graph of /(x) = 2X
reflected across the x-axis and expanded vertically by a factor of 3 (Figure 3.1.3).
StudyTip
Analyzing Graphs Notice that the
transformations of f(x) given by
g(x), h(x), and j(x) do not affect
the location of the horizontal
asymptote, the x-axis. However,
the transformations given by h(x)
and g(x) do affect the y-intercept
of the graph.
Figure 3.1.1
Figure 3.1.2
Figure 3.1.3
► GuidedPractice
Use the graph o f /(x ) = 4 Vto describe the transform ation that results in each function.
2A. k(x) = 4 X - 2
2B. m(x) = - 4* + 2
2C. p(x) = 2(4~x)
It may surprise you to learn that for most real-world applications involving exponential functions, the
most commonly used base is not 2 or 1 0 but an irrational number e called the natural base, where
e = lim ( l + i ) .
* —>00 \
-W
By calculating the value of ( l +
for greater and greater values
of x, we can estimate that the value of this expression approaches
a number close to 2.7183. In fact, using calculus, it can be shown
that this value approaches the irrational number we call e, named
after the Swiss mathem atician Leonhard Euler who com puted e
to 23 decimal places.
e = 2.718281828...
i
The number e can also be defined as lim (l + x )*, since for fractional
x—* 0
1
values of x closer and closer to 0, (1 + x)* = 2.718281828... or e.
Figure 3.1.4
160
1
2
10
2.59374...
100
2.70481...
1000
2.71692...
10,000
2.71814...
100,000
2.71827...
1,000,000
2.71828...
The function given by/(x) = ex, is called the natural base exponential function (Figure 3.1.4) and has
the same properties as those of other exponential functions.
| Lesson 3-1 | Exponential Functions
ReadingM ath
Base e Expressions with base e
are read similarly to exponential
expressions with any other base.
For example, the expression e 4* is
M ^ j J ^ ^ j G r a p h Natural Base Exponential Functions
>
Use the graph of f(x ) — ex to describe the transform ation that results in the graph of each
function. Then sketch the graphs of the functions.
a. a(x) = e4*
This function is of the form a(x) = / (4x). Therefore, the graph of a(x) is the graph of
f( x ) = e x com pressed horizontally by a factor of 4 (Figure 3.1.5).
read as e to the four x.
b. b(x) = e~x +
3
This function is of the form b(x) = f ( —x) + 3. Therefore, the graph of b(x) is the graph of
f ( x ) = e x reflected in the j/-axis and translated 3 units up (Figure 3.1.6).
c(x) = j e x
C.
This function is of the form c(x) = 4-/(x). Therefore, the graph of c(x) is the graph of
1
f( x ) = e x com pressed vertically by a factor of — (Figure 3.1.7).
p GuidedPractice
3B. r(x) = e x - 5
3A. q (x) = e~x
Exponential Growth and Decay
3C. t(x) = 3ex
A common application of exponential growth is
mm compound interest. Suppose an initial principal P is invested into an account with an annual
interest rate r, and the interest is compounded or reinvested annually. At the end of each year, the
interest earned is added to the account balance. This sum becomes the new principal for the next year.
Year
3
CL,
2
o
1
II
0
Account Balance After Each Com pounding
P = original investment or principal
A l = A 0 + A 0r
= A 0( l + r)
= P (1
A0 = P
+
r)
A 2 = A j ( 1 + r)
Interest from year 1 is added.
= P ( 1 + r ) ( l + r)
A^ = P(1 + r)
= P( 1 + r) 2
Simplify.
A 3 = A2(l + r)
=
P ( 1 + r)2( l + r)
= P(1 + r)3
4
Interest from year 0, A0r, is added.
=
A3( l + r)
A2 = P( 1 + r)2
Simplify.
= P ( 1 + r ) 3( 1 + r)
/13 = P(1 + r)3
=
Simplify.
P(1 + r)3
The pattern that develops leads to the follow ing exponential function w ith base (1 + r).
A(t) = P(1 + r)‘
Account balance after /years
To allow for quarterly, monthly, or even daily com poundings, let n be the num ber of tim es the
interest is com pounded each year. Then
• the rate per com pounding
is a fraction of the annual rate r, and
• the num ber of com poundings after t years is nt.
Replacing r with — and t w ith nt in the formula A(t) = P( 1 + r)f, we obtain a general form ula for
com pound interest.
K eyC oncept Compound Interest Formula
If a principal Pis invested at an annual interest rate r(in decimal form) compounded n times a year, then the balance A in the
account after t years is given by
m r n m m Use Compound Interest
FINANCIAL LITERACY Krysti invests $300 in an account with a 6% interest rate, m aking no other
deposits or withdrawals. W hat will Krysti's account balance be after 20 years if the interest is
compounded:
a. semiannually?
For sem iannually com pounding, n = 2.
A = p (l +
Compound Interest Formula
= 300^1 + M & j2<20)
P - 300, r = 0.06, n = 2, and t - 20
~ 978.61
Simplify.
W ith sem iannual com pounding, K rysti's account balance after 20 years will be $978.61.
b. monthly?
For m onthly com pounding, n = 12, since there are 12 m onths in a year.
A =
1 + j^ jnt
= 3 0 0 (l + ^ jy -j12<20)
P = 300, r = 0 .0 6 ,/? = 1 2 , and? = 2 0
a; 9 9 3 .0 6
Simplify.
With m onthly com pounding, K rysti's account balance after 20 years will be $993.06.
StudyTip
c. daily?
For daily com pounding, n = 365.
Daily Compounding In this text,
for problems involving interest
A = p (l +
compounded daily, we will
assume a 365-day year.
/
n n f i \ 365(20)
= 300(1 + ^ j
P = 300, r = 0,06, f = 20, and n = 365
« 9 9 5 .9 4
Simplify.
W ith daily com pounding, K rysti's account balance after 20 years will be $995.94.
k GuidedPractice
4.
FINANCIAL LITERACY If $1000 is invested in an online savings account earning 8 % per year,
how much will be in the account at the end of 1 0 years if there are no other deposits or
withdraw als and interest is com pounded:
A.
sem iannually?
\
B. quarterly?
C. daily?
...................
Notice that as the number of com poundings increases in Exam ple 4, the account balance also
increases. However, the increase is relatively small, only $995.94 — $993.06 or $2.88.
162
| Lesson 3-1 j Exponential Functions
The table below shows the am ount A com puted for several values of n. N otice that while the
account balance is increasing, the am ount of increase slows dow n as n increases. In fact, it appears
that the am ount tends towards a value close to $996.03.
Compounding
n
A = 300^1 + M ® . ) 20"
annually
1
$962.14
semiannually
2
$978.61
quarterly
4
$987.20
monthly
12
$993.06
daily
365
$995.94
hourly
8760
$996.03
Suppose the interest were com pounded continuously so that there w as no waiting period betw een
interest payments. We can derive a form ula for continuous com pound interest by first using
algebra to m anipulate the regular com pound interest formula.
(
1 +
1 \nt
Compound interest formula with
1 \ xrt
1
Let
n
x= j
and
r
written as
1
n = xr.
= P{1+j)
= p[(> + i ) :
Power Property of Exponents
X
The expression in brackets should look familiar. Recall from page 160 that lim ( l + j-5-1
) = e.
Since r is a fixed value and x = y , x — > o o as n —> o o . Thus,
&
p( > + ^ r - t e ' i ( i + i ) T = R ’'-
This leads us to the formula for calculating continuous com pounded interest show n below.
K eyC oncept Continuous Compound Interest Formula
If a principal P is invested at an annual interest rate r (in decimal form) compounded continuously, then the balance A in the
account after t years is given by
A = Pen.
r n m
m
Use Continuous Compound Interest
FINANCIAL LITERACY Suppose Krysti finds an account that will allow her to invest her $300 at
a 6% interest rate com pounded continuously. If there are no other deposits or withdrawals,
what will Krysti's account balance be after 20 years?
A = P ert
=
300e(°-°6)<20)
~ 996.04
Real-W orldLink
Continuous Compound Interest Formula
P = 300,
r=
0.06, and
t=
20
Simplify.
W ith continuous com pounding, K rysti's account balance after 20 years will be $996.04.
The prime rate is the interest rate
that banks charge their most
credit-worthy borrowers. Changes
in this rate can influence other
rates, including mortgage interest
rates.
Source: Federal Reserve System
p GuidedPractice
5.
ONLINE BANKING If $1000 is invested in an online savings account earning 8 % per year
com pounded continuously, how m uch will be in the account at the end of 1 0 years if there
are no other deposits or withdraw als?
a
connectED.m cgraw-hill.com |
163
In addition to investments, populations of people, animals, bacteria, and am ounts of radioactive
material can also change at an exponential rate. Exponential growth and decay models apply to any
situation where growth is proportional to the initial size of the quantity being considered.
KeyC oncept Exponential Growth or Decay Formulas
If an initial quantity N0 grows or decays at an exponential rate ro r /c (as a decimal), then the final amount N after a time fis given
by the following formulas.
Exponential Growth or Decay
Continuous Exponential Growth or Decay
N = W0ew
« = W 0(1 +r)<
If r is a growth rate, then r > 0.
If k is a continuous growth rate, then k > 0.
If r is a decay rate, then r < 0.
If k is a continuous decay rate, then k < 0.
v,............
J
Continuous growth or decay is sim ilar to continuous com pound interest. The growth or decay is
com pounded continuously rather than ju st yearly, monthly, hourly, or at some other time interval.
Population growth can be m odeled exponentially, continuously, and by other models.
Real-World Example 6 Model Using Exponential Growth or Decay
POPULATION M exico has a population of approxim ately 110 million. If M exico's population
continues to grow at the described rate, predict the population of M exico in 10 and 20 years.
a. 1.42% annually
Use the exponential growth formula to write an equation that models this situation.
N = N 0 (1 + r)1
= 110,000,000(1 + 0.0142)'
=
110,000,000(1.0142)'
Exponential Growth Formula
N0 =
110,000,000 and
r=
0.0142
Simplify.
Use this equation to find N when t = 10 and t = 20.
N = 110,000,000(1.0142)*
N = 110,000,000(1.0142)*
Modeling equation
=
110,000,000(1.0142)10
f = 10 or t = 20
=
110,000,000(1.0142)20
~
126,656,869
Simplify.
=
145,836,022
If the population of M exico continues to grow at an annual rate of 1.42%, its population in
10 years will be about 126,656,869; and in 20 years, it will be about 145,836,022.
Real-W orldLink
In 2008, the population of Mexico
b.
was estimated to be growing at a
1.42% continuously
Use the continuous exponential growth form ula to write a m odeling equation.
rate of about 1.42% annually.
Source:
CIA-TheWorldFactBook
N = N 0e kt
=
110,000,000e° 0142(
Continuous Exponential Growth Formula
N0 =
110,000,000 and
k=
0.0142
Use this equation to find N when t = 10 and t = 20.
N = 110,000,000e00142t
Modeling equation
N = mOOOÔOe1-0142*
= 1 1 0 ,0 0 0 ,0 0 0 e ° 0142<10)
f = 10 and f = 20
=
,
= 126,783,431
Simplify.
~ 146,127,622
1 1 0 0 0 0
, 0 0 0 e ° ° 142(20)
If the population of M exico continues to grow at a continuous rate of 1.42%, its population in
10 years will be about 126,783,431; in 20 years, it will be about 146,127,622.
WatchOut!
Using Rates of Decay Remember
to write rates of decay as negative
values.
164
p GuidedPractice
6.
POPULATION The population of a town is declining at a rate of 6 %. If the current population is
12,426 people, predict the population in 5 and 10 years using each model.
A. annually
| Lesson 3-1 | E xponential Functions
B.continuously
After finding a model for a situation, you can use the graph of the model to solve problems.
Real-World Example 7 Use the Graph of an Exponential Model
DISEASE The table shows the num ber of reported cases of chicken pox in the United States in
1980 and 2005.
U.S. Reported Cases of
Chicken Pox
Cases (thousands)
190.9
Source: U.S. Centers fo r Disease Control and Prevention
a. If the num ber of reported cases of chicken pox is decreasing at an exponential rate,
identify the rate of decline and write an exponential equation to model this situation.
If we let N(t) represent the num ber of cases t years after 1980 and assum e exponential decay,
then the initial num ber of cases N 0 = 190.9 and at time t = 2005 — 1980 or 25, the number of
reported cases N(25) = 32.2. Use this inform ation to find the rate of decay r.
Real-W orldLink
A chicken pox vaccine was first
N(t) = N 0( 1 + r)f
Exponential Decay Formula
32.2 = 190.9(1 + r) 2 5
A/(25) = 32.2, N0 = 190.9, and f = 25
32.2
=
190.9
licensed for use in the United
States in 1995.
Source: Centers fo r Disease Control
25/153
190.'
V
s / I!]
190.' f
V
-
(1
1
+ r)
+ r
1
0.069
Divide each side by 190.9.
Take the positive 25th root of each side.
Simplify.
The number of reported cases is decreasing at a rate of approxim ately 6.9% per year.
Therefore, an equation m odeling this situation is N(f) = 190.9[1 + (—0.069)]* or
N(t) = 190.9(0.931)'.
b. Use your model to predict w hen the num ber of cases will drop below 20,000.
To find w hen the num ber of cases will drop below 20,000,
find the intersection of the graph of N(f) = 190.9(0.931) 1 and
the line N(f) = 20. A graphing calculator show s that value
of t for w hich 190.9(0.931)f = 20 is about 32.
Since f is the number of years after 1980, this model suggests
that after the year 1980 + 32 or 2012, the num ber of cases
will drop below 2 0 , 0 0 0 if this rate of decline continues.
[-5 , 50] scl: 5 by [-25, 200] scl: 25
GuidedPractice
7. POPULATION Use the data in the table and assum e that the population of M iami-Dade County
is growing exponentially.
Estim ated Population of
M iam i-D ade County, Florida
Year
Population (million)
1990
1.94
2000
2.25
A. Identify the rate of growth and write an exponential equation to model this growth.
B. Use your model to predict in w hich year the population of M iam i-Dade County will
surpass 2.7 million.
j f c connectED.m cgraw-hill.com 1
165
Exercises
Sketch and analyze the graph of each function. Describe its
domain, range, intercepts, asymptotes, end behavior, and
where the function is increasing or decreasing. (Example 1)
1
26. FINANCIAL LITERACY Katrina invests $1200 in a certificate
of deposit (CD). The table shows the interest rates offered
by the bank on 3- and 5-year CDs. (Examples 4 and 5)
2. r{x) = 5 X
. fix ) = 2~x
3. /2 (x) = 0.2* + 2
4. k(x) = 6X
5. m(x) = —(0.25)*
6
7-
8
- (I)’
. SM = ( I ) '
Use the graph of fix ) to describe the transformation that results
in the graph o f g ix ). Then sketch the graphs o f f i x ) and g ix ).
(Examples 2 and 3)
11. fix ) = 4X; g(x) = 4* — 3
12 ./ W = ( | ) ; ,( x ) = ( f + 4
Compounded
monthly
POPULATION Copy and complete the table to find the
population N of an endangered species after a time t given
its initial population N 0 and annual rate r or continuous
rate k of increase or decline. (Example 6)
E
16 ,y ( x ) = ex; g(x) = e *
continuously
b. How m uch would her investm ent be w orth if the
5-year CD w as com pounded continuously?
fix ) = 2X; g(x) = 2* ~ 2 + 5
15. f(x ) = W x;g {x ) = W ~x + 3
5
a. How m uch would her investm ent be w orth w ith each
option?
13. /(x) = 3*;s(x) = -2 (3 *)
$
3
Interest
. p(x) = 0.1~x
10. d(x) = 5~ x + 2
9. c(x) = 2 X — 3
CD Offers
Years
s
10
15
20
50
27. N 0 = 15,831, r = -4 .2 %
28. N 0 = 23,112, r = 0.8%
29. N 0 = 17,692, k = 2.02%
30. N 0 = 9689, k = -3 .7 %
31. WATER W orldwide water usage in 1950 was about
17. f(x ) = ex; g(x) = ex + 2 — 1
19. f{x ) = ex;g ix ) = 3ex
294.2 million gallons. If water usage has grown at the
described rate, estimate the amount of water used in 2 0 0 0
and predict the am ount in 2050. (Example 6)
20. fix ) = ex; g(x) = ~ (ex) + 4
a. 3% annually
18.j fix ) = ex; gix) = e~x +
1
b. 3.05% continuously
32. WAGES Jasm ine receives a 3.5% raise at the end of each
FINANCIAL LITERACY Copy and complete the table below to
find the value of an investm ent A for the given principal P,
rate r, and time t if the interest is compounded n times
year from her em ployer to account for inflation. W hen
she started working for the com pany in 1994, she was
earning a salary of $31,000. (Example 6)
annually. (Examples 4 and 5)
a. W hat w as Jasm ine's salary in 2000 and 2004?
4
1
12
365
continuously
21 . P = $500, r = 3%, t = 5 years
22. P = $1000, r = 4.5%, t = 10 years
23. P = $1000, r = 5%, f = 20 years
24. P = $5000, r =
6
33. PEST CONTROL Consider the termite guarantee m ade by
Exterm -inc in their ad below.
i
Termite & Pest Control
SINCE 1995
%, t = 30 years
FINANCIAL LITERACY Brady acquired an inheritance of
$ 2 0 , 0 0 0 at age 8 , but he will not have access to it until
he turns 18. (Examples 4 and 5)
a. If his inheritance is placed in a savings account earning
4.6% interest com pounded monthly, how m uch will
Brady's inheritance be w orth on his 18th birthday?
b. How much will Brady's inheritance be w orth if
it is placed in an account earning 4.2% interest
compounded continuously?
166
b. If Jasm ine continues to receive a raise at the end of
each year, how m uch m oney will she earn during her
final year if she plans on retiring in 2024?
| Lesson 3-1 ; Exponential Functions
|
FREE TERMITE INSPECTION
■
&
5 Termite Guarantee: 60% of termite colony
eliminated with each treatment. All but
eliminated after just 3 treatments!
1%
FINANCING AVAILABLE
555-3267
If the first statem ent in this claim is true, assess the
validity of the second statement. Explain your
reasoning. (Example 6)
34. INFLATION The Consum er Price Index (CPI) is an index
number that measures the average price of consumer
goods and services. A change in the CPI indicates the
growth rate of inflation. In 1958 the CPI was 28.6, and in
2008 the CPI was 211.08. (Example 7)
a. Determine the growth rate of inflation betw een 1958
and 2008. Use this rate to write an exponential
equation to model this situation.
RADIOACTIVITY The half-life of a radioactive substance is
7the am ount of tim e it takes for half of the atoms of the
substance to disintegrate. Uranium-235 is used to fuel a
commercial power plant. It has a half-life of 704 million years.
a. H ow m any gram s of uranium -235 will remain after
1 m illion years if you start w ith 2 0 0 grams?
b. H ow many gram s of uranium -235 will remain after
4540 m illion years if you start with 200 grams?
b. W hat will be the CPI in 2015? At this rate, w hen will
the CPI exceed 350?
35) GASOLINE Jordan wrote an exponential equation to model
the cost of gasoline. He found the average cost per gallon
of gasoline for two years and used these data for his
model. (Example 7)
Average Cost
per Gallon of Gasoline
Year
Cost($)
38. BOTANY Under the right growing conditions, a particular
species of plant has a doubling time of 12 days. Suppose a
pasture contains 46 plants of this species. How many
plants will there be after 20, 65, and x days?
39. RADIOACTIVITY Radiocarbon dating uses carbon-14 to
estimate the age of organic m aterials found commonly
at archaeological sites. The half-life of carbon-14 is
approxim ately 5.73 thousand years.
1990
1.19
a. Write a m odeling equation for the exponential decay.
2007
3.86
b. How many grams of carbon-14 will remain after
a. If the average cost of gasoline increased at an
exponential rate, identify the rate of increase.
Write an exponential equation to model this situation.
12.82 thousand years if you start with 7 grams?
C.
Use your m odel to estim ate when only 1 gram of the
original 7 gram s of carbon-14 will remain.
b. Use your model to predict the average cost of a gallon
of gasoline in 2011 and 2013.
C.
W hen will the average cost per gallon of gasoline
exceed $7?
d.
W hy might an exponential m odel not be an accurate
representation of average gasoline prices?
MICROBIOLOGY A certain bacterium used to treat oil spills
has a doubling time of 15 minutes. Suppose a colony
begins w ith a population of one bacterium.
a. Write a m odeling equation for this exponential growth.
b. About how m any bacteria w ill be present after 55
minutes?
PHYSICS The pressure of the atmosphere at sea level
is 15.191 pounds per square inch (psi). It decreases
continuously at a rate of 0.004% as altitude increases
by x feet.
a. Write a modeling function for the continuous
exponential decay representing the atmospheric
pressure a(x).
b. Use the model to approximate the atmospheric
pressure at the top of M ount Everest.
Highest point above sea,
level— peak of M ount Everest
C.
A population of 8192 bacteria is sufficient to clean a
sm all oil spill. Use your m odel to predict how long it
will take for the colony to grow to this size.
4 1 . ENCYCLOPEDIA The num ber of articles making up an
online open-content encyclopedia increased exponentially
during its first few years. The num ber of articles, A(t),
t years after 2001 can be m odeled by A(t) = 16,198 •2.13f.
a. According to this m odel, how many articles made up
the encyclopedia in 2001? A t w hat percentage rate is
the number of articles increasing?
b. During w hich year did the encyclopedia reach
1
29,035 ft
m illion articles?
c. Predict the num ber of articles there will be at the
beginning of 2 0 1 2 .
Sea Level: 15.191 psi
C.
If a certain rescue helicopter can fly only in
atmospheric pressures greater than 5.5 pounds per
square inch, how high can it fly up M ount Everest?
42. RISK The chance of having an autom obile accident
increases exponentially if the driver has consumed
alcohol. The relationship can be modeled by A(c) = 6 eU 8c,
where A is the percent chance of an accident and c is the
d river's blood alcohol concentration (BAC).
a. The legal BAC is 0.08. W hat is the pendent chance of
having a car accident at this concentration?
b. W hat BAC would correspond to a 50% chance of
having a car accident?
K
I
167
43. GRAPHING CALCULATOR The table shows the number of
blogs in millions sem iannually from Septem ber 2003 to
March 2006.
1
7
13
19
25
31
0.7
2
4
8
16
31
D eterm ine the tran sform ations o f the given parent function
that produce each graph.
a. Using the calculator's exponential regression tool, find
a function that models the data.
b. After how many months did the num ber of blogs
reach 2 0 million?
C.
Predict the number of blogs after 48 months.
44. LANGUAGES Glottochronology is an area of linguistics
that studies the divergence of languages. The equation
c = e-L f, where c is the proportion of words that remain
unchanged, t is the time since two languages diverged,
and L is the rate of replacement, models this divergence.
a. If two languages diverged 5 years ago and the rate of
replacement is 43.13%, what proportion of words
remains unchanged?
b. After how many years will only 1% of the words
remain unchanged?
58. #gl MULTIPLE REPRESENTATIONS In this problem , you will
investigate the average rate of change for exponential
functions.
a. GRAPHICAL Graph/(x) = bx for b = 2 ,3 ,4 , or 5.
b. ANALYTICAL Find the average rate of change of each
function on the interval [0 , 2 ],
c. VERBAL W hat can you conclude about the average rate
of change of/(x) = bx as b increases? How is this
show n in the graphs in part a?
d. GRAPHICAL Graph/(x) = b~x for b = 2 , 3 ,4 , or 5.
45. FINANCIAL LITERACY A couple just had a child and wants
to immediately start a college fund. Use the information
below to determine how much money they should invest.
$ 6 0 ,0 0 0
e. ANALYTICAL Find the average rate of change of each
function on the interval [0 , 2 ].
f. VERBAL W hat can you conclude about the average rate
of change of/(x) = b~x as b increases. How is this
show n in the graphs in part d?
H.O.T. Problems
Interest Rate: 9 %
Compounding: dally
18 years old
0 years old
GRAPHING CALCULATOR D eterm ine the value(s) of x that m akes
each equation or inequality below true. Round to the nearest
hundredth, if necessary.
46. 2* < 4
47. e2* = 3
48. - e * > - 2
49. 2 —4 x
.
D escribe the dom ain, range, continuity, and increasing/
decreasing behavior for an exponential function w ith the
given intercept and end behavior. Then graph the function.
50. /(0) = —1, lim f(x ) = 0, lim f(x ) = —oo
x —> - o c r
x —>oc
51. f(0) = 4, lim fix ) = o o , lim f(x ) = 3
x — >— 00
59) ERROR ANALYSIS Eric and Sonja are determ ining the w orth
of a $550 investm ent after 12 years in a savings account
earning 3.5% interest com pounded monthly. Eric thinks
the investm ent is w orth $837.08, while Sonja thinks it is
w orth $836.57. Is either of them correct? Explain.
REASONING State w heth er each statem ent is true or fa ls e .
60. Exponential functions can never have restrictions on
the domain.
61. Exponential functions always have restrictions on the
range.
62. Graphs of exponential functions always have an
asymptote.
63. OPEN ENDED W rite an exam ple of an increasing
exponential function w ith a negative y-intercept.
x —* 0 0 -'
52. /(0) = 3, lim f(x ) = 2, lim /(x) = o o
X—>—oo
x—»ooy
Determ ine the equation of each function after the given
transform ation of the parent function.
53. f(x ) = 5Xtranslated 3 units left and 4 units down
54. /(x) = 0 .25* compressed vertically by a factor of 3 and
translated 9 units left and 12 units up
55. f(x ) = 4 Xreflected across the x-axis and translated 1 unit
left and 6 units up
1 68
Lesson 3-1 j E xponential Functions
64. CHALLENGE Trina invests $1275 in an account that
com pounds quarterly at 8 %, but at the end of each year
she takes 100 out. How m uch is the account w orth at the
end of the fifth year?
65. REASONING Two functions of the form/(x) = b x sometimes,
always, or never have at least one ordered pair in com mon.
66. WRITING IN MATH Com pare and contrast the domain,
range, intercepts, symmetry, continuity, increasing/
decreasing behavior, and end behavior of exponential and
pow er parent functions.
Spiral Review
Solve each inequality. (Lesson 2-6)
(x — 3)(x + 2) < 0
67.
6 8
. x2 +
6
69. 3x 2 + 15 > x 2 + 15x
x < —x — 4
Find the domain of each function and the equations of any vertical or horizontal asymptotes,
noting any holes. (Lesson 2-5)
70. f{ x ) = -
73.
71. f( x ) =
:2 - 4x + 4
x - l
x2
72. f( x ) =
+ 4x - 5
x2 — 8x + 16
x —4
TEMPERATURE A formula for converting degrees Celsius to Fahrenheit is
Fix) = f- x + 32. (Lesson 1-7)
a. Find the inverse F _ 1 (x). Show that F(x) and F - 1 (x) are inverses.
b. Explain what purpose F _1(x) serves.
74.
SHOPPING Lily wants to buy a pair of inline skates that are on sale for 30% off the original
price of $149. The sales tax is 5.75%. (Lesson 1-6)
a. Express the price of the inline skates after the discount and the price of the inline skates
after the sales tax using function notation. Let x represent the price of the inline skates,
p(x) represent the price after the 30% discount, and s(x) represent the price after the sales
tax.
b. W hich composition of functions represents the price of the inline skates, p[s(x)] or s[p(x)]?
C. How much will Lily pay for the inline skates?
75. EDUCATION The table shows the num ber of freshmen
who applied to and the number of freshmen attending
selected universities in a certain year. (Lesson 1-1)
Applied
Attending
Auburn University
13,264
4184
a. State the relation as a set of ordered pairs.
University of California-Davis
27,954
4412
b. State the domain and range of the relation.
University of Illinois-Urbana-Champaign
21,484
6366
Florida State University
13,423
4851
State University of New York-Stony Brook
16,849
2415
The Ohio State University
19,563
5982
Texas A&M University
17,284
6949
University
C. Determine whether the relation is a function. Explain.
d.
Assuming the relation is a function, is it reasonable
to determine a prediction equation for this situation?
Explain.
Source:
HowtoGetIntoCollege
76.
SAT/ACT A set of n num bers has an average (arithmetic
mean) of 3k and a sum of 12m, where k and m are
positive. W hat is the value of n in terms of k and m l
A
4m
k
B 36km
77.
C —
E
m
D
78.
REVIEW If 4 X+ 2 = 48, then 4 X = ?
A 3.0
C 6.9
B 6.4
D 12.0
4m
79. REVIEW W hat is the equation
of the function?
m
4k
The number of bacteria in a colony were growing
exponentially. Approxim ately how many bacteria
were there at 7 p . m ?
F 15,700
Time
G 159,540
2 P.M.
100
H 1,011,929
4 P.M.
4000
Number of Bacteria
F y = 2(3)*
\
|
y
G » = #
H * =
#
V
' 0
X
J y = 3(2)*
J 6,372,392
Lg ---- - wmmRHNm..............—
f l | connectED.m cgraw-hill.corn |
m
169
Financial Literacy:
Exponential Functions
OOOO
OOOO
o o o o
CDOO
In Lesson 3-1, you used exponential functions to calculate compounded interest. In the compounding
•
Calculate future values of
annuities and monthly
payments.
formula, you assume that an initial deposit is made and the investor never deposits nor w ithdraw als
any money. Other types of investments do not follow this simple compounding rule.
When an investor takes out an annuity, he or she makes identical deposits into the account at regular
intervals or periods. The compounding interest is calculated at the tim e of each deposit. We can
determine the future value of an annuity, or its value at the end of a period, using the form ula below.
StudyTip
Future Value Formula The
payments must be periodic and of
equal value in order for the
Because solving this equation by hand can be tedious, you can use the finance application on a TI-84.
The time value of money solver can be used to find any unknown value in this formula. The known
variables are all entered and zeros are entered for the unknown variables.
formula to be accurate.
Activity 1
Find a Future Value of an Annuity
An investor pays $600 quarterly into an annuity. The annuity earns 7.24% annual interest.
W hat will be the value of the annuity after 15 years?
ETHTBI Select Finance in the A PPS M enu. Then select CALC, TVM Solver.
■ w a in ia t!
l nance...
! RLG1CH5
3: flLGlF'RT 1
4 : R reaForn
5 : C a b riJ r
6 : CBL^CBR
74-Cel Sheet.
_
VRRS
M So Iyer...
: tw 't-P i'r t
3 : typi-IJ;
4:tvM _PV
5 : t-ypi-N
6 : tupi_FV
74npy(
CflSffW Enter the data.
Payments are made quarterly over 15 years, so there
are 4 • 15 or 60 payments. The present value, or
amount at the beginning, is $0. The future value is
unknown, 0 is used as a placeholder. Interest is
com pounded quarterly, so P /Y and C /Y are 4.
(C /Y and P /Y are identical.) Paym ent is m ade at the
end of each month, so select end.
N=60
I * = 7 .2 4
PU=0
PMT=600
FV=0
P/Y=4
O Y=4
PNT:l*2H BEGIN
Calculate.
Quit the screen then go back into the Finance
application. Select tvm _FV to calculate the
future value. Then press enter). The result is the
future value subtracted from the present value,
so the negative sign is ignored.
After 15 years, the value of the annuity will be about $64,103.
V......................................................................................................................................
170
| Lesson 3-1
tu N _ F V
-6 4 1 0 2 .9 1 4 0 2
When taking out a loan for a large purchase like a home or car, consumers are typically concerned with
StudyTip
>how much their monthly payment will be. While the exponential function below can be used to
determine the monthly payment, it can also be calculated using the finance application in the TI-84.
DownPayments When a
consumer makes a down
payment, that amount is
present value
PV-
subtracted from the present value
PM T = ■
of the loan before anything else is
calculated.
v
~ ( 1
v
Activity 2
C /Y
+ c/y)
Calculate Monthly Payment
You borrow $170,000 from the bank to purchase a home. The 30-year loan has an annual
interest rate of 4.5%. Calculate your m onthly paym ent and the am ount paid after 30 years.
EflSfln Select Finance in the APPS M enu. Then select CALC,
TVM Solver.
N = 360
I * = 4 .5
PV=1 7 0 0 0 0
PMT=0
FU=0
P/Y=12
C/V = 1 2
P M U f lf lj BEGIN
Enter the data.
The num ber of paym ents is N = 30 • 12 or 360.
The interest rate I is 4.5%.
The present value of the loan P V is $170,000.
The m onthly paym ent and future value are unknown.
The num ber of paym ents per year P / Y and C /Y is 12.
Paym ent is m ade at the end of month, so select end.
iv n _ P m t
- 8 6 1 .3 6 5 0 2 6 7
fin s *3 6 0
-3 1 0 0 9 1 .4 0 9 6
Calculate.
Select tvm_Pmt to calculate the m onthly payment.
Then press e n t e r |. M ultiply the m onthly paym ent by 360.
Your monthly paym ent will be $861.37 and the total that will be repaid is $310,091.41.
Exercises
Calculate the future value of each annuity.
1.
$800 semiannually, 12 years, 4%
2.
$400 monthly,
3.
$200 monthly, 3 years, 7%
4.
$1,000 annually, 14 years, 6.25%
5.
$450, quarterly,
6
.
$300 bimonthly, 18 years, 4.35%
8
years, 5.5%
6
years, 5.5%
Calculate the monthly paym ent and the total amount to be repaid for each loan.
7.
10.
13.
$220,000,30 years, 5.5%
8. $140,000, 20 years, 6.75%
$5,000, 5 years, 4.25%
11. $ 45,000,10 years, 3.5%
9. $20,000,5 years, 8.5%
12. $180,000, 30 years, 6.5%
CHANGING VALUES Changing a value of any of the variables m ay dram atically affect the loan
payments. The m onthly paym ent for a 30-year loan for $150,000 at 6 % interest is $899.33, with
a total paym ent am ount of $323,757.28. Calculate the m onthly paym ent and the total amount
of the loan for each scenario.
a. Putting down $20,000 on the purchase.
b.
Paying 4% interest instead of 6 %.
c. Paying the loan off in 20 years instead of 30.
d.
M aking 13 payments per year.
e. W hich saved the m ost m oney? W hich had the low est m onthly payment?
[connectED.mcgraw-hill.com §
§
171
•
m
You graphed and
•
analyzed exponential
functions. (Lesson 3-1)
NewVocabulary
logarithmic function with
base b
logarithm
common logarithm
natural logarithm
■# Evaluate expressions •
I involving logarithms.
_ . ,
,
.
O S k e tc h and analyze
4
, ..
’ .
**■ fa p h s of logarithmic
functions.
The intensity level of sound is measured in decibels.
A whisper measures 20 decibels, a normal conversation
60
decibels,and a vacuum cleaner at 80 decibels. The
. , . . .
. .
u.
music
playingin headphones
maximizesat100 decibels.
s
Logarithmic Functions and Expressions Recall from Lesson 1-7 that graphs of
functions that pass the horizontal line test are said to be one-to-one and have inverses that are
also functions. Looking back at the graphs on page 159, you can see that exponential functions of
the form /(x) = b x pass the horizontal line test and are therefore one-to-one w ith inverses that are
functions.
1
The inverse o if ( x ) = bx is called a logarithmic function
with base b, denoted log,, x and read log base b o fx . This
m eans that if/(x) = bx, b > 0 and b =/= 1 , then/- 1 (x) = log;, x,
as shown in the graph of these two functions. N otice that the
graphs are reflections of each other in the line y = x.
This inverse definition provides a useful connection betw een exponential and logarithmic
equations.
K eyC oncept Relating Logarithmic and Exponential Forms
If
b>
0,
b±
1 ■and
x>
Wwi
0, then
Logarithmic Form
T
base J
= x.
if and only if
lo g „ * = y
i
Exponential Form
A
I
L exponent
base
ft
-■ L exponent
The statement above indicates that lo g b x is the exponent to w hich b m ust be raised in order to
obtain x. Therefore, w hen evaluating logarithm s, remember that a logarithm is an exponent.
Evaluate Logarithms
Evaluate each logarithm,
b.
a. log3 81
log 3 81 = y
= 81
Let log3 81
= y.
Write in exponential form.
log5 V5
Let log5 \ / 5
5V = V 5
1
3y = 34
81 = 34
y = 4
Equality Prop, of Exponents
= y.
iog 5 V s = y
5^ = 5 2
1
5 2 = a/5
Equality Prop, of Exponents
Therefore,log 3 81 = 4, because 3 4 = 81.
y | 172
Lesson 3-2
Therefore, logg V 5 = y , because 5 2 = V 5 .
c.
lo g 7 i
d. log2 2
log 7 ^
- - 2 , because 7 - 2 =
or
log 2 2 = 1, because 2 1 = 2.
► GuidedPractice
1A. log 8 512
1B' log 4
1C- lo§ 2 ^
4 3 2
V
1D- !o g i6 V 2
............
Exam ple 1 and other exam ples suggest the follow ing basic properties of logarithms.
K eyC oncept Basic Properties of Logarithms
If b
StudyTip
Inverse Functions The inverse
log„1 = 0
•
logft6 = 1
•
log „ b x= x
■ Inverse Properties
• b'09^ = x, x > 0
properties of logarithms also
follow from the inverse
> 0, b £ 1, and x is a real number, then the following statements are true.
•
V
relationship between logarithmic
and exponential functions and the
definition of inverse functions. If
These properties follow directly from the statem ent relating the logarithm ic and exponential forms
of equations.
f(x) = t f and M { x ) = logfi x,
then the following statements
are true.
log fe 1 =
0
, because b ° =
logb b =
1
, because b 1 = b.
1
logb b- = y, because b y = by.
.
b l° 8bX =
x, because logb x = logb x.
f-i[f(x)] = \ogb bx = x
V
f [ f - \ x ) ] = b'°W = x
You can use these basic properties to evaluate logarithmic and exponential expressions.
_________ _______
■
Apply
Properties of Logarithms
Evaluate each expression.
b. 12loSi24-7
a. lo g 5 125
log 5 125 = log 5 5 3
1 2 loS i2 4 7 = 4 .7
5 3 = 125
= x
log „bK= x
f GuidedPractice
2B. 3 l o g 3
2A. log 9 81
A logarithm w ith base 10 or log 1 0 is called a com m on logarithm
and is often w ritten w ithout the base. The com m on logarithm
function y = log x is the inverse of the exponential function y = 1 0 *.
Therefore,
y = log x
if and only if
1 0
-l/= x, for all .t >
0
.
The properties for logarithm s also hold true for com m on logarithms.
KeyC oncept Basic Properties of Common Logarithms
If x is a real number, then the following statements are true.
•
log 1 = 0
•
log 10 = 1
•
Iog10* = x
•
1olog X= x ,x > 0
Inverse Properties
I
connectED.m cgraw-hill.coin 1
173
Com m on logarithms can be evaluated using the basic properties described above. Approxim ations
of com m on logarithm s of positive real num bers can be found by using |LOGl on a calculator.
B E E S E E E Common Logarithms
Evaluate each expression,
b. 10loss
a. log 0.001
log
0 .0 0 1
= log
1 0 _3
0 .0 01
103
= -3
TechnologyTip
or 1 0 -
log 26 = 1.42
the common logarithm of a
negative number, your calculator
d.
Use a calculator.
message ERR: N O N R E A L
A N S or an imaginary number.
10
l09/= x
log ( - 5 )
Since f ( x ) = lo g b x is only defined when
x > 0 , log (—5) is undefined on the set of real
CHECK Since 26 is betw een 10 and 100,
log 26 is betw een log 1 0 and
log 100. Since log 10 = 1 and
log 1 0 0 = 2 , log 26 has a value
betw een 1 and 2 . ✓
will display either the error
loS 5 = 5
log 1 0 * = x
C. log 26
Error Message If you try to take
1 0
numbers.
f GuidedPractice
3A. log 10,000
3B. log 0.081
3C. log —0
3D. 10los"
A logarithm with base e or logc is called a natural logarithm and
is denoted In. The natural logarithmic function y = In x is the
inverse of the exponential function y = e x. Therefore,
y = In x
if and only if
e y = x, for all x >
0
.
The properties for logarithms also hold true for natural logarithms.
KeyC oncept Basic Properties of Natural Logarithms
If x is a real number, then the following statements are true.
•
In 1 = 0
• In e = 1
•
In e x =
x
■ Inverse Properties
• eln * = x, x > 0
J
V,
Natural logarithms can be evaluated using the basic properties described above. Approxim ations of
natural logarithms of positive real num bers can be found by using [LN] on a calculator.
B E S H S I E Natural Logarithms
a.
In e 0
73
In e0 -7 3 = 0.73
c.
b.
\nex = x
d.
e ln6
eln 6 = 6
In ( - 5 )
In (—5) is undefined.
In 4
In 4 = 1.39
eln* = x
Use a calculator.
►GuidedPractice
4A. In 32
4B.e l n 4
4C. In
V___________________________________________ ___ __________
174
Lesson 3-2 | L og a rith m ic Functions
4D.
-In 9
Graphs of Logarithmic Functions You can use the inverse relationship betw een
exponential and logarithmic functions to graph functions of the form y = logb x.
Graphs of Logarithmic Functions
Sketch and analyze the graph of each function. D escribe its dom ain, range, intercepts,
asym ptotes, end behavior, and where the function is increasing or decreasing.
a.
f(x ) = log3 x
Construct a table of values and graph the inverse of this logarithm ic function,
the exponential function/_ 1 (x) = 3X.
-4
-2
-1
0
1
2
0.01
0.11
0.33
1
3
9
Since/(x) = log 3 x and/_ 1 (x) = 3* are inverses, you can obtain the graph of/(x) by plotting
the points (/_ 1 (x), x).
f-H x )
0.01
0.11
0.33
1
3
9
-4
-2
-1
0
1
2
X
The graph of/(x) = log 3 x has the follow ing characteristics.
Domain: (0, oo)
Range: (—0 0 , 0 0 )
x-intercept: 1
Asym ptote: y-axis
End behavior: lim
x^ 0+
f(x ) — —0 0 and lim f(x ) =
00
x —>oo
Increasing: ( 0 , 0 0 )
StudyTip
Graphs To graph a logarithmic
function, first graph the inverse
using your graphing calculator.
>
b. g(x) =
logi x
Construct a table of values and graph the inverse of this logarithm ic function, the exponential
Then, utilize the T A BLE function
to quickly obtain multiple
function g _ 1 (x) =
.
coordinates of the inverse. Use
these points to sketch the graph
I
of the logarithmic function.
“ 4
-2
0
1
2
4
4
1
0.5
0.25
0.06
Graph g(x) by plotting the points (g 1(x), x).
<rV>
1
M M
1
-
4
4
1
0.5
0.25
0.06
-2
0
1
2
4
The graph of g(x) = logi x has the follow ing characteristics.
Domain: (0,
0 0
)
Range: (—0 0 ,
x-intercept: 1
End behavior: lim ?(x) =
x^ o+
Decreasing: (0, 0 0 )
0 0
)
Asym ptote: t/-axis
00
and lim ?(x) =
— 00
p GuidedPractice
5A. h{x) = log 2 x
5B. ;(x ) = lo g ix
M
connectED.mcgraw-hill.com J
175 g |
The characteristics of typical logarithmic growth, or increasing logarithm ic functions, and logarithmic
decay, or decreasing logarithmic functions, are sum m arized below.
K eyC oncept Properties of Logarithmic Functions
Logarithmic Growth
Logarithmic Decay
y
y
fix) = logf i x
b> 1
fix) = lo g „ x
0< b< 1
V
\(1 ,0 )
0
(0, oo)
(1 ,0 )
0
X
X
R a n g e : (—oc , 0°)
D o m a in :
y - ln t e r c e p t: none
x - ln t e r c e p t: 1
y - ln t e r c e p t: none
x - ln t e r c e p t: 1
E x tre m a : none
A s y m p to te : y-axis
E x tre m a : none
A s y m p to te : y-axis
D o m a in :
E n d B e h a v io r: lim f(x) = —oo and
JT-.0+
lim f(>t) =
X—
>oo
C o n tin u ity : continuous on
R a n g e : (—oo, oo)
E n d B e h a v io r: lim +f(x) =
x->0
oo
(0, oo)
(0, oo)
lim ftx) =
X—
>oo
C o n tin u ity : continuous on
oo and
-oo
(0, oo)
L
J
The same techniques used to transform the graphs of exponential functions can be applied to the
graphs of logarithm ic functions.
jiy T T n f f f f f f l Graph Transformations of Logarithmic Functions
WatchOut!
Transformations
Remember that
horizontal translations are
dependent on the constant inside
the parentheses, and vertical
translations are dependent on the
constant outside of the
>
Use the graph of fix ) = log x to describe the transform ation that results in each function.
a. k(x ) = log (x + 4)
This function is of the form k(x) = f i x + 4). Therefore, the graph of k{x) is the graph of fi x )
translated 4 units to the left (Figure 3.2.1).
parentheses.
b. mix ) =
—log x — 5
The function is of the form mix) = —fi x ) — 5. Therefore, the graph of mix) is the graph of fi x )
reflected in the x-axis and then translated 5 units dow n (Figure 3.2.2).
C. pix ) = 3 log ix -I- 2)
The function is of the form p{x) = 3f i x + 2). Therefore, the graph of pix) is the graph of/(x)
expanded vertically by a factor of 3 and then translated 2 units to the left. (Figure 3.2.3).
Figure 3.2.2
w GuidedPractice
Use the graph o f /(x ) = In x to describe the transform ation that results in each function.
6A. aix) = In (x — 6 )
176
| Lesson 3-2
j
L o g a rith m ic Functions
6B. bix) = 0.5 In x - 2
6C. c(x) = In (x + 4) + 3
Logarithm s can be used in scientific calculations, such as w ith pH acidity levels and the intensity
level of sound.
Real-World Example 7 Use Logarithmic Functions
SOUND The in ten sity level o f a sound, m easured in d ecibels, can be m odeled by
d(iv ) = 1 0 log -Ijj-, w here w is the in ten sity o f the sound in watts per square m eter and
w 0 is the constant 1.0 X 10~12 watts per square meter.
a. If the intensity of the sound of a person talk in g loud ly is 3.16 X 10 - 8 watts per square
m eter, w hat is the intensity level o f the sound in d ecibels?
Evaluate d(w) w hen w = 3.16 x 10 ~8.
d(iu) =
1 0
lo g ^ -
1 0
log
Original function
3.16 X 10
_ 12
45
Real-W orldCareer
w = 3.16 x 1 0 " 8 and w0 = 1.0 x 10“ 12
1.0 x 10
Use a calculator.
The intensity level of the sound is 45 decibels.
Sound Engineer Sound
engineers operate and maintain
sound recording equipment. They
b.
also regulate the signal strength,
If the threshold of hearing fo r a certain person w ith hearing loss is 5 d ecibels, w ill a sound
w ith an intensity level o f 2 . 1 x 1 0 - 1 2 watts per square m eter be aud ible to that person?
clarity, and range of sounds of
Evaluate d(w) when w = 2.1 x 10
recordings or broadcasts. To
-12
become a sound engineer, you
should take high school courses
d(w) =
1 0
log -
=
1 0
log
Original function
in math, physics, and electronics.
3.22
2.1 x 1 0 ~ 12
1.0 x 10“ 12
w=
2.1
x
1 0 ~ 12
and w0 =
1.0
x
10
Use a calculator.
Because the person can only hear sounds that are 5 decibels or higher, he or she would not be
able to hear a sound w ith an intensity level of 3.22 decibels.
C.
Sounds in excess of 85 decibels can cause hearing dam age. D eterm ine the intensity of
a sound w ith an in ten sity level of 85 decibels.
Use a graphing calculator to graph d(w) = 10 log
and d(w ) = 85 on the same screen
1 x I Q " 12
and find the point of intersection.
[0, 0.001] scl: 0.0001 by [5 0 ,1 0 0 ] scl: 10
W hen the intensity level of the sound is 85 decibels, the intensity of the sound is
3.1623 x 1 0 - 4 watts per square meter.
p GuidedPractice
7.
TECHNOLOGY The num ber of m achines infected by a specific com puter virus can be modeled
by c(d) = 6 . 8 + 2 0 . 1 In d, where d is the num ber of days since the first m achine was infected.
A. About how m any m achines were infected on day 12?
B. How many more m achines were infected on day 30 than on day 12?
C. On about what day will the num ber of infected machines reach 75?
$
con nectE D .m cg raw -h ill.co m I
177
Exercises
Evaluate each expression. (Examples 1-4)
1. log 2
3.
7.
2. log 1 0
8
log 6 l
5. logn
[ '
6
121
loS V 9 8 1
1 0
4. 4 log4 1
6. log 2
8. log
2 3
10. H x X2
11. log 5275
12. In e~ u
13. 3 In e4
14. In (5 — V 6 )
15.
16. 4 In (7 - V 2 )
S ketch and analyze the graph o f each fu nction. D escribe its
dom ain, range, intercepts, asym ptotes, end behavior, and
w here the fu nction is increasing or decreasing. (Example 5)
28. fix ) = log 4 x
29. g(x) = log 5 x
30. h(x) = log 8 x
31. j(x) = logi x
32. m(x) = logi x
33. n(x) = logi x
4
In 2
In 7
17. log 635
18.
19. In ( - 6 )
20.
21. In
22. !° g ^ 4
7
23.
In e
MEMORY The students in Mrs. Ross' class were tested on
exponential functions at the end of the chapter and then
were retested m onthly to determ ine the am ount of
inform ation they retained. The average exam scores can
be m odeled by fix ) = 85.9 — 9 In x, where x is the num ber
of m onths since the initial exam. W hat was the average
exam score after 3 months? (Example 4)
0 .0 1
9. log 42
8
27.
Use the graph of fix ) to describe the transformation that results
in the graph of g(x). Then sketch the graphs o f/(x ) andg(x).
(Example 6 )
ta ( £ )
24. log
64
1 0 0 0
34. f{ x ) = log 2 x; gix) = log 2 (x + 4)
35. fi x ) = log 3 x; gix) = log 3 (x - 1)
36. fi x ) = log x; g(x) = log 2x
25. LIGHT The amount of light A absorbed by a sample
solution is given by A = 2 — log 100T, where T is the
fraction of the light transmitted through the solution as
shown in the diagram below. (Example 3)
37. fi x ) = In x; g(x) = 0.5 In x
^38. f( x ) = log x; g(x) = - l o g (x -
2
)
39. fi x ) = In x; g{x) = 3 In (x) + 1
40. fi x ) = log x; gix ) = - 2 log x + 5
41. fi x ) = In x; gix) = In (—x)
Percent of light
transmitted T =
Intensity of light
Intensity of light
leaving solution
entering solution
Sample solution
In an experiment, a student shines light through two
sample solutions containing different concentrations
of a certain dye.
a. If the percent of light transmitted through the first
sample solution is 72%, how much light does the
sample solution absorb to the nearest hundredth?
b.
If the absorption of the second sample solution is
0.174, what percent of the light entering the solution
is transmitted?
26. SOUND W hile testing the speakers for a concert, an audio
engineer notices that the sound level reached a relative
intensity of 2.1 x 10 8 watts per square meter. The
equation D = log I represents the loudness in decibels
D given the relative intensity I. W hat is the level of the
loudness of this sound in decibels? Round to the nearest
housandth if necessary. (Example 3)
Lesson 3-2
L o g a rith m ic Functions
42. INVESTING The annual growth rate for an investm ent can
be found using r = -j- In ■£-, where r is the annual growth
-'o
rate, t is tim e in years, P is the present value, and P 0 is the
original investm ent. An investm ent of $10,000 w as made
in 2002 and had a value of $15,000 in 2009. W hat was the
average annual growth rate of the investment? (Example 7)
Determine the domain, range, x-intercept, and vertical
asymptote of each function.
43. y = log (x + 7)
44. y
45. y = In (x — 3)
46. y
Find the inverse o f each equation.
47. y = e3x
48. y
(4 9 ) i/ = 4e2x
50. y
51. y = 20x
52. y
D eterm ine the dom ain and range of the inverse of each
function.
53. y = log x — 6
54. y = 0.25ex +
2
55. COMPUTERS Gordon Moore, the cofounder of Intel, made
a prediction in 1975 that is now know n as M oore's Law.
He predicted that the number of transistors on a
computer processor at a given price point would double
every two years.
a. Write M oore's Law for the predicted num ber of
transm itters P in terms of time in years t and the initial
number of transistors.
b. In October 1985, a specific processor had 275,000
transistors. About how many years later would you
expect the processor at the same price to have about
4.4 m illion transistors?
68. BACTERIA The function t = lnB 2 *nA models the amount
of tim e t in hours for a specific bacteria to reach amount B
from the initial am ount A.
a. If the initial num ber of bacterial present is 750, how
m any hours would it take for the num ber of bacteria
to reach 300,000? Round to the nearest hour.
b. Determ ine the average rate of change in bacteria per
hour for the bacterial am ounts in part a.
69. f£» MULTIPLE REPRESENTATIONS In this problem , you will
com pare the average rates of change for an exponential, a
power, and a radical function.
a. GRAPHICAL Graph f{x) = 2 X and g{x) = x2 for 0 < x <
Describe the domain, range, symmetry, continuity, and
increasing/decreasing behavior for each logarithmic
function with the given intercept and end behavior. Then
sketch a graph of the function.
56. / (I) - 0; lim f( x ) = —o o ; lim / ( x ) = o o
x—
0
x
- kx/
*
oo
X —*0
x
f. VERBAL Compare the growth rates of the functions
from part d as x increases.
>00
Use the parent graph of fix ) — log x to find the equation of
each function.
y
to = logx
j—
X
° (
—r
r
(
f
/to
n /r
c. VERBAL Com pare the growth rates of the functions
from part a as x increases.
x —>0
59. / ( I ) = 0; lim i(x) = o o ; lim j(x) = — o o
'
b. ANALYTICAL Find the average rate of change of each
function from part a on the interval [4, 6 ],
e. ANALYTICAL Find the average rate of change of each
function from part d on the interval [4, 6 ],
X —>oo
58. h (—1) = 0; lim h(x) = o o ; lim h(x) = —o o
x
.
d. GRAPHICAL Graph f(x ) = In x and g(x) = Vx.
57. g (—2) = 0; lim g (x) = —o o ; lim g(x) = o o
x —>—3
8
H.O.T. Problem s
70. WRITING IN MATH Com pare and contrast the domain,
range, intercepts, symmetry, continuity, increasing/
decreasing behavior and end behavior of logarithmic
functions w ith a(x) = x n, b(x) = x _1, c(x) = a x, and
d(x) = ex.
(7 ?) REASONING Explain why b cannot be negative in
/(x) = log b x.
i
fix)
f(x) = log x
\
:3*
X
O f
/
X
k(x) -
72. CHALLENGE For/(x) = log 1 0 (x — k), where A: is a constant,
w hat are the coordinates of the x-intercept?
73. WRITING IN MATH Com pare the large-scale behavior of
exponential and logarithm ic functions w ith base b for
b = 2 , 6 , and 1 0 .
f
REASONING D eterm ine w hether each statement is true
or fa ls e .
GRAPHING CALCULATOR Create a scatter plot of the values
shown in the table. Then use the graph to determine
whether each statement is true or fa ls e .
1
3
0
1
9
2
64. y is an exponential function of x.
27
3
74. Logarithm ic functions w ill always have a restriction on
the domain.
75. Logarithm ic functions will never have a restriction on the
range.
76. Graphs of logarithm ic functions always have an
asymptote.
65. x is an exponential function of y.
66. y is a logarithmic function of x.
67. y is inversely proportional to x.
77. WRITING IN MATH Use words, graphs, tables, and
equations to com pare logarithm ic and exponential
functions.
179
Spiral Review
78. AVIATION W hen kerosene is purified to m ake jet fuel, pollutants are rem oved by passing the
kerosene through a special clay filter. Suppose a filter is fitted in a pipe so that 15% of the
impurities are removed for every foot that the kerosene travels. (Lesson 3-1)
a. Write an exponential function to model the percent of im purity left after the kerosene
travels x feet.
c. About what percent of the im purity remains after the kerosene travels 12 feet?
d. Will the impurities ever be com pletely removed? Explain.
Solve each inequality. (Lesson 2-6)
79. x 2 — 3x — 2 >
8
82. - - 3)(A - 4), < 0
(x - 5)(x - 6y
80.4 > —(x — 2 ) 3 + 3
8 1 .| + 3 > ^
83.V 2x + 3 - 4 < 5
84. V * - 5 + V * + 7 < 4
85 , 2a
a —9
»
a _ _ — ^6—
a+9
_ 2 g ----------- 2 g _ _ ^
a2 —81
2q + 3
g7 _ 4 -------- z + 6 _ ^
2q - 3
z-2
z+ 1
Graph and analyze each function. Describe its domain, range, intercepts, end behavior,
continuity, and where the function is increasing or decreasing. (Lesson 2-1)
i
-88. f(x ) = - ~ x 7
89. g(x) = 3x ~ 6
90. h{x) = 2x 4
>
MICROBIOLOGY One model for the population P of bacteria in a sample after f days is
91.
1
given by P(t) = 1000 — 19.75f -I- 20f 2 — i f 3. (Lesson 1-2)
a. W hat type of function is P(f)?
b. W hen is the bacteria population increasing?
t
c. W hen is it decreasing?
92. SAT/ACT The table below shows the per unit revenue
and cost of three products at a sports equipm ent
factory.
Revenue
per Unit ($)
Cost
per Unit ($)
football
f
4
baseball
b
3
soccer ball
6
y
Product
If profit equals revenue minus cost, how m uch profit
do they make if they produce and sell two of each
item?
94. REVIEW The curve represents
a portion of the graph of
w hich function?
A y = 50 — x
B y = log x
C y = e ~x
95.
REVIEW A radioactive elem ent decays over time
according to
y = x (\ )A
93.
A 2 f + 2 b — 2y — 2
C f + b - y - 1
B 2y — 2b — 2/ — 2
D b+ 2/ +y-7
W hat is the value of n if log 3 3 4 ' 1 ~ 1 = 11?
F 3
G 4
H
6
J 12
V_____________________________________________
180
w here x = the num ber of grams present initially and
t = time in years. If 500 grams w ere present initially,
how m any grams will remain after 400 years?
| Lesson 3 -2 j L o g a rith m ic Functions
F 12.5 grams
H 62.5 grams
G 31.25 gram s
J 125 grams
•
You evaluated
logarithmic
expressions with
different bases.
(Lesson 3-2) s
•
Plants take in carbon-14 through photosynthesis, and animals
and humans take in carbon-14 by ingesting plant material.
When an organism dies, it stops taking in new carbon, and
the carbon-14 already in its system starts to decay. Scientists
can calculate the age of organic materials using a logarithmic
function that estimates the decay of carbon-14. Properties of
logarithms can be used to analyze this function
■f Apply properties of
I logarithms.
2
Apply the Change of
Base Formula.
........
Properties of Logarithms
Recall that the follow ing properties of exponents, where b, x,
and y are positive real numbers.
Product Property
b x .b v = b x + y
Quotient Property
Power
¥ - = bx ~y
bv
Property
(bx)y = b xv
Since logarithm s and exponents have an inverse relationship, these properties of exponents imply
these corresponding properties of logarithms.
K eyC oncept Properties of Logarithms
If b, x, and y are positive real numbers, b j= 1, and p is a real number, then the following statements are true.
Product Property
log^xy = lo g „ x + log6 y
Quotient Property
ln"
iog6 f* -= iog6 * - i o g 6 y
Power Property
log6 / ' , = p lo g i, /
You w ill prove the Quotient and Power Properties in Exercises 113 and 114.
To show that the Product Property of Logarithm s is true, let m = log b z and n = log(j y.
Then, using the definition of logarithm, b m = x and bn = y.
logfc xy = logb b mb n
x = 6 raand y = b"
= log b bm + "
Product Property of Exponents
= rn + n
Inverse Property of Logarithms
= lo g b x + lo g b y
m = log ^xa n d n ~ logf iy
These properties can be used to express logarithm s in terms of other logarithms.
m
m
m
Use the Properties of Logarithms
Express each logarithm in terms of In 2 and In 3.
b. I n f
a. In 54
In 54 = In (2 •3 3)
f
54 = 2 * 3
In |- = In 9 — In
8
Quotient Property
= In 2 + In 3 3
Product Property
= In 3 2 - In 2 3
32 = 9 and 2 3 = 8
= In 2 + 3 In 3
Power Property
= 2 1 n 3 -3 1 n 2
Power Property
GuidedPractice
Express each logarithm in terms of log 5 and log 3.
1A. log 75
1B.log 5.4
181
The Product, Quotient, and Power Properties can also be used to sim plify logarithms.
Simplify Logarithms
Evaluate each logarithm.
a. log4\/64
Since the base of the logarithm is 4, express V 6 4 as a power of 4.
log4\^64 = log 4 645
l
= log 4 (4:3'5
Rewrite using rational exponents.
4 3 = 64
3
Math HistoryLink
Joost Burgi
(1550-1617)
= log4 45
= |l o g 44
Power Property of logarithms
Iv o r I
A Swiss mathematician, Burgi
logx x = 1
was a renowned clockmaker who
also created and designed
b.
astronomical instruments. His
5 In e2 — In e3
greatest works in mathematics
5 In e2 — In e3 = 5(2 In e) — 3 In e
came when he discovered
logarithms independently
from John Napier.
f
Power Property of Logarithms
= 10 In e — 3 In e
Multiply.
= 10(1) - 3(1) or 7
in e — 1
GuidedPractice
2A. log6^ 3 6
2B. In e9 + 4 In e3
The properties of logarithms provide a w ay of expressing logarithm ic expressions in forms that
use simpler operations, converting m ultiplication into addition, division into subtraction, and
powers and roots into multiplication.
B A S I S S ® * Expand Logarithmic Expressions
Expand each expression.
a. log 12xsy ~2
The expression is the logarithm of the product of 12, x 5, and y 2
log 12x 5 y -
b.
In
2
= log 12 + log x 5 + log y ~ 2
Product Property
= log 12 + 5 log x — 2 log y
Power Property
*r 2
V4.v +
1
The expression is the logarithm of the quotient of x 2 and V 4x + 1.
In
V4 j + 1
: = In x 2 — In \j4x +
1
Quotient Property
= In x 2 — In (4x + 1)2
V 4 x + 1 = (4
= 2 In x — —In (4x + 1)
Power Property
x
+ 1 )£
►GuidedPractice
3A. log 1 3 6 a 3bc4
182
j Lesson 3-3 | P roperties o f Logarithm s
3B. In
3y + 2
4</y
The same m ethods used to expand logarithm ic expressions can be used to condense them.
.1 Condense Logarithmic Expressions
Condense each expression,
a. 4 log3 x - | log3 (x + 6)
I
1
4 log3 * - 3 lo§3 (* + 6) = log3 x4
-
= l°g3 x 4 ~
log3 (X + 6) 3
Power Property
1o83 V x + 6
(x-f 6)3 = 1 /7 + 6
A
= 1°g3
WatchOut!
Logarithm of a Sum The
logarithm of a sum or difference
does not equal the sum or
difference of logarithms. For
Quotient Property
\Jx + 6
= lo g s'
>
b.
6 In (x — 4)
6
example, In (x ± 4) ± in x ± In 4.
+
Rationalize the denominator.
x + 6
3 In x
In (x — 4) + 3 In x = In (x - 4 ) 6 + In x 3
= In x 3(x — 4 ) 6
Power Property
Product Property
y GuidedPractice
4B. In (3x + 5) — 4 In x — In (x — 1)
4A. —5 log 2 (x + 1) + 3 log 2 (6 x)
C hange of Base Form ula
Som etimes you m ay need to w ork w ith a logarithm that has an
inconvenient base. For exam ple, evaluating log 3 5 presents a challenge because calculators
have no key for evaluating base 3 logarithm s. The Change of Base Form ula provides a way of
changing such an expression into a quotient of logarithm s w ith a different base.
2
K eyC oncept Change of Base Formula
For any positive real numbers a, 6, and
1, b
x, a
1,
loga*
log„*= log
ab'
You w ill prove the Change of Base Formula in Exercise 115.
M ost calculators have only two keys for logarithm s, lLO G | for base 10 logarithm s and [LN] for base e
logarithms. Therefore, you will often use the Change of Base Formula in one of the follow ing two
forms. Either method will provide the correct answer.
log*
logb
StudyTip
Check for Reasonableness
You can check your answer
£,tau y
3 1,47. Because 3147 = 5,
In x
Tnfc
Use the Change of Base Formula
Evaluate each logarithm,
b. log] 6
a. log3 5
2
in Example 5a by evaluating
the answer is reasonable.
log b x =
l
s
10835
1115
Change of Base Formula
lo g i 6 =
log 6
log j
~ 1 .4 7
Use a calculator.
-2 .5 8
Use a calculator.
p GuidedPractice
5A. l o g 78 4 2 1 2
5B. log 15 33
5C. log l 10
You can use properties of logarithm s to solve real-world problem s. For exam ple, the ratio of the
frequencies of a note in one octave and the same note in the next octave is 2 :1 . Therefore, further
octaves will occur at 2 " times the frequency of that note, w here n is an integer. This relationship
can be used to find the difference in pitch betw een any two notes.
H J 2 B S S B Use the Change of Base Formula
MUSIC The musical cen t («t) is a unit of relative pitch. One octave consists of 1200 cents.
V«
M u s ic a l C e n ts
IV
F
f
I I I I
G
A
B
c
2f
semitone
1000whole tone -2 0 0 0 minor third - 3 0 0 0 - 4 0 0 0 major th ird - 5 0 0 ? fo u rth -------- 7 0 0 0 fifth ----------12000 octave-------
The formula to determine the difference in cents betw een two notes with beginning
frequency a and ending frequency b is n = 1200|log2 -2-j. Find the difference in pitch
Standard pitch, also called concert
pitch, is the pitch used by
orchestra members to tune their
between each of the following pairs of notes,
a. 493.9 Hz, 293.7 Hz
instruments. The frequency of
standard pitch is 440 hertz, which
Let a = 493.9 and b = 293.7. Substitute for the values of a and b and solve.
is equivalent to the note A in the
fourth octave.
n = 1200|log 2
Source: Encyclopaedia Britannica
=
1200
=
1200
KS?)
Original equation
a = 493.9 and b = 293.7
log 493.9
293.7
log 2
« 899.85
Simplify.
The difference in pitch betw een the notes is approxim ately 899.85 cents.
b. 3135.9 Hz, 2637 Hz
Let a = 3135.9 and b = 2637. Substitute for the values of a and b and solve.
n=
1 2 0 0
(log 2 f )
-1200(log2f f l | S )
Original equation
a = 3 1 3 5 .9 and b = 2637
3135.9
=
1200
lo8 2637
log 2
: 299.98
Simplify.
The difference in pitch betw een the notes is approxim ately 299.98 cents.
►GuidedPractice
6.
PHOTOGRAPHY In photography, exposure is the am ount of light allow ed to strike the film.
Exposure can be adjusted by the num ber of stops used to take a photograph. The change in
the number of stops n needed is related to the change in exposure c by n = log 2 c.
A. How many stops would a photographer use to triple the exposure?
B. How many stops would a photographer use to get i the exposure?
184
| Lesson 3-3 ; Properties o f Logarithm s
Express each logarithm in terms of In 2 and In 5. (Example 1)
1
.
4
2. In
2 0 0
4. In 12.5
5.
6.
°-8
7. ln
4
8. In
2 0 0 0
30. In
31.
log3 l 7 ^ L =
y j3 q - 1
32. In
33.
lo g jj ab~ 4c 12d 7
34. log 7 h2j n k~ 5
35.
log 4 lOt uv~
36. log^ a b
W+2
Mf5
\/l - 3d
1 .6
Express each logarithm in terms of In 3 and In 7. (Example 1)
e
10. i 49
ln 8l
9. In 63
11. i n f
12. In 147
13. In 1323
14. In 3 ^
729
15.
log 9 6 x 3 y 5z
29.
3. In 80
~2 ~
Expand each expression. (Example 3)
, 2401
ln ^ r
37. In If b?c
\lb - 9
38. log2 > + 2
“ V l —5x
Condense each expression. (Example 4)
16. In 1701
39. 31og5 * - I l o g 5 ( 6 - x )
17. CHEMISTRY The ionization constant of water Kw is the
product of the concentrations of hydrogen (H +) and
hydroxide (O H - ) ions.
Nonionized
After Ionization
0
5 log 7^2x) - 1 log 7 (5x + 1) '
V £ %
4 1 . 7 log^ a + lbg 3 b - 2 log 3 (8 c)
4 In (x + 3)4 - | In (4x + 7)
Water Molecule
43. 2 log 8 (9x) - log 8 (2x - 5)
^
~ >’i&
h 2°
qf
Hydroxide
Hydrogen
Ion
Ion
The formula for the ionization constant of water is
K w = [H +][O H - ], where the brackets denote
concentration in moles per liter. (Example 1)
In 13 + 7 In a — 1 1 In b + In c
45. 2 log 6 (5a) + log 6 b + 7 log 6 c
log 2 ^ - log 2 y — 3 log 2
47. J i n
(2
a- b ) -
2
In (3 b + c)
l o g 3 4 - | l ° g 3 (6 x — 5)
a. Express log K w in terms of log [H +] and log [O H - ].
b.
C.
The value of the constant K w is 1 x 1 0 -14. Simplify
your equation from part a to reflect the num erical
value of K w.
If the concentration of hydrogen ions in a sample
of water is 1 x 1 0 - 9 moles per liter, what is the
concentration of hydroxide ions?
18. TORNADOES The distance d in miles that a tornado travels
w — 65
is d = 1 0 9 3 , where w is the wind speed in m iles per
hour of the tornado. (Example 1)
Evaluate each logarithm . (Example 5)
49. log 6
^ 0 ? log 3
1 4
51. l° g 7 5
(
10
log 1 2 8
2
53. log ] 2 145
54. log 2 2 400
55. l° S io o
56. l o g i f
log 1 3 0 0 0 13
57. lo g - 2 8
a. Express w in terms of log d.
b.
If a tornado travels 100 miles, estimate the wind speed.
Evaluate each logarithm. (Example 2)
In e 2 — In e 12
19. log5 \/25
20.
21. 9 In e3 + 4 In e 5
22. log2 V 32
23. 2 log 3 \/27
24. 3 log 7 \/49
25. 4 log 2 V 8
26. 50 log 5 V l2 5
27. log 3 V 243
28. 36 In e0 5 — 4 In e:
8
59) COMPUTERS Com puter programs are written in sets of
instructions called algorithms. To execute a task in a
com puter program, the algorithm coding in the program
m ust be analyzed. The running time in seconds R that it
takes to analyze an algorithm of n steps can be modeled
by R = log 2 n. Example 6)
a. Determ ine the running time to analyze an algorithm
of 240 steps.
b.
To the nearest step, how many steps are in an
algorithm w ith a running time of 8.45 seconds?
Si,
'.'1.
-...
I
185
60. TRUCKING Bill's Trucking Service purchased a new
Condense each expression.
delivery truck for $56,000. Suppose f = log^ _ r) ^
76.
4
4
In x +
In y + f - In z
4 ^ 4
represents the time t in years that has passed since the
purchase given its initial price P, present value V, and
annual rate of depreciation r. (Example 6)
77. log 2 15 +
a. If the truck's present value is $40,000 and it has
78. In 14 - | In 3x - | In (4 - 3x)
depreciated at a rate of 15% per year, how much time
has passed since its purchase to the nearest year?
b. If the truck's present value is $34,000 and it has
depreciated at a rate of 1 0 % per year, how m uch time
has passed since its purchase to the nearest year?
6
log 2 x — 1 ^
2
* - } l o g 2 (x + 3)
79. 3 log 6 2x + 9 log 6 y - f log 6 * - f !og 6 y 80. log 4 25 - | log 4 x - f log4 y - f log 4 (z + 9)
81. ■| In x +
In (y +
8
) - 3 In y - In
(1 0
- x)
63. log 3
64.
66.
l° g i2 1 77
65.
lo § 5
124
O
Cvj
OO
67.
Use the properties of logarithm s to rewrite each logarithm
below in the form a In 2 + b In 3, where a and b are
constants. Then approximate the value of each logarithm
given that In 2 w 0.69 and In 3 w 1.10.
o
10
68.
lo®4 1^5
69. In \Jx\x + 3)
70. log 5
11
@
73.
lo S l4
72. In
\Jx5(8x -
1
‘7 /x 3 y2(z -
)
1
x2y5
In 4
In 48
84. In 162
85. In 216
86.
87. I n f
- I
88.
89. i n f
9xLyz
Determine the graph that corresponds to each
( y - 5) 4
74. log 1 2
)
CO
CJ
62. log 2 13
crc
61. l° g 4 5
o
Estimate each 1
b.
5x
yjx7(x + 13)
75. EARTHQUAKES The Richter scale measures the intensity of
an earthquake. The m agnitude M of the seismic energy in
joules E released by an earthquake can be calculated by
M = | l o g —T73 & jo 4 4
c.
T h e R ic h te r S c a le
0 -1 .9
a
Detectable
only by
seismograph
2 -2 .9
3 -3 .9
4 -4 .9
5 -5 .9
i
Hanging
objects
may
swing
Similar to
passing
truck
vibrations
Small
unstable
objects fall
e
Furniture
moves
a. Use the properties of logarithms to expand the equation.
e.
b. W hat m agnitude would an earthquake releasing
7.94 x 10 1 1 joules have?
c. The 2007 Alum Rock earthquake in California released
4.47 x 10 1 2 joules of energy. The 1964 Anchorage
earthquake in Alaska released 1.58 x 10 1 8 joules of
energy. How m any times as great was the magnitude
of the Anchorage earthquake as the m agnitude of the
Alum Rock earthquake?
d. Generally, earthquakes cannot be felt until they reach a
magnitude of 3 on the Richter scale. How many joules
of energy does an earthquake of this m agnitude
release?
186 | Lesson 3-3 |
P roperties o f Logarithm s
£
90.
/(x) = In x + In (x + 3)
91 . f( x ) = In x — In (x + 5)
92.
f( x ) = 2 In (x + 1)
93. f( x ) = 0.5 In (x - 2)
94. f(x) = In (2 - x) + 6
95. / ( x ) = In 2x — 4 In x
Write each set of logarithmic expressions in increasing order.
96- loS3 T ' log3 T +
lo § 3 4 ' l o § 3 1 2 - 2
Simplify each expression.
m
lo g 3 4
(log 3
98. BIOLOGY The generation time for bacteria is the time that
it takes for the population to double. The generation time
G can be found using G =
-, where t is the time
3.3 log b f
period, b is the number of bacteria at the beginning of the
experiment, and / is the number of bacteria at the end of
the experiment. The generation time for m ycobacterium
tuberculosis is 16 hours. How long will it take 4 of these
bacteria to m ultiply into 1024 bacteria?
111. (log 5
7)(log 5 2)
1 2
) + (log 8
1 2
)
a. The intensity of the light perceived by a moviegoer
who sits at a distance d from the screen is given by
I = ~ , where
100.
y
2
112. MOVIES Traditional movies are a sequence of still pictures
which, if show n fast enough, give the viewer the
im pression of motion. If the frequency of the stills shown
is too small, the m oviegoer notices a flicker betw een each
picture. Suppose the m inim um frequency/at which the
flicker first disappears is given b y / = K log I, where I is
the intensity of the light from the screen that reaches the
viewer and K is the constant of proportionality.
Write an equation for each graph.
4
@ (lo g
)(log 6 13)
110. (log 4 9) + (log 4 2)
97. log 5 55, logs VlOO, 3 log 5 V 75
99.
6
k
is a constant of proportionality.
Show that/ = -K(log k — 2 log
V
d ).
projector
fU )
2
1u, u
O
1
12
,0 )
screen
\(1, J)
X
O
-2
TVJ
X
Uj . —
y(x J
f
101.
y
I
h(x)
V
I I
\<-, >
b. Suppose you notice the flicker from a movie projection
102 .
y
d ,(V
O
(1 o
O
X
and m ove to double your distance from the screen. In
terms of K, how does this m ove affect the value of/?
■
(x)J-(15(30. 1)
-I*
*
400 800 1200 1600
H.O.T. Problems
PROOF Investigate graphically and then prove each of the
following properties of logarithms.
103. CHEMISTRY pK a is the logarithmic acid dissociation
constant for the acid HF, which is com posed of ions H +
[H+][F~]
and F . The pK a can be calculated by pK a = —log — p -^ —-,
where [H +] is the concentration of H + ions, [F ] is the
concentration of F _ ions, and [HF] is the concentration of
the acid solution. All of the concentrations are measured
in moles per liter.
a. Use the properties of logs to expand the equation for
pK a.
b. W hat is the p K a of a reaction in which [H +] = 0.01
moles per liter, [F _ ] = 0 . 0 1 m oles per liter, and
[HF] = 2 moles per liter?
115.
logs*
PROOF Prove that logb x ■
l°g a b~
116. REASONING How can the graph of g ( x ) = log 4 x be
obtained using a transform ation of the graph of
f{x )
m +][F-
[H F ]
d. Aldehydes are a com m on functional group in organic
molecules. Aldehydes have a p K a around 17. To what
K a does this correspond?
104. In ln(ee6)
105. 10loge'"4
106. 4 log17 17logl0 100
107. c ^ 4'"2
x
can In x not be
118. ERROR ANALYSIS Omar and N ate expanded log 2 (-y-)
using the properties of logarithm s. Is either of them
correct? Explain.
-. If a substance has a
pK a = 25, what is its K a?
X?
117. CHALLENGE If x £ N, for w hat values of
sim plified?
c. The acid dissociation constant K a of a substance can be
calculated by K a =
114. Power Property
113. Q uotient Property
Omar:
4 log 2 x + 4 log 2 y — 4 log 2 z
Nate:
2 log 4 x + 2 log 4 y — 2 log 4 z
PROOF Use logarithm ic properties to prove
log 5 (nt)2
2 log n log 4 + 2 log f log 4
lo S 4 7
log 5 log t - log 5 log r
120. WRITING IN MATH The graph of g ( x ) = l o g b x is actually
a transform ation of/(x) = log x. Use the Change of Base
Form ula to find the transform ation that relates these two
graphs. Then explain the effect that different values of b
have on the com m on logarithm graph.
Lr~
187
Spiral Review
Sketch and analyze each function. Describe its domain, range, intercepts, asym ptotes, end
behavior, and where the function is increasing or decreasing. (Lesson 3-2)
1 2 1 ./(x) = log 6 x
122.
123.
g(x) = lo g i x
h(x) = log 5 x - 2
Use the graph o f /(x ) to describe the transform ation that yields the graph of g ( x ) . Then sketch
the graphs of f ( x ) and g ( x ) . (Lesson 3-1)
124. /( x ) = 2X; g(x) = —2X
125. f( x ) = 5*; g(x) = 5* + 3
126. /( x ) = (I )* ; * (x) = ( J ) * -
2
127. GEOMETRY The volume of a rectangular prism with a square base is fixed at 120 cubic
feet. (Lesson 2-5)
a. Write the surface area of the prism as a function A(x) of the length of the side of the
square x.
b.
Graph the surface area function.
C. W hat happens to the surface area of the prism as the length of the side of the square
approaches 0 ?
Divide using synthetic division. (Lesson 2-3)
128. (x 2 - x + 4) -r (x - 2)
129. (x 3 + x 2 - 17x + 15) + (x + 5)
130. (x 3 - x 2 + 2) h- (x + 1)
Show th a t/a n d g are inverse functions. Then graph each function on the same graphing
calculator screen. Lesson 1-7)
1 3 1 ./( x ) = - | x + I
1 3 2
g (x) = - § * + \
1 3 3 ./ ( x ) = (x — 3 ) 3 +
./ W = _ L _
g(x) = \ - 2
4
g(x) = \/x — 4 + 3
134. SCIENCE Specific heat is the amount of energy per unit of m ass required to raise the
temperature of a substance by one degree Celsius. The table lists the specific heat
in joules per gram for certain substances. The amount of energy transferred is
given by Q = cmT, where c is the specific heat for a substance, m is its mass, and
T is the change in temperature. (Lesson 1-5)
a. Find the function for the change in temperature.
Substance
aluminum
Specific Heat (j/g )
0.902
gold
0.129
mercury
0.140
b.
iron
0.45
W hat is the parent graph of this function?
ice
2.03
C.
What is the relevant domain of this function?
water
4.179
air
1.01
135. SAT/ACT If b + 0, let a A b =
If x A y = 1, then
8
— 2 log 5 3 equal?
which statem ent must be true?
A log 5 2
C log 5 0.5
A x= y
D x > 0 and y > 0
B log 5 3
D 1
B x = -y
E x = |y|
C x 2 —y 2 = 0
136. REVIEW Find the value of x for log 2 (9x + 5) = 2 +
log 2 (x 2 - 1 ).
F - 0 .4
G 0
188
137. To w hat is 2 log 5 12 — log 5
H 1
J 3
| Lesson 3-3 | P roperties o f Logarithm s
138. REVIEW The w eight of a bar of soap decreases by
2.5% each tim e it is used. If the bar of soap weighs
95 gram s when it is new, what is its weight to the
nearest gram after 15 uses?
F 58 g
G 59 g
H 65 |
J
93 3
Mid-Chapter Quiz
Lessons 3-1 through 3-3
Sketch and analyze the graph of each function. Describe its domain,
range, intercepts, asymptotes, end behavior, and where the function is
©
Evaluate each expression. (Lesson 3-2)
10.' log2 64
fix) = 5 ~*
:* .- ( ! ) ' + 3
3.) log 0.001
Use the graph of f(x) to describe the transformation that results in the
graph of g(x). Then sketch the graphs of f(x) and g{x). (Lesson 3-1)
14. TECHNOLOGY The number of children infected by a virus can be
modeled by c(d) = 4.9 + 11.2 In d, where d is the number of days
since the first child was infected. About how many children are
infected on day 8? (Lesson 3-2)
3- f(x) = (I)" ; g(x) = ( f p
f c \ f ( x ) = 3/ ; g(x) = 2 - 3 x ~ 2
f 5X f(x) = ex\ g(x) = - e x - 6
6.
Evaluate each function for the given value. (Lesson 3-2)
f(x) = 10*; g(x) = 10 2*
7. MULTIPLE CHOICE In the formula for compound interest
A = p (l +
jj ) nf, which variable has NO effect on the
amount of time it takes an
16.
H{a) = 4 log
2a
8; a = 2 5
investment to double? (Lesson 3-1)
A P
Cn
B r
Dt
8. FINANCIAL LITERACY Clarissa has saved $1200 from working
summer jobs and would like to invest it so that she has some extra
money when she graduates from college in 5 years. ;Lesson 3-1)
a. How much money will Clarissa have if she invests at an annual
rate of 7.2% compounded monthly?
b.
15. T(x) = 2 In (X + 3); x = 18
Express each logarithm in terms of In 3 and In 4. {Lesson 3-3)
17. In 48
18. In 2.25
64
27
19.
In
20.
in A
How much money will Clarissa have if she invests at an annual
rate of 7.2% compounded continuously?
21.
1 . MULTIPLE CHOICE The parent function for the graph shown is
f(x) = log 2 x.
y +
i
i
i 4, 1)-f
0
a. If there is initially 75 grams of the substance, how much of the
substance will remain after 14 years?
b.
i
—
CHEMISTRY The half-life of a radioactive isotope is 7 years.
(Lesson 3-3)
After how many years will there be ^ of the original amount
remaining?
c. The time it takes for a substance to decay from N0 to N can be
i'x
-4
modeled by t - 7 log05 -^-.Approximately how many years will
H
0
'r
The graph contains the given point and has the vertical asymptote
shown. Which of the following is the function for the graph?
it take for any amount of the radioactive substance to decay to -j
its original amount?
Expand each expression. (Lesson 3-3)
22. log 3 V * 2/ 3'?5
23. log9 ^
(Lesson 3-2)
F
G
f(x) =
f(x) =
log2 ( x + 3 ) + 1
log2 ( x - 4)+ 1
Condense each expression. (Lesson 3-3)
H
f(x) =
-lo g 2 ( x - 3 ) + 1
24. 5 log4 a + 6 log4 b - i log4 7c
J
f(x ) =
log2 (x - 3)- I - 1
25. 2 log ( x + 1) - log (x2 - 1)
connectED .m cgraw -hill.com I
§
189
•
You applied the
inverse properties of
exponents and
logarithm s to
sim plify expressions.
(Lesson 3-2)
•
A pply the O ne-to-
The intensity o f an earthquake can be
One Property of
calculated using /? = log j + B, w here R
1
Exponential Functions
is the R ichter scale number, a is th e a m plitude
to solve equations.
o f th e ve rtical ground m otion, 7“ is th e period
o f th e seism ic w ave in seconds, and 6 is a
Apply the O ne-to-O ne
2
fa c to r th a t accounts fo r th e w e akening o f the
Property of
seism ic w aves.
Logarithm ic Functions
to solve equations.
i | One-to-One Property of Exponential Functions
In Lesson 3-2, exponential functions
1 were show n to be one-to-one. Recall from Lesson 1-7 that if a fu nction/ is one-to-one, no
i/-value is matched w ith more than one x-value. That is,/(a) = f( b ) if and only if a = b. This leads
us to the following One-to-One Property of Exponential Functions.
K eyC oncept One-to-One Property of Exponential Functions
W ords
For b > 0 and b ± 1, b x = b y if and only if x = y.
E xam ples
I f 3 * = 3 5, th e n * = 5 .
If log x = 3, then 10IO9x= 103.
This is also known as the Property of Equality for exponential functions.
The if and only if wording in this property implies tw o separate statements. One of them, bx = by if
x = y, can be used to solve some sim ple exponential equations by first expressing both sides of the
equation in terms of a com m on base.
Solve Exponential Equations Using One-to-One Property
Solve each equation,
a. 3 6 T+ 1 =
6
T+ 6
36* + 1 = 6X+ 6
(62) x + 1 = 6X+ 6
folx + 2 _fox + 6
2x -f- 2 — x
6
Original equation
62 = 36
Power of a Power
One-to-One Property
x + 2 = 6
Subtract
x from each side.
x = 4
Subtract
2 from each side. Check this solution in the original equation.
„. ( i f = 6 « 5
Hn
'vO
II
u
5 B
Original equation
I
—C
2
= (2
2~c
=
2~ 1 = i 26 = 64
6) 2
Power of a Power
2 3
—c = 3
c
►
= -3
V .............. .......
Lesson 3-4
Solve for c. Check this solution in the original equation.
GuidedPractice
1A. 16* + 3 =
190
One-to-One Property
4 4x
+7
..............
( t r 5= ( #
Another statem ent that follow s from the O ne-to-One Property of Exponential Functions, if x = y,
then bx = by, can be used to solve logarithmic equations such as log 2 x = 3.
log 2 x = 3
2 lo§ 2
Original equation
x —23
One-to-One Property
Inverse Property
This application of the One-to-One Property is called exponentiating each side of an equation. Notice
that the effect of exponentiating each side of log 2 x = 3 is to convert the equation from logarithmic
to exponential form.
B J 2 H S I3 I3 0 l^ o*ve Logarithmic Equations Using One-to-One Property
Solve each logarithmic equation. Round to the nearest hundredth,
a.
StudyTip
In x = 6
Method 1
Use exponentiation.
In x = 6
Original equation
gin x _ £6
Exponentiate each side.
x = e6
Write in exponential form,
x = e6
Inverse Property
x ~ 403.43
Use a calculator.
x ~ 403.43
Use a calculator.
CHECK In 403.43 ~
6
Method 2
In x =
Original equation
6
✓
Solutions to Logarithmic
Equations While it is always a
good idea to check your solutions
to equations, this is especially true
b. 6 + 2 log 5x = 18
6
+ 2 log 5x = 18
of logarithmic equations, since
2 log 5x = 12
logarithmic functions are only
defined on the set of positive real
log 5x =
Original equation
Divide each side by 2.
6
numbers.
5x = 10 6
T_ i° i
5
x = 200,000
Simplify. Check this solution in the original equation.
c. log8 x 3 = 12
logg x 3 =
Original equation
1 2
Power Property
3 log 8 x = 12
log8 x = 4
x=
or 4096
8 4
Write in exponential form and simplify. Check this solution.
GuidedPractice
2A. - 3 In x = - 2 4
2C. log 3 (x 2 — 1) = 4
2B. 4 - 3 log (5x) = 16
i One-to-One Property of Logarithmic Functions
Logarithm ic functions are also onei to-one. Therefore, we can state the follow ing O ne-to-One Property of Logarithm ic Functions.
StudyTip
Property of Equality The One-to-
K eyC on cept One-to-One Property of Logarithmic Functions
Words
For
Examples
If log2 x = log2 6, th e n * = 6 .
b > 0 and
1, log()x = log,, y if and only if
x = y.
One Property of Logarithmic
Functions is also known as the
If ey = 2, then In
ey = In 2.
Property of Equality for
Logarithmic Functions.
One statem ent implied by this property is that log 6 x = log b y if x = y. You can use this statement to
solve som e sim ple logarithmic equations by first condensing each side of an equation into
logarithms w ith the same base.
A
v 3 ■
]
191
Solve Exponential Equations Using One-to-One Property
a. log4 x = log4 3
+
log4 (x
log 4 X = log 4 3
+
log 4 (x - 2)
log 4 x
=
log 4 3(x -
log 4 X = log 4 (3x
- 6
I
=
Original equation
2
)
Product Property
6
)
vO
II
*
CO
X
—2 x
-
2)
-
One-to-One Property
Subtract 3a-from each side.
Divide each side by —2. Check this solution.
x= 3
b.
log3 (x 2
3) = log3 52
+
log 3 (x 2 + 3)
=
Original equation
log 3 52
One-to-One Property
x 2 + 3 = 52
x2
=
49
x
=
±7
Take the square root of each side. Check this solution.
► GuidedPractice
v 3A. log 6 2x
log 6 (x 2 - x
=
+
3B. log 1 2 (x + 3) = log 1 2 x + log 1 2 4
2)
Another statem ent that follows from the O ne-to-One Property of Logarithm ic Functions,
if x = y, then logb x = log 6 y, can be used to solve exponential equations such as ex = 3.
Original equation
ex = 3
In ex = In 3
x = In 3
One-to-One Property
Inverse Property
This application of the One-to-One Property is called taking the logarithm o f each side of an equation.
While natural logarithm s are more convenient to use when the base of the exponential expression
is e, you can use logarithms to any base to help solve exponential equations.
Solve Exponential Equations
StudyTip
Alternate Solution The problem
in Example 4a could also have
Solve each equation. Round to the nearest hundredth,
a. 4* = 13
been solved by taking the log4 of
4X = 13
Original equation
each side. The result would be
log 4* = log 13
x = log4 13. Notice that when the
Change of Base Formula is applied,
Take the common logarithm of each side.
x log 4 = log 13
this is equivalent to the solution
Power Property
log 13
or about 1.85
log 4
log 13
b.
e4~
3* =
Divide each side by log 4 and use a calculator.
6
e4 ~ 3x =
Original equation
6
In e4 ~ 3x = In
6
Take the natural logarithm of each side.
4 — 3x = In
6
Inverse Property
_ In 6 —4
or about 0.74
-3
Solve for x and use a calculator.
p- GuidedPractice
4A.
192
8
* = 0.165
| Lesson 3 -4 j Exponential and L o g a rith m ic Equations
4B. 1.43" + 3.1 = 8.48
4C.
e 2 + 5w =
12
B E 2 S H E E S°lve in Logarithmic Terms
Solve 4 3*
x. Round to the nearest hundredth.
1 = 32
Solve Algebraically
^ 3 x — 1 _o l — x
In 4 3x -
1
Original equation
= In 3 2 - x
(3x — 1 ) In 4 = (2 — x) In 3
Power Property
3 x ln 4 — In 4 = 2 1 n 3 — x l n 3
3x In 4 + x In 3 = 2 In 3 + In 4
Isolate the variables on the left side of the equation.
x(3 In 4 + In 3) = 2 In 3 + In 4
x(ln 4 3 + In 3) = In 3 2 + In 4
Power Property
x In [3(43)] = In 36
WatchOut!
Product Property
x In 192 = In 36
Sim plifying Notice that the
x =
In 192
Quotient Property cannot be used
to further simplify
3(43)
or about
= 192
Divide each side by In 192.
0 .6 8
Confirm Graphically
L
Graph y = 4 3x “ 1 and y — 3 2 ~ x. The point of intersection
of these two graphs given by the calculator is approximately
0 .6 8 , which is consistent w ith our algebraic solution.
In te rs e c tio n
K =.fiB ifi0i92 V=H.2S6J2H£
►GuidedPractice
1 0 ,1 0 ] scl: 1 by [- 1 0 ,1 0 ] scl: 1
Solve each equation. Round to the nearest hundredth.
5A. 6 lx + 4 = 5~ x + 1
5B. 43* + 2 = 62 * - i
Equations involving m ultiple exponential expressions can be solved by applying quadratic
techniques, such as factoring or the Q uadratic Formula, Be sure to check for extraneous solutions.
So,ve Exponential Equations in Quadratic Form
Solve e2* + 6ex — 16 = 0.
e2* + 6ex - 16 =
0
Original equation
+ 6u — 16 =
0
Write in quadratic form by letting
« 2
(u + 8)(u - 2) = 0
Factor.
u=
U = —c
TechnologyTip
pX _
<r
= _- »
Finding Zeros You can confirm
In e x = In
x = ln ( - 8 )
e2x+ 6 e * - 16 = 0 graphically
2
e* = 2
In e* = In ( - 8 )
the solution of
0 = 6*.
Replace
u with ex.
2
x = In 2 or about 0.69
Inverse Property
by using a graphing calculator to
locate the zero of
The only solution is x = In 2 because In (—8 ) is extraneous. Check this solution.
y = e 2x+ 6 e x - 16. The
>
graphical solution of about 0.69 is
consistent with the algebraic
solution of In 2 a 0.69.
CHECK
a 2x
„ 2 ( ln 2)
e1" 2* +
'
f
'
+ 6ex - 16 =
Jn 2
6
0
eln 2 — 16 = 0
2 2 + 6(2) — 16 = 0
'
Original equation
Replaces w ith In 2.
Power Property
Inverse Property
►GuidedPractice
Z tK *
K=.fi53iH7iH
v=o
[ - 5 , 5] scl: 1 by
[- 4 0 ,4 0 ] scl: 5
6A. e2* + 2ex =
J
8
6
B. 4 eix + 8e2x = 5
V_________________ ___ __
[7=
&
connectED.mcgraw-hill.com 1
193
Equations having m ultiple logarithmic expressions m ay be solved by first condensing expressions
using the Power, Product, and Q uotient Properties, and then applying the One-to-One Property.
Solve Logarithmic Equations
Solve In (x + 2) + In (3 x — 2) = 2 In 2x.
In (x + 2) + In (3x — 2) = 2 In 2x
Original equation
In (x + 2)(3x - 2) = In (2x ) 2
Product and Power Property
In (3x 2 + 4x — 4) = In 4x 2
Simplify.
3x 2 + 4x — 4 = 4x 2
One-to-One Property
0 = x 2 — 4x + 4
= (x -
0
x =
2
)(x -
Simplify.
2
)
Factor.
2
CHECK You can check this solution in the original equation,
or confirm graphically by locating the intersection
of the graphs of y = In (x + 2) + In (3x — 2) and
y = 2 In 2 x.
V
I n t e r s e c t io n
K = i.9 9 9 9 9 ? 7
►CheckYour Progress
I'
V=2.772SB6H
[-3 , 3] scl: 1 by [-3 , 3] scl: 1
7B. In (2x + 1) + In (2x - 3) = 2 In (2x - 2)
7A. In (7x + 3) — In (x + 1) = In (2x)
It m ay not be obvious that a solution of a logarithm ic equation is extraneous until you check it in
the original equation.
■
n
r n
i M
Extraneous Solutions
Solve log12 12* 4- log12 (x — 1) = 2.
log 1 2
1
)=
2
Original equation
1
)=
2
Product Property
l° g i 2 ( 1 2 x 2 — 1 2 x) =
2
1 2
x + log 1 2 (x log 1 2
1 2
x(x -
log 1 2 ( 1 2 x 2 -
1 2
x) = log 1? 1 2 2
l° g i 2 (12x 2 — 12x) = log 1 2 144
12x 2 - 12x = 144
12x 2 - 12x - 144 = 0
12(x - 4)(x + 3) = 0
x = 4 or x = —3
StudyTip
CHECK
Inverse Property
122 = 144
One-to-One Property
Factor.
log 1 2 12x + log 1 2 (x — 1) = 2
lo §12
12x +
lo §12
(X -
1) =
Identify the Domain of an
Equation Another way to check
for extraneous solutions is to
identify the domain of the
log 1 2 12(4) + lo g i 2 (4 — 1) = 2
lo g ^ 48 + log i 2 3 = 2
lo g i 2
1 2
(—3) + lo g i 2 ( - 3 -
1
)=
2
2
logl2 ( ~ 36) + lo§12 ( - 4 ) - 2
log i 2 48 •3 = 2
equation. In Example 8, the
domain of log, 2 12x is x > 0
lo g i2 144 = 2 /
while the domain of log12 (x — 1)
Since neither lo g j 2 (—36) nor lo g i 2 (—4) is
defined, x = —3 is an extraneous solution.
is x > 1. Therefore, the domain
of the equation is x > 1. Since
- 3 1, - 3 cannot be a solution
of the equation.
)►GuidedPractice
8A. In (6 y + 2) — In (y + 1) = In (2y — 1)\
194
| Lesson 3 -4 | Exponential and L o g a rith m ic Equations
8B. log (x - 12) = 2 + log (x - 2)
You can use inform ation about growth or decay to w rite the equation of an exponential function.
j2 J 5 2 3 2 S 0 2 E E D i E
Exponential Growth
INTERNET The table shows the num ber of hits
a new Web site received by the end of January
and the end of April of the same year.
Web Site Traffic
Number of Hits
a. If the num ber of hits is increasing at an exponential rate,
identify the continuous rate of growth. Then w rite an
exponential equation to model this situation.
January
April
125
2000
Let N(t) represent the num ber of hits at the end of t m onths and assume continuous
exponential growth. Then the initial num ber N 0 is 125 hits and the num ber of hits N after a
time of 3 m onths, the num ber of months from January to April, is 2000. Use this information
to find the continuous growth rate k.
N (t) = N 0e kt
Exponential Growth Formula
2000 = 125ek(3)
N(3) = 2000, Na = 125, and f = 3
16 = e3k
In 16 = In e 3k
In 16 = 3 k
Inverse Property
- k
~ K
0.924 « k
1x116
3
Use a calculator.
The num ber of hits is increasing at a continuous rate of approxim ately 92.4% per month.
Therefore, an equation m odeling this situation is N (t) = 125eo m it.
b.
Use your model to predict the num ber of m onths it will take for the Web site to receive
2 m illion hits.
N (t) = 125eom it
2,000,000 = 125ea924'
16,000 = e 0924t
Exponential growth model
N(t) = 2,000,000
In 16,000 = In e 0SZ4t
In 16,000 = 0.924f
Inverse Property
In 16,000
= t
0.924
10.48
t
Divide each side by 0.924.
Use a calculator.
According to this m odel, the Web site will receive 2,000,000 hits in about 10.48 months.
^ GuidedPractice
9. MEMORABILIA The table shows revenue from sales of T-shirts and other memorabilia sold by
two different vendors during and one w eek after the World Series.
Real-W orldLink
Championship hats and shirts are
printed for both teams before a
World Series M em orabilia Sales
Days after
Series
major athletic contest like the
Vendor A
Vendor B
(S)
Sales ($)
Sales
Bowl Championship Series. The
0
300,000
200,000
losing team’s merchandise is
7
37,000
49,000
often donated to nonprofit
organizations that distribute it to
families in need in other countries.
In 2007, an estimated $2.5 million
of unusable sports clothing was
donated.
Source: World Vision
A. If the sales are decreasing at an exponential rate, identify the continuous rate of decline for
each vend or's sales. Then write an exponential equation to model each situation.
B. Use your m odels to predict the World Series mem orabilia sales by each vendor 4 weeks
after the series ended.
C. Will the two vendors' sales ever be the same? If so, at what point in time?
K»connectED.m cgraw-hill.com ]
195
Exercises
4. 32J£- 1 = 4 X + 5
6. 123*
\j
+ 11
= 1442* + 7
\5 ( r - © r
7. 2 5 3 = 5
9. INTERNET The number of people P in m illions using two
different search engines to surf the Internet t weeks after
the creation of the search engine can be m odeled by
P j(f) = 1.5* + 4 and P 2 (f) = 2.25f ~ 33 , respectively. During
which week did the same number of people use each
search engine? :Example1)
10. FINANCIAL LITERACY Brandy is planning on investing
$5000 and is considering two savings accounts. The
first account is continuously com pounded and offers
a 3% interest rate. The second account is annually
compounded and also offers a 3% interest rate, but the
bank will match 4% of the initial investment. (Example 1)
a. Write an equation for the balance of each savings
account at time t years.
b. How many years will it take for the continuously
com pounded account to catch up w ith the annually
com pounded savings account?
c. If Brandy plans on leaving the m oney in the account
for 30 years, which account should she choose?
Solve each logarithmic equation. (Example 2)
11. In a = 4
vl2)
- 8
log b = - 6 4
13. In (—2) = c
14. 2 + 3 log 3d = 5
15. 14 + 20 In 7x = 54
^
17. 7000 In h = -2 1 ,0 0 0
18. - 1 8 log0 / = - 1 2 6
19. 12,000 log 2 k = 192,000
20. log 2 m4 = 32
100 + 500 lo g j g = 1100
32.
8
003
y^
1
= 51
33. 2e7x = 84
V- 1 = 3.4
34. 8.3e9-v = 24.9
35. eZx + 5 = 16
36. 2.5e* + 4 = 14
37. 0.75e3Ax - 0.
38. GENETICS PCR (Polymerase Chain Reaction) is a
technique com m only used in forensic labs to amplify
DNA. PCR uses an enzyme to cut a designated nucleotide
sequence from the DN A and then replicates the sequence.
The num ber of identical nucleotide sequences N after
t m inutes can be m odeled by N(t) = 100 • 1.17*. (Example 4)
a. At what tim e will there be 1 x 10 4 sequences?
b. A t what time will the DNA have been amplified to
1 m illion sequences?
y2x
+1_
+3
40.
1 1
_ 2$x - 4
42.
4
+ 3 _ g -x + 2
44.
5 3
gx + 2
rîx
yc
46.
6 x ~ 2 = 5 2x + 3
2$x + 6 _
^2x +
1
* + 1 —7 * ~ 1
* - 3 _ fr2x - 1
* —1 _
4
X+ 1
48. 6 X~ 2 = 9 X~
49. ASTRONOMY The brightness of two celestial bodies as seen
from Earth can be com pared by determ ining the variation
in brightness betw een the two bodies. The variation in
brightness V can be calculated by V = 2.512"^ mb,
where
is the m agnitude of brightness of the fainter
body and m b is the m agnitude of brightness of the
brighter body. (Example 5)
Celestial
Bodies
(2l| CARS If all other factors are equal, the higher the
displacement D in liters of the air /fuel mixture of an
engine, the more horsepower H it will produce. The
horsepower of a naturally aspirated engine can be
modeled by H = lo g j
31. e3x +
30. 3eix = 45
13 \5x + 4
1
X
I8 - x
CJ1
7
V+ 4 = 3 2 3t
1
5.
*+4_
8
II
4 9
2.
1
3.
^
% ^
X
4
N>
■|
Solve each equation. Round to the nearest hundredth.
(Example 4)
C\
29. 1.8* = 9.6
28. 6 X = 28
OO
Sun
803
C
S
Find the displacement
when horsepower is 200. (Example 2)
Full Moon
Venus
i
Jupiter
I
Mercury
|
Neptune
a. The Sun has m = —26.73, and the full M oon has
m = —12.6. Determ ine the variation in brightness
betw een the Sun and the full Moon.
22. log 6 (xz + 5) = log 6 41
23. log 8
( * 2
+ 11) = log 8 92
24. log 9 (x 4 - 3) = log 9 13
25. log 7 6x == log 7 9 + log 7 (x - 4)
26. loS 5x = log 5 (* +
6
) - log 5
4
27. logu 3* = logn (x + 5) - log u 2
196
Lesson 3 -4 | E xponential and L o g a rith m ic Equations
b. The variation in brightness betw een M ercury and
Venus is 5.25. Venus has a m agnitude of brightness
of —3.7. Determ ine the m agnitude of brightness of
Mercury.
C.
N eptune has a m agnitude of brightness of 7.7, and the
variation in brightness of N eptune and Jupiter is
15,856. W hat is the m agnitude of brightness of Jupiter?
Solve each logarithm ic equation. (Example 8)
50. e2* + 3ex - 130 = 0
51. e2* - - I5 e x + 56 = 0
75. log (29,995x + 40,225) = 4 + log (3x + 4)
52. e2* + 3e* =: -
2
53. b e2* - 5ex =
76. logi (|x) = - l o g , (x +
54. 9e2x — 3ex ■
=
6
55. 8eix - 15e2* + 7 = 0
77. log x = 3 — log (lOOx + 900)
e- x =
0
Solve each logarithm ic equation. (Example 7)
60. In x + In (x + 2) = In 63
62. In (3x + 1) + In (2x — 3) = In 10
63. In {x — 3) + In (2x + 3) = In ( - 4 x 2)
64. log (5x2 + 4) = 2 log 3x 2 — log (2x 2 — 1)
6
78- log 5 y - 3 = loS 5 ^ j
79. log 2x + log
(4
—
= 2 log (x - 2)
80. TECHNOLOGY A chain of retail com puter stores opened
2 stores in its first year of operation. After 8 years of
operation, the chain consisted of 206 stores. (Example 9)
61. In x + ln (x + 7) = l n l 8
65. log (x +
)- |
0
II
2 0
1
59. lie * - 51 -
8
*
58. 10ex - 15 -- 45e~x = 0
1
1
57. 2e5x
0
in
=
t-H
1
1
56. 2eSx + e4x --
6
) = log (8 x) — log (3x + 2)
a. Write a continuous exponential equation to model the
num ber of stores N as a function of year of operation t.
Round k to the nearest hundredth.
b. U se the m odel you found in part a to predict the
num ber of stores in the 12th year of operation.
66. In (4x 2 — 3x) = In (16x — 12) — In x
67. In (3x 2 - 4) + In (x 2 + 1) = In (2 - x 2)
68. SOUND Noise-induced hearing loss (NIHL) accounts for
25% of hearing loss in the United States. Exposure to
sounds of 85 decibels or higher for an extended period
can cause NIHL. Recall that the decibels (dB) produced
by a sound of intensity I can be calculated by
dB =
1 0
Sound
fireworks
31.623
jet plane
82. 5 + 5 log 100 x = 20
83. 6 + 2 loge2 x = 30
3.162
ambulance
84. 5 — 4 logi x = —19
85. 36 + 31og3 x = 60
0.316
rock concert
0.032
headphones
0.003
hair dryer
a. W hich of the sounds listed in the table produce
enough decibels to cause NIHL?
C.
Solve each logarithm ic equation.
316.227
Source: D angerous D ecibels
b.
a. Write a continuous exponential equation to model the
price of stock P as a function of year of trading f.
Round k to the nearest ten-thousandth.
b. Use the model you found in part a to predict the price
of the stock during the ninth year of trading.
V
log (------ -— — (Example 7)
° \ 1 x 1 0 “12/
Intensity (W /rrr)
81. STOCK The price per share of a coffee chain's stock was
$0.93 in a m onth during its first year of trading. During
its fifth year of trading, the price per share of stock was
$3.52 during the same month. (Example 9)
86. ACIDITY The acidity of a substance is determ ined by its
concentration of H + ions. Because the H + concentration
of substances can vary by several orders of magnitude,
the logarithmic pH scale is used to indicate acidity. pH
can be calculated by pH = —log [H +], where [H +] is the
concentration of H + ions in m oles per liter.
Determine the number of hair dryers that would
produce the same num ber of decibels produced by a
rock concert. Round to the nearest w hole number.
How many jet planes would it take to produce the
same number of decibels as a firework display? Round
to the nearest whole number.
Item
pH
ammonia
11.0
baking soda
8.3
human blood
7.4
water
7.0
milk
6.6
Solve each logarithmic equation. (Example 8)
apples
3.0
69. log 2 (2x — 6 ) = 3 + log 2 x
lemon juice
2.0
70. log (3x + 2) = 1 + log 2x
a. Determ ine the H + concentration of baking soda.
(7 1 ) log x = 1 - log (x - 3)
b. How m any times as acidic is m ilk than human blood?
72. log 50x = 2 + log (2x — 3)
C.
By how many orders of m agnitude is the [H +] of
lem on juice greater than [H +] of ammonia?
d.
How m any m oles of H + ions are in 1500 liters of
hum an blood?
73. log 9 9x - 2 = —log 9 x
74. log (x - 10) = 3 + log (x - 3)
I
197
GRAPHING CALCULATOR Solve each equation algebraically, if
possible. If not possible, approximate the solution to the
nearest hundredth using a graphing calculator.
87. x 3 = 2X
89. 3X= .x(5T)
104. 27 =
88. log 2 x = log 8 x
106. 1000 =
90. logj.5 = log 5 x
12
105. 22 =
10,000
107. 300 =
1 + 19e~f
pX
108. 16X + 4 X 91. RADIOACTIVITY The isotopes phosphorous-32 and
sulfur-35 both exhibit radioactive decay. The half-life
of phosphorous-32 is 14.282 days. The half-life of
sulfur-35 is 87.51 days.
In R
,
.
400
1 + 3e~2k
p -x
109. e, + e
e" — e
= 0
111 .
110. ! n (4* + ^ = 3
In (4x - 2)
a. Write equations to express the radioactive decay of
phosphorous-32 and sulfur-35 in terms of time t in
days and ratio R of remaining isotope using the
general equation for radioactive decay, A = t • _ Q^
where A is the num ber of days the isotope has
decayed and t is the half-life in days.
6
1 + L z l e -15
3
=6
ex — e
e + e-X
1
2
112. POLLUTION Som e factories have added filtering systems
called scrubbers to their smokestacks in order to reduce
pollution em issions. The percent of pollution P removed
after/ feet of length of a particular scrubber can be
0.9
m odeled by P =
1 + 70e~O28f
b. At what value of R will sulfur-35 have been decaying
5 days longer than phosphorous-32?
Scrubbers
Solve each exponential inequality.
92.
2 < 2* < 32
94.
1 < 8P < ■1
4096
~ 64
93. 9 < 3 y < 27
i -
95. — — <
2197
4000 > 5^ > 125
96.
10 < I0 d < 100,000
97.
98.
49 < 7 2 < 1000
9 9 .10,000 < 10° < 275,000
-i. > 4b > —
15 “
- 64
100.
-Chimney
stack
100
102. F0RENSICS Forensic pathologists perform autopsies to
determine time and cause of death. The time t in hours
I _____
T —R
since death can be calculated by f = — 1 0 In
98.6 - R,
where T is the temperature of the body and R, is the room
temperature.
a. A forensic pathologist measures the body temperature
to be 93°F in a room that is 72°F. W hat is the time of
death?
b. A hospital patient passed away 4 hours ago. If the
hospital has an average temperature of 75°F, w hat is
the body temperature?
c. A patient's temperature was 89°F 3.5 hours after the
patient passed away. Determine the room temperature.
103. MEDICINE Fifty people were treated for a virus on the
same day. The virus is highly contagious, and the
individuals m ust stay in the hospital until they have no
symptoms. The number of people p who show symptoms
after f days can be modeled by p = ----- 32'76n_-,.
3
3 r
1 + 0.03e
a. Graph the percent of pollution removed as a function
of scrubber length.
b. Determ ine the m axim um percent of pollution that can
be rem oved by the scrubber. Explain your reasoning.
C. Approxim ate the m axim um length of scrubber that a
factory should choose to use. Explain.
H.O.T. Problem s
113. REASONING W hat is the m axim um num ber of extraneous
solutions that a logarithmic equation can have? Explain
your reasoning.
114. OPEN ENDED Give an exam ple of a logarithmic equation
w ith infinite solutions.
1 $ CHALLENGE If an investm ent is m ade w ith an interest rate
r com pounded monthly, how long will it take for the
investm ent to triple?
116. REASONING H ow can you solve an equation involving
logarithmic expressions w ith three different bases?
117. CHALLENGE For w hat x values do the dom ains of
f i x ) = log (x 4 — x 2) and g(x) = log x + log x + log
(x — 1 )+ log (x + 1 ) differ?
a. How many show symptoms after 5 days?
b. Solve the equation for t.
C. How many days will it take until only one person
shows symptoms?
198
| Lesson 3 -4 | Exponential and L o g a rith m ic Equations
118. WRITING IN MATH Explain how to algebraically solve for
t in P = ■
1
+
M
’
Spiral Review
Evaluate each logarithm. Lesson 3-3)
119.
122.
log8 15
120. log2 8
121. log5 625
SOUND An equation for loudness L, in decibels, is L = 10 log 10 R, where R is the relative
intensity of the sound. (Lesson 3-2)
a. Solve 130 = 10 log10 R to find the relative intensity of a fireworks display w ith a
loudness of 130 decibels.
b. Solve 75 = 10 log 10 R to find the relative intensity of a concert w ith a loudness of
75 decibels.
c. How many times as intense is the fireworks display as the concert? In other words,
find the ratio of their intensities.
For each function, (a) apply the leading term test, (b) determ ine the zeros and state the
multiplicity of any repeated zeros, (c) find a few additional points, and then (d) graph
the function. (Lesson 2-2)
123./(x) = x3 - 8x2 + 7x
124./(x) = x 3 + 6 x 2 + 8x
125.
,
1
126. j( 1 2 a ) 3 = 1
„ _____
127. \Jx - 4 = 3
I
128. (3y)3 + 2 = 5
f( x ) = —x 4 + 6x3 — 32x
Use logical reasoning to determine the end behavior or limit of the function as x approaches
infinity. Explain your reasoning. (Lesson 1 -3)
129.
f( x ) = x w — x 9 + 5x8
130. g(x) = ^ +^ 2
131. h(x) = |( x - 3)2 - 1|
Find the variance and standard deviation of each population to the nearest tenth. Lesson 0-8)
132. (4 8 ,3 6 ,4 0 ,2 9 ,4 5 ,5 1 ,3 8 ,4 7 ,3 9 ,3 7 )
133. {321, 322, 3 2 3 ,3 2 4 ,3 2 5 , 326, 3 2 7 ,3 2 8 ,3 2 9 ,3 3 0 }
134. (43,56, 78, 8 1 ,4 7 ,4 2 ,3 4 , 22, 7 8 ,9 8 , 38, 4 6 ,5 4 , 67, 5 8 ,9 2 ,5 5 )
135.
SAT/ACT In a movie theater, 2 boys and 3 girls are
randomly seated together in a row. W hat is the
probability that the 2 boys are seated next to each
other?
B I
136.
REVIEW W hich equation is equivalent to log4 j t = x?
c
l 4
4
F 16=*
G
(£ )*-
H 4X = i h
16
J
416
= X
137. If 2 4 = 3X, then w hat is the approximate value of x l
A 0.63
C
2.52
B 2.34
D
2.84
138. REVIEW The pH of a person's blood is given by
pH = 6.1 -I- lo g 10 B — lo g 10 C, where B is the
concentration base of bicarbonate in the blood and
C is the concentration of carbonic acid in the blood.
D eterm ine w hich substance has a pH closest to a
person's blood if their ratio of bicarbonate to carbonic
acid is 17.5:2.25.
F lem on juice
G baking soda
H m ilk
J amm onia
Substance
pH
lemon juice
2.3
milk
6.4
baking soda
8.4
ammonia
:: 1 ' :
': 'r' : . ...:
[^conn ectE D .m cgraw -hill.com |
11.9
j
199 P ^ '
I
~
—
— —
Solving Exponential and
Logarithmic Inequalities
In Lesson 3-4, you solved exponential equations algebraically and confirmed solutions graphically. You can
Solve exponential and
logarithm ic inequalities
algebraically and
graphically.
use similar techniques and the following properties to solve inequalities involving exponential functions.
KeyConcept Properties of Inequality for Exponential Functions
Words
If b > 1, then b* > by if and only if
Example
If 5 / < 54, then
x<
x>
y, and bf1 < by if and only if
x < y.
4.
This property also bolds for < and > .
Activity 1
Exponential inequalities
Solve 52x~6 > 0.04* _ 3
Solve Algebraically
52* - 6 > 0 .0 4 *~3
\x~3
Original inequality
Rewrite 0.04 as
52* “ 6 > (5-2)* " 3
Rewrite
xp.x - 6 > $-2x + 6
Power of a Power
2 x - 6 > -2 x + 6
Property of Inequality for Exponential Functions
25
as
1
25'
y
or 5'2 so each side has the same base.
4x > 12
Addition Property of Inequalities
x > 3
Division Property of Inequalities
The solution set is [x \ x > 3, x e R or (3, oo).
Confirm Graphically
E T f m Replacing each side of the inequality w ith y
yields the system of inequalities y > 0.04* “ 3
and y < 52x~ 6.
Enter each boundary equation and select
the appropriate shade option.
ETHTO Graph the system. The x-values of the points
in the region where the shadings overlap is the
solution set of the original inequality. Using
the INTERSECT feature, you can conclude that
the solution set is (3, oo), w hich is consistent
w ith our algebraic solution set.
P lo ti
Plo tE
Plo ts
,' V i B . 0 4 A< X - 3 )
k V z B 5 A < 2 X -6 >
W j=
sVh =
W e=
W fi =
-0.5,4.5] scl: 0.5 by [-2, 3] scl: 0.5
Exercises
1.
16* < 8* + 1
4. 92* - 1 > 32* + 8
199A
| Lesson 3-4
2. 325* + 2 > 165*
3. 24* - 5 > 0.5* - 5
5. 343* ~ 2 < 49
6. 100* < 0.013* - 4
To solve inequalities that involve logarithms, use the following property.
K eyC oncept Logarithmic to Exponential Inequality
Words
b > 1, x > 0, and
b > 1, x > 0, and
then 0 < x < by.
If
log^ x >
If
log6 x <
y, then x > by.
y,
Example
log3 x < 5
0 < x < 35
log2 x > 3
x > 23
s . . ...........
J
This property also holds for < and > .
Activity 2
Logarithmic Inequalities
Solve log x < 2.
Solve Algebraically
log x < 2
Original inequality
0 < x < 102
Logarithmic to Exponential Inequality
0 < x < 100
Simplify.
The solution set is jx | 0 < x < 100, i e l
-25,225] scl: 25 by [-1, 4] scl: 0.5
Figure 3.4.1
1 or (0,100].
Confirm G raphically Graph the system of inequalities y < 2 and y > log x (Figure 3.4.1).
Using the TRACE and INTERSECT features, you can conclude that the solution set is (0,100]. v''
V_
To solve inequalities that involve logarithms with the same base on each side, use the following property.
K eyC oncept Properties of Inequality for Logarithmic Functions
Words
If
b > 1, then logft x >
x < y.
log6 y if and only if
x>
y, and logfi x < log6 y
if and only if
Example
If log2 x > log2 9, then
x>
9.
J
V
This property also holds for < and > .
Vl=ln(K+fi):
Activity 3
inequalities with Logarithms on Each Side
Solve In ( 3 x - 4) < In ( x + 6).
Solve Algebraically
In (3x - 4) < In (x + 6)
3x - 4 < x + 6
x <5
Original inequality
Property of Inequalities for Logarithmic Functions
Division Property of Inequalities
,1 and
Exclude all values of x such that 3x - 4 < 0 or x + 6 < 0. Thus, the solution set is x > 1—
Intersection
>
-6 and x < 5. This com pound inequality sim plifies to jx |1-|- < x
<
5, xe e| or | lp 5j.
H=5 llim illllllll IV=M S9?H 9£3
x
-1 ,9 ] scl: 1 by [-1 ,4 ] scl: 0.5
Confirm G raphically G raph the system of inequalities y < In (x + 6) and y > In (3x - 4)
(Figure 3.4.2). Using the TRACE and INTERSECT features, you can conclude that the solution set
Figure 3.4.2
Exercises
7.
10.
In (2x - 1) < 2
log (5x + 2) < log (x — 4)
8.
11.
log (3x — 8) > 6
In (3x — 5) > In (x + 7)
9.
12.
In 2x < - 1
log (x2 — 6) > log x
c o n n e c tE D .n ic g ra ^ iiT u o m 'l
1 9 9B
Modeling with Nonlinear Regression
•
You modeled data
using polynomial
functions.
•
(Lesson 2-1)
•
Model data using
exponential,
logarithmic, and
logistic functions.
While exponential growth is not a perfect
model for the growth of a human population,
government agencies can use estimates
from such models to make strategic plans
that ensure they will be prepared to meet
the future needs of their people.
I Linearize and analyze
data.
B
NewVocabulary
logistic growth function
linearize
° | Exponential, Logarithm ic, and Logistic M o delin g
In this lesson, we will use the
i exponential regression features on a graphing calculator, rather than algebraic techniques, to
model data exhibiting exponential or logarithmic growth or decay.
B B S IE Q 3 D Exponential Regression
POPULATION M esa, Arizona, is one of the fastest-grow ing cities in the United States. Use
exponential regression to model the M esa population data. Then use your model to estimate
the population of M esa in 2020.
'opulation of M esa, Arizona (thousands)
1
1900
1910
1920
1930
1940
1950
1960
1970
1980
1990
2000
1.6
3.0
3.7
7.2
16.8
33.8
63
152
288
396
Population
EflSflWI M ake a scatter plot.
Let P(f) represent the population in thousands of M esa t years after 1900. Enter and
graph the data on a graphing calculator to create the scatter plot (Figure 3.5.1). Notice
that the plot very closely resem bles the graph of an exponential growth function.
RTTjffln Find an exponential function to model the data.
W ith the diagnostic feature turned on and using ExpReg from the list of regression
models, we get the values shown in Figure 3.5.2. The population in 1900 is represented
by a and the growth rate, 6.7% per year, is represented by b. Notice that the correlation
coefficient r ~ 0.9968 is close to 1, indicating a close fit to the data. In the |Y=I menu, pick
up this regression equation by entering VARS 1, Statistics, EQ, RegEQ.
PTTTffl Graph the regression equation and scatter plot on the same screen.
Notice that the graph of the regression fits the data fairly well. (Figure 3.5.3).
E xpR e9
y = a * b Ax
a = .6 901414149
b = 1 .0 6 7 1 1 4 4 5 8
r i = .9 9 3 5 3 7 2 7 9 S
r = .9967634021
[0 ,1 3 0 ] scl: 10 by [- 5 0 , 500] scl: 50
[0 ,1 3 0 ] scl: 10 bv [- 5 0 . 500] scl: 50
Figure 3.5.1
Figure 3.5.2
EflSffn Use the model to make a prediction.
To predict the population of Mesa in 2 0 2 0 ,1 2 0 years after
1900, use the C A LC feature to evaluate the function for
P(120) as shown. Based on the m odel, Mesa will have
about 1675 thousand or 1.675 million people in 2020.
200
Lesson 3-5
Figure 3.5.3
►GuidedPractice
1. INTERNET The Internet experienced rapid growth in the 1990s. The table shows the number of
users in m illions for each year during the decade. Use exponential regression to model the
data. Then use your model to predict the num ber of users in 2008. Let x be the number of
years after 1990.
Year
1991
1992
1993
1994
1995
1996
1
1.142
1.429
4.286
5.714
10
Internet Users
1997
1998
1999
2000
21.429 34.286 59.143 70.314
W hile data exhibiting rapid growth or decay tend to suggest an exponential model, data that grow
or decay rapidly at first and then more slowly over time tend to suggest a logarithmic model
calculated using natural logarithm ic regression.
M
2 1 S
Logarithmic Regression
BIRTHS Use logarithm ic regression to model the data in the table about twin births in the
United States. Then use your model to predict w hen the num ber of twin births in the U.S.
will reach 150,000.
Num ber of Twin Births in the United States
Year
Births
1995
1997
1998
2000
2002
2004
2005
96,736
104,137
110,670
118,916
125,134
132,219
133,122
Let B(t) represent the num ber of twin births t years after 1990. Then create a scatter plot
(Figure 3.5.5). The plot resem bles the graph of a logarithm ic growth function.
LnR e9
y = a + b ln x
a = 3 8 4 2 8 .9 6 3 0 8
b = 3 5 0 0 0 .1 6 7 9
r 2= . 9 8 9 7 4 7 2 4 8 7
r = .9948604167
E3SH Calculate the regression equation using LnReg. The correlation coefficient r ~
0.9949
indicates a close fit to the data. Rounding each value to three decim al places, a natural
logarithm function that models the data is B(f) = 38,428.963 + 35,000.168 In x.
In the |Y=| menu, pick up this regression equation. Figure 3.5.4 shows the results of the
regression B(t). The number of tw in births in 1990 is represented by a. The graph of B(t)
fits the data fairly w ell (Figure 3.5.6).
Figure 3.5.4
ETTSm To find when the num ber of twin births will reach 150,000, graph the line y = 150,000
and the m odeling equation on the same screen. Calculating the point of intersection
(Figure 3.5.7), we find that according to this m odel, the num ber of tw in births will reach
150,000 when t ~ 24, w hich is in 1990 + 24 or 2014.
StudyTip
Rounding Remember that
the rounded regression equation
is not used to make our
prediction. A more accurate
predication can be obtained by
using the entire equation.
>
Ihttrsectien.uû^dûi.
v=isocmo
[- 1 , 2 0 ] scl: 1 by
[-2 0 ,0 0 0:1 5 0 ,0 0 0 ] scl: 20,000
[- 1 , 2 0 ] scl: 1 by
-20,000:150,000] scl: 20,000
Figure 3.5.5
[- 1 , 3 0 ] scl: 2 by
-20,000; 200,000] scl: 20,000
Figure 3.5.6
Figure 3.5.7
►GuidedPractice
2. LIFE EXPECTANCY The table shows average U.S. life expectancies according to birth year.
Use logarithm ic regression to model the data. Then use the function to predict the life
expectancy of a person born in 2020. Let x be the num ber of years after 1900.
1950
1960
1970
1980
1990
1995
2000
2005
68.2
69.7
70.8
73.7
75.4
75.8
77.0
77.8
co ^ 'ertiam c^ w ^ lT lM n n i
201
Exponential and logarithmic growth is unrestricted, increasing at an ever-increasing rate w ith no
upper bound. In many growth situations, however, the amount of growth is limited by factors that
sustain the population, such as space, food, and water. Such factors cause growth that w as initially
exponential to slow down and level out, approaching a horizontal asymptote. A logistic growth
function models such resource-limited exponential growth.
KeyConcept Logistic Growth Function
StudyTip
Logistic Decay If
then
A logistic growth function has the form
b < 0,
f(t) =
f(t) =
1 + ae -“
would represent logistic decay.
1 + ae -“
J
b, and care positive
c is the limit to growth.
Unless otherwise stated, all
where f is any real number, a,
logistic models in this text will
constants, and
represent logistic growth.
m
=
1 + aei—b t
Logistic growth functions are bounded by two horizontal asym ptotes, y ; : 0 and y = c. The lim it to
growth c is also called the carrying capacity of the function.
Logistic Regression
BIOLOGY Use logistic regression to find a logistic grow th function to model the data in the
table about the num ber of yeast grow ing in a culture. Then use your model to predict the
limit to the grow th of the yeast in the culture.
Yeast Population in a Culture
Time (h)
Yeast
E 0H
1
10
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
31
45
68
120
172
255
353
445
512
561
597
629
641
653
654
658
Let Y(t) represent the num ber of yeast in the culture after t hours. Then create a scatter
plot (Figure 3.5.8). The plot resem bles the graph of a logistic growth function.
EflSflFl Calculate the regression equation using Logistic (Figure 3.5.9). Rounding each value
to three decimal places, a logistic function that m odels the data is
_____661.565_____
1 + 131.178e-°'5551'
The graph of Y(t) = -
661.565
fits the data fairly well (Figure 3.5.10).
1 + 1 3 1 .1 7 8 f> - a555f
ETTfltl The lim it to growth in the m odeling equation is the num erator of the fraction or
661.565. Therefore, according to this m odel, the population of yeast in the culture
will approach, but never reach, 662.
StudyTip
Correlation Coefficients
Logistic regressions do not have
a corresponding correlation
L o g is tic
y = c / < l + a e ^ < -bx3>
a = 1 3 1 .1781232
b = .5545874884
c = 6 6 1 .5 6 4 9 4 5 6
>
coefficient due to the nature
of the models.
[ - 2 , 22] scl: 2 by [- 5 0 , 700] scl: 50
Figure 3.5.10
Figure 3.5.9
Figure 3.5.8
!► GuidedPractice
3.
FISH Use logistic regression to model the data in the table about a lake's fish population.
Then use your model to predict the lim it to the growth of the fish population.
Time (mo)
Fish
202
| Lesson 3-5 | M o d e lin g w ith N o n lin e a r Regression
0
4
8
12
16
20
24
125
580
2200
5300
7540
8280
8450
StudyTip
Logistic Regression Notice how
the logistic graph at the right
W hile you can use a calculator to find a linear, quadratic, power, exponential, logarithmic, or logistic
regression equation for a set of data, it is up to you to determine which model best fits the data by
looking at the graph and/or by examining the correlation coefficient of the regression. Consider the
graphs of each regression model and its correlation coefficient using the same set of data below.
represents only the initial part of
the graph. Therefore, it is a little
more difficult to assess the
Linear Regression
Quadratic Regression
Power Regression
accuracy of this regression
r = 0.74
r = 0.94
r = 0.94
without expanding the domain.
The complete graph of the logistic
regression is shown below.
[0 ,1 0 ] scl: 1 by [ - 2 , 1 0 ] scl: 1
Exponential Regression
Logarithm ic Regression
r = 0.99
r = 0.58
[0 ,1 0 ] scl: 1 by [ - 2 , 1 0 ] scl: 1
Over the domain displayed, the exponential and logistic regression m odels appear to most
accurately fit the data, w ith the exponential model having the strongest correlation coefficient.
■ S H U c h o o s 0 a Regression
EARTHQUAKES Use the data below to determ ine a regression equation that best relates the
distance a seismic wave can travel from an earthquake's epicenter to the time since the
earthquake. Then determ ine how far from the epicenter the w ave will be felt 8.5 minutes
after the earthquake.
Travel Time (min)
Distance (km)
1
2
5
7
10
12
13
400
800
2500
3900
6250
8400
10,000
ETHTn From the shape of the scatter plot, it appears that these data could best be modeled by
any of the regression models above except logarithmic. (Figure 3.5.10)
W U iU Use the LinReg(ax+b), QuadReg, CubicReg, QuartReg, LnReg, ExpReg, PwrReg,
and Logistic regression features to find regression equations to fit the data,
noting the corresponding correlation coefficients. The regression equation with
a correlation coefficient closest to 1 is the quartic regression with equation rounded
to y = 0.702x4 — 16.961x3 + 160.826x2 — 21.045% + 293.022. Rem em ber to use 1 VARS 1
to transfer the entire equation to the graph.
CTTHiffl The quartic regression equation does indeed fit the data very well. (Figure 3.5.11)
StudyTip
C T !m Use the CALC feature to evaluate this regression equation for x = 8.5. (Figure 3.5.12)
Since y ~ 4981 w hen x = 8.5, you would expect the wave to be felt approximately
4981 kilom eters away after 8.5 minutes.
Y1=.m?H7?JH3*2B3X"N+ -
Using a Regression Equation
Some models are better than
others for predicting long-term
behavior, while others are a better
fit for examining short-term
behavior or interpolating data.
K=B.S
-1 ,1 5 ] scl: 1 by [0,12,0 00] scl: 1000
Figure 3.5.10
[ - 1 , 1 5 ] scl: 1 by [0,12,0 00] scl: 1000
Figure 3.5.11
,
-1 ,1 5 ] scl: 1 by [0,12,0 00] scl: 1000
Figure 3.5.12
3
connectED.m cgraw-hill.com 1
203
^ GuidedPractice
4. INTERNET Use the data in the table to determ ine a regression equation that best relates the
cumulative number of domain nam es that were purchased from an Internet provider each
month. Then predict how many dom ain nam es will be purchased during the 18th month.
Tim e (mo)
Domain Names
Time (mo)
Domain Names
1
2
3
4
5
6
7
8
211
346
422
468
491
506
522
531
»
|
540
10
538
11
551
■...■■■
12
542
13
14
15
16
565
571
588
593
Linearizing Data The correlation coefficient is a m easure calculated from a linear regression.
mm How then do graphing calculators provide correlation coefficients for nonlinear regression?
The answer is that the calculators have linearized the data, transforming it so that it appears to
cluster about a line. The process of transforming nonlinear data so that it appears to be linear is
called linearization.
To linearize data, a function is applied to one or both of the variables in the data set as shown in the
example below.
Original Data
0
LinearizedData
PV
0
1
In y
In y
0
1
1.4
1
0.3
2
1.9
2
0.6
3
2.7
3
1.0
4
3.7
4
1.3
5
5.2
5
1.6
6
7.2
6
2.0
7
10.0
7
2.3
X
By calculating the equation of the line that best fits the linearized data and then applying inverse
functions, a calculator can provide you w ith an equation that models the original data. The
correlation coefficient for this nonlinear regression is actually a measure of how well the calculator
was able to fit the linearized data.
Data m odeled by a quadratic function are linearized by applying a square root function to the
y-variable, while data m odeled by exponential, power, or logarithm ic functions are linearized
by applying a logarithmic function to one or both variables.
KeyC oncept Transformations for Linearizing Data
StudyTip
To linearize data modeled by:
Linearizing Data Modeled by
Other Polynomial Functions
To linearize a cubic function
y = ax3 + bx2 + cx + d, graph
(x,\ Jy). To linearize a quartic
function y = ax4 + bx3 + cx 2 +
dx + e, graph ( x, ^ / y ).
>
• a quadratic function y = ax2 + b x + c, graph (x, \/y ).
• an exponential function y = a b x, graph (x, In y).
• a logarithmic function y = a In x + b, graph (In x, y).
• a power function y = a x b, graph (In x, In y).
J
L
You w ill justify two of these linear transformations algebraically in Exercises 34 and 35.
204
|
Lesson 3-5
M o d e l i n g w i t h N o n l i n e a r R e g r e s s io n
A graph of the data below is shown at the right. Linearize the
data assum ing a power model. Graph the linearized data, and
find the linear regression equation. Then use this linear model
to find a model for the original data.
StudyTip
Semi-Log and Log-log Data
When a logarithmic function is
applied to the x- or y-values of a
B
0.5
1
1.5
2
2.5
3
3.5
4
0.13
2
10.1
32
78.1
162
300.1
512
[0 ,5 ] scl: 0.5 by [0,1000 ] scl: 100
ETHin Linearize the data.
To linearize data that can be m odeled by a pow er function, take the natural log of both
the x - and y-values.
data set, the new data set
is sometimes referred to as
In x
the semi-log of the data (x, In y)
In y
- 0 .7
0
0.4
0.7
0.9
1.1
1.3
1.4
-2
0.7
2.3
3.5
4.4
5.1
5.7
6.2
or (In x, y). Log-log data refers to
data that have been transformed
by taking a logarithmic function
of both the x- and y-values,
(In x, In y).
Graph the linearized data and find the linear regression
equation.
The graph of (In x , In y) appears to cluster about
a line. Let x = In x and y = In y. Using linear
regression, the approxim ate equation modeling
the linearized data is y = 4 x + 0.7.
[0, 5] scl: 0.5 by [0 ,1 0 ] scl: 1
PTHTTil Use the model for the linearized data to find a model for the original data.
StudyTip
Replace x w ith In x and y w ith In y, and solve for y.
Comparing Methods Use the
calculator to find an equation that
models the data in Example 5.
How do the two compare?
Equation for linearized data
t? = 4 x + 0.7
power regression feature on a
£ = ln jfa n d p = ln y
In y = 4 In x + 0.7
£>ln
How does the correlation coefficient
from the linear regression in Step 2
compare with the correlation
coefficient given by the power
y
_
^ 4 In
y — e4
In
* + 0 .7
y —e 4
In
x e0.7
e 0-7
y = eln
regression?
x + 0 .7
y = x 4 e 0-7
y = 2x4
e0J a 2
Therefore, a power function that m odels these data
is y = 2 x 4. The graph of this function with the scatter
plot of the original data shows that this model is a
good fit for the data.
p GuidedPractice
Make a scatter plot of each set of data, and linearize the data according to the given model.
Graph the linearized data, and find the linear regression equation. Then use this linear
model for the transform ed data to find a model for the original data.
5A. quadratic model
B
0
1
2
3
4
5
6
7
1
2
9
20
35
54
77
2
3
4
5
6
7
8
7.1
8.3
9.5
9.8
10.4
10.8
11.2
5B. logarithmic model
B
'
s
205
You can linearize data to find models for real-world data w ithout the use of a calculator.
m
m
m
& m
Use Linearization
m
SPORTS The table shows the average professional football player's salary for several years.
Find an exponential model relating these data by linearizing the data and finding the linear
regression equation. Then use your model to predict the average salary in 2012.
Year
I
1990
1995
2000
2002
2003
2004
2005
2006
Average Salary ($1000)
j
354
584
787
1180
1259
1331
1400
1700
Source: NFL Players Association
ETBT1 M ake a scatter plot, and linearize the data.
Let x represent the num ber of years after 1900 and y the average salary in thousands.
H
NFL player salaries are regulated
by a salary cap, a maximum
90
95
100
102
354
584
787
1180 1259 1331 1400 1700
103
104
105
106
amount each franchise is allowed
B
to spend on its total roster each
The plot is nonlinear and its shape suggests that
the data could be m odeled by an exponential function.
Linearize the data by finding (x, In y).
season. In 2008, the salary cap
per team was $116 million.
Source: NFL
90
95
100
102
103
104
105
106
5.9
6.4
6.7
7.1
7.1
7.2
7.2
7.4
[8 0 ,1 2 0 ] scl: 5 by [0, 2000] scl: 200
Graph the linearized data, and find a linear regression equation.
A plot of the linearized data appears to form a
straight line. Letting y = In y, the rounded regression
equation is about y = 0.096x — 2.754.
FHTfln Use the model for the linearized data to find a
model for the original data.
Replace y w ith In y, and solve for y.
y
= 0.096* - 2.754
J
h
II
>>
e to y =
j> = In y
£ 0 .0 9 6 x - 2 .7 5 4
e 0 .0 9 6 x - 2 .7 5 4
e 0 .0 9 6 x e - 2 . 7 5 4
0 .0 6 e°O96x
e~2m
y =
y =
y =
Equation for linearized data
0.096x - 2.754
w o ,0 6
Therefore, an exponential equation that m odels these data is y = 0.06e 0.096.V
WatchOut!
k W 'H Use the equation that m odels the original data to solve the problem.
Using the Wrong Equation
Be careful not to confuse the
equation that models the
linearized data with the equation
that models the original data.
To find the average salary in 2012, find y when x = 2012 — 1900 or 112. According
to this model, the average professional football player's salary in 2012 will be
0.06eao96(112) w $2803 thousand or about $2.8 million.
^ GuidedPractice
6. FALLING OBJECT Roger drops one of his shoes out of a hovering helicopter. The distance d in
feet the shoe has dropped after t seconds is show n in the table.
t
0
1
1.5
2
2.5
3
4
5
d
0
15.7
35.4
63.8
101.4
144.5
258.1
404.8
Find a quadratic model relating these data by linearizing the data and finding the linear
regression equation. Then use your model to predict the distance the shoe has traveled after
7 seconds.
206
| Lesson 3-5
M o d e lin g w ith N o n lin e a r Regression
Exercises
For Exercises 1 -3 , complete each step.
a. Find an exponential function to model the data.
b. Find the value of each model at x = 20. (Example 1)
1' D
D
2 W M KM
For Exercises 7 -9 , complete each step.
a. Find a logarithm ic function to model the data.
3■ F ™ M
7' —
—
M
9' —
1
7
0
1
0
25
1
50
2
8.6
1
40
2
11
1
6
1
6
2
42
4
7.2
2
49.9
3
25
2
23
2
1.6
3
37
6
6.4
3
55.8
4
47
3
124
3
0.4
4
33
8
5.8
4
59.9
5
96
4
620
4
0.09
5
31
10
5.4
5
63.2
6
193
5
3130
5
0.023
6
28
12
5.0
6
65.8
7
380
6
15,600
6
0.006
7
27
14
4.7
7
68.1
4. GENETICS Drosophila melanogaster, a species of fruit fly,
are a com mon specimen in genetics labs because they
reproduce about every 8.5 days, allow ing researchers
to study several generations. The table show s the
population of drosophila over a period of days. (Example 1)
Generation
Drosophila
Generation
Drosophila
1
80
5
1180
2
156
6
2314
3
307
7
4512
4
593
8
8843
10. CHEMISTRY A lab received a sam ple of an isotope of cobalt
in 1999. The am ount of cobalt in grams remaining per
year is show n in the table below. (Example 2)
Year
2000
2001
2002
2003
2004
2005
2006
2007
Cobalt (g)
877
769
674
591
518
454
398
349
a. Find a logarithm ic function to model the data. Let
x = 1 represent 2000.
b. Predict the amount of cobalt remaining in 2020.
b. Use the function to predict the population of drosophila
after 93.5 days.
( J ) SHARKS Sharks have numerous rows of teeth embedded
directly into their gums and not connected to their jaws.
As a shark loses its teeth, teeth from the next row move
forward. The rate of replacement of a row of teeth in days
per row increases w ith the water temperature. (Example 1)
For Exercises 11-13, complete each step.
a. Find a logistic function to m odel the data.
m
n
■ an
,3 .
■m
1
3
3
21
67
2
5
6
25
4
80
3
7
9
28
0
50
2
6
89
4
8
12
31
Temp. (°C)
|
20
21
22
23
24
25
26
27
8
94
5
13
15
33
Days per Row
I
66
54
44
35
28
22
18
16
10
97
6
16
18
34
12
98
7
19
21
35
14
99
8
20
24
35
b. Use the function to predict the temperature at which
sharks lose a row of teeth in 12 days.
6. WORDS A word fam ily consists of a base word and all of
its derivations. The table shows the percentage of words
in an average English text com prised of the m ost com m on
word families. (Example 2)
Word Families
Percentage of Words
1000
73.1
2000
79.7
3000
84.0
4000
86.7
14. CHEMISTRY A student is perform ing a titration in lab. To
perform the titration, she uses a burette to add a basic
solution of N aO H to a neutral solution. The table shows
the pH of the solution as the N aOH is added. (Example 3)
5000
NaOH (mL)
88.6
pH
-
1
2
3
5
7.5
10
10.4
10.6
11.0
11.3
11.5
11.5
a. Find a logarithmic function to model the data.
a. Find a logistic function to model the data.
b. Predict the number of word fam ilies that m ake up 95%
b. Use the m odel to predict the pH of the solution after
of the words in an average English text.
12 milliliters of NaOH have been added.
coniiectED~m eg raw -h ill com |
207
2010
1.357
For Exercises 18-21, complete each step.
a. Linearize the data according to the given model.
b. Graph the linearized data, and find the linear regression
equation.
C. Use the linear model to find a model for the original data.
Check its accuracy by graphing. (Examples 5 and 6)
2015
1.389
18. exponential
(l5| CENSUS The table shows
the projected population
of Maine from the 2000
census. Let x be the number
of years after 2000.
(Example 3)
Population
(m illions)
Year
2000
1.275
2005
1.319
a. Find a logistic function
to model the data.
2020
1.409
2025
1.414
b. Based on the model, at
2030
1.411
11
0
1.0
1
32
1
6.6
2
91
2
17.0
0
what population does
the 2000 census predict
M aine's growth to level off?
c. Discuss the effectiveness of the model to predict the
population as time increases significantly beyond the
domain of the data.
16. SCUBA DIVING Scuba divers search for dive locations with
good visibility, which can be affected by the murkiness of
the water and the penetration of surface light. The table
shows the percent of surface light reaching a diver at
different depths as the diver descends. (Example 4)
3
268
3
32.2
4
808
4
52.2
5
2400
5
77.0
6
7000
6
106.6
7
22,000
7
141.0
Light (%)
15
89.2
21. power
20. logarithmic
2
Depth (ft)
19. quadratic
80.0
1
5
4
83.5
2
21
6
85.5
3
44
30
79.6
8
87.0
4
79
45
71.0
10
88.1
5
120
60
63.3
12
89.0
6
180
75
56.5
14
90.0
7
250
16
90.5
8
320
90
50.4
105
44.9
120
40.1
a. Use the regression features on a calculator to determine
the regression equation that best relates the data.
b. Use the graph of your regression equation to
approximate the percent of surface light that reaches
the diver at a depth of 83 feet.
17. EELS The table shows the average length of female king
snake eels at various ages. (Example 4)
22. TORNADOES A tornado w ith a greater wind speed near
the center of its funnel can travel greater distances. The
table show s the wind speeds near the centers of tornadoes
that have traveled various distances. (Example 6)
Distance (m i)
W ind Speed (mph)
0.50
37
0.75
53
1.00
65
1.25
74
81
Age (yr)
Length
(in.)
Age (yr)
Length
(in.)
1.50
1.75
88
4
8
14
17
2.00
93
6
11
16
18
2.25
98
8
13
18
18
2.50
102
10
15
20
19
2.75
106
12
16
a. Linearize the data assum ing a logarithmic model.
a. Use the regression features on a calculator to
determine if a logarithmic regression is better than a
logistic regression. Explain.
b. Use the graph of your regression equation to
approximate the length of an eel at 19 years.
208
Lesson 3-5
M o d e lin g w ith N o n lin e a r Regression
b. Graph the linearized data, and find the linear
regression equation.
C. Use the linear model to find a model for the original
data, and approximate the wind speed of a funnel that
has traveled 3.7 miles.
23. HOUSING The table shows the appreciation in the value of
a house every 3 years since the house was purchased.
(Example 6)
Years
Value ($)
0
3
6
9
78,000
81,576
85,992
90,791
12
15
95,859 101,135
a. Linearize the data assum ing an exponential model.
b. Graph the linearized data, and find the linear
regression equation.
C. Use the linear model to find a model for the original
data, and approximate the value of the house 24 years
after it is purchased.
24. COOKING Cooking times, temperatures, and recipes are
often different at high altitudes than at sea level. This is
due to the difference in atmospheric pressure, w hich
causes boiling points for varius substances, such as
water, to be lower at higher altitudes. The table shows
the boiling point of water at different elevations above
sea level.
Elevation (m)
Boiling Point (°C)
0
100
1000
99.29
2000
98.81
3000
98.43
4000
98.10
5000
97.80
6000
97.53
7000
97.28
8000
97.05
9000
96.83
10,000
96.62
a. Linearize the data for exponential, power, and
logarithmic models.
b. Graph the linearized data, and determ ine which model
best represents the data.
Linearize the data in each table. Then determine the most
appropriate model.
27 D D
28B
B
1
29' poVHQfl
2
2.5
6
1
37.8
4
7.3
2
29
2
17.0
6
13.7
3
42
3
7.7
8
21.3
4
52
4
3.4
10
30.2
5
59
5
1.6
12
40.0
6
65
6
0.7
14
50.8
7
70
7
0.3
16
62.5
8
75
8
0.1
30. FISH Several ichthyologists are studying the smallmouth
bass population in a lake. The table shows the smallmouth
bass population of the lake over time.
Year
2001
2002
2003
2004
2005
2006
2007
2008
Bass
673
891
1453
1889
2542
2967
3018
3011
a. Determ ine the m ost appropriate model for the data.
b. Find a function to model the data.
c. Use the function to predict the smallmouth bass
population in 2012.
d. Discuss the effectiveness of the model to predict the
population of the bass as time increases significantly
beyond the dom ain of the data.
H.O.T. Problem s
31. REASONING W hy are logarithm ic regressions invalid when
the dom ain is 0?
32. CHALLENGE Show that y
t o y = a ekx.
= a b x
can be converted
33. REASONING Can the graph of a logistic function ever have
any intercepts? Explain your reasoning.
C. Write an equation to model the data based on your
analysis of the linearizations.
Determine the model most appropriate for each scatter plot.
25.
26.
PROOF Use algebra to verify that data m odeled by each type
of function can be linearized, or expressed as a function
y = mx + b for some values m and b, by replacing (x, y) with
the indicated coordinates.
34. exponential, (x , In y)
(35) power, (In x, In y)
36. REASONING How is the graph of g(x) = -—
to the graph of f( x ) = -
+ a related
g_X1 Explain.
37. WRITING IN MATH Explain how the param eters of an
exponential or logarithm ic model relate to the data set
or situation being modeled.
j
209
Spiral Review
38.
3te = 33 - 1
39. 3 5x. 8 1 1 “ a; = 9 x - 3
40. 49* = 7*2- 15
41.
log5 (Ax - 1) = log5 (3 x + 2)
42. log 10z + log10 (z + 3) = 1
43. log 6 (a2 + 2) + log6 2 = 2
44.
ENERGY The energy E, in kilocalories per gram m olecule, needed to transport a substance from
the outside to the inside of a living cell is given by £ = 1.4(log10 C2 — log 10 C x), where C 2 and
C2 are the concentrations of the substance inside and outside the cell, respectively. (Lesson 3-3)
a. Express the value of E as one logarithm.
b. Suppose the concentration of a substance inside the cell is twice the concentration
outside the cell. How much energy is needed to transport the substance on the outside of
the cell to the inside? (Use log10 2 ~ 0.3010.)
c. Suppose the concentration of a substance inside the cell is four times the concentration
outside the cell. How much energy is needed to transport the substance from the outside
of the cell to the inside?
45. FINANCIAL LITERACY In 2003, M aya inherited $1,000,000 from her
grandmother. She invested all of the m oney and by 2015, the
amount will grow to $1,678,000. (Lesson 3-1)
Investment
a. Write an exponential function that could be used to model the
amount of money y. Write the function in terms of x, the
number of years since 2003.
b. Assume that the amount of money continues to grow at the
same rate. Estimate the amount of m oney in 2025. Is this
estimate reasonable? Explain your reasoning.
k
JS
1,800,000
Q 1,400,000
c
V
■jg 1,000,000
Sim plify. (Lesson 0-2)
46. (—2*)(—6i)(4i)
47. 3 i(—5i)
49. (1 — 4i)(2 + i)
50.
4i
3+ i
48. i;1 3
51.
5 + 3i
52. SAT/ACT A recent study showed that the num ber of
Australian hom es with a com puter doubles every
8 months. Assuming that the num ber is increasing
continuously, at approximately what m onthly rate
must the number of Australian com puter owners be
increasing for this to be true?
A 6.8%
C 12.5%
B 8.66%
D 8.0%
E 2%
Hours
0
1
2
3
4
Bacteria
5
8
15
26
48
Approximately how much time will it take the culture
to double after hour 4?
210
F 1.26 hours
H 1.68 hours
J 1.76 hours
a. Graph v(t) for 0 < t < 10.
b. Describe the domain and range of v(t). Explain your
reasoning.
53. The data below gives the number of bacteria found in a
certain culture. The bacteria are growing exponentially.
G 1.35 hours
54. FREE RESPONSE The speed in miles per hour at which
a car travels is represented by v(t) = 6o(l — e-f2) where
f is the time in seconds. Assum e the car never needs to
stop.
| Lesson 3-5 | M o d e lin g w ith N o n lin e a r Regression
C. W hat type of function is v(t )?
d. W hat is the end behavior of v(t)? W hat does this
m ean in the context of the situation?
e. Let d(t) represent the total distance traveled by the
car. W hat type of function does d(t) represent as f
approaches infinity? Explain.
f. Let a(t) represent the acceleration of the car. W hat
is the end behavior of a(t)? Explain.
Study Guide
^ V o c a b u la ry
KeyConcepts
Exponential Functions (Lesson 3-1)
algebraic function
•
common logarithm
•
Exponential functions are of the form f(x) = abx, where a =/= 0, b is
positive and £> =/= 1. For natural base exponential functions, the base is
the constant e.
If a principal Pis invested at an annual interest rate r(in decimal form),
then the balance A in the account after /years is given by
A = p (l + - t y 1, if compounded n times a year or
A = Pert, if compounded continuously.
•
logarithmic function with
base b (p. 1 7 2 )
(p. 1 5 8 )
(p. 1 7 3 )
logistic growth function p. 202)
continuous compound
interest (p. 1 6 3 )
exponential function
linearize
p. 2 0 4 )
logarithm
(p. 1 7 2 )
natural base
p. 1 6 0 )
natural logarithm
(p. 1 5 8 )
(p. 17 4 )
transcendental function
(p. 15 8 )
If an initial quantity N0 grows or decays at an exponential rate ror k
(as a decimal), then the final amount N after a time t is given by
N = N0(1 + r)1or N = N0 ekt, where r is the rate of growth per time t
and k is the continuous rate of growth at any time t.
Logarithm ic Functions (Lesson 3-2)
•
The inverse of f(x) = bx, where b > 0 and
function with base b, denoted M ( x ) = log6 x.
•
If b > 0, b 1, and x > 0, then the exponential form of
log6 x = y is by = x and the logarithmic form of by = x is
log6 x = y. A logarithm is an exponent.
•
Common logarithms: log10 x or log x
•
Natural logarithms: log e x or In x
b ± 1, is the logarithmic
Choose the correct term from the list above to complete each sentence.
1.
A logarithmic expression in which no base is indicated uses the
2. ______________ are functions
Properties of Logarithm s (Lesson 3 - 3 )
•
Product Property: log6 xy = log6 x + log6 y
•
Quotient Property: logbj = log6 x - log„y
•
Power Property: log 6 ■v*’ = p • log 6 x
•
Change of Base Formula: log b x =
log**
Exponential and Logarithm ic Equations (Lesson 3 - 4 )
•
One-to-One Property of Exponents: For b > 0 and b =/= 1,
bx = bv if and only if x = y.
•
One-to-One Property of Logarithms: For b > 0 and b =/= 1,
logfix = logfiy if and only if x = y.
M odeling w ith Nonlinear Regression (Lesson 3 - 5)
in which the variable is the exponent.
3. Two examples o f___________ are exponential functions and
logarithmic functions.
4. The inverse of f(x) = bx is called a(n)____________
5. The graph of a(n)___________ contains two horizontal asymptotes.
Such a function is used for growth that has a limiting factor.
6. Many real-world applications use th e _____________e, which,
like 7r or V 5 , is an irrational number that requires a decimal
approximation.
7. To______________ data,a
function
is applied to one or both of the
variables in the data set, transforming the data so that it appears to
cluster about a line.
8. Power, radical, polynomial, and rational functions are examples of
To linearize data modeled by:
•
a quadratic function y = ax2 + b x + c, graph {x, yfy).
•
an exponential function y = abx, graph (x, In y).
•
a logarithmic function y = a In x + b, graph (In x, y).
•
a power function y = axb, graph (In x, In y).
occurswhenthereisno
9.
waitingperiod
between
interest payments.
10. The_____________ is
denotedbyIn.
211
jfp |j9 , - ffijjljSSlSSI
Continued
Lesson-by-Lesson Review
—
Exponential Functions (pp. 158-169)
range, intercepts, asymptotes, end behavior, and where the function
is increasing or decreasing.
Example 1
11. f(x) = 3X
12. f(x) = 0.4*
Use the graph of f(x) = 2 x to describe the transformation that
results in the graph of g(x) = —2 X_ 5. Then sketch the graphs
of g and f.
13. f(x) = ( f ) X
14. « - ( ! ) '
This function is of the form g(x) = —f(x — 5).
Use the graph of f(x) to describe the transformation that results in
the graph of g(x). Then sketch the graphs of f(x) and g(x).
15. f(x) = 4X; g(x) = 4* + 2
So, g(x) is the graph of
f(x) — 2 Xtranslated 5 units
to the right and reflected
in the x-axis.
16. f(x) = 0.1*;g(x) = 0.1x ~ 3
17. f(x) = 3X; g(x) = 2 • 3* - 5
Example 2
18. W = ({)*■, g(x) = ( $ * + 4 + 2
Copy and complete the table below to find the value of an
investment A for the given principal P, rate r, and time f if the
interest is compounded n times annually.
4
I
12
365
continuously
What is the value of $2000 invested at 6.5% after 12 years if the
interest is compounded quarterly? continuously?
A = P ^ + $ nt
=
2000^1
0 .0 6 5 \4<12)
= $4335.68
A = Pen
19. P = $250, r = 7%, f = 6 years
20. P = $1000, r = 4.5%, t = 3 years
P = 2000, r = 0.065, n = 4, t - 12
Simplify.
Continuous Interest Formula
= 2000e°°65(12)
P = 2000, r = 0.065, f = 12
= $4362.94
Simplify.
Logarithmic Functions (pp. 172- 180)
Example 3
24.
25. In e11
CM
log 80
O
CO
log 25 5
l093 8T
CO
22.
CO
CVJ
21. log2 32
26. 3 log39
28. eln12
Use the graph of f[x) to describe the transformation that results in
the graph of g[x). Then sketch the graphs of f(x) and g(x).
29. f(x) = log x; g(x) = -lo g (X + 4)
30. f(x) = log2 x; g(x) = log2 x + 3
31. f(x) = In x; g(x) = 1 In x - 2
212
C h a p te r 3
Study G uide and Review
Use the graph of f(x) = In xto describe the transformation
that results in the graph of g(x) = - I n (x - 3). Then sketch the
graphs of g(x) and f(x).
This function is of the form g(x) = - f ( x - 3). So, g(x) is the
graph of f(x) reflected in the x-axis translated 3 units to the right.
Properties of Logarithms (pp. 181-188)
Example 4
32. log3 9x3y 3z 6
Condense 3 log3 x + log3 7 — 1 log3 x.
33. log5 x 2a7v^b
3 log3 x + log3 7 - j log3 x
34. In-
x 2y 3z
35. log
\fgFk
= log3 x 3 + log3 7 - log3V x
Power Property
= log3 7x3 - log3V x
Product Property
=lot^
100
Quotient Property
36. 3 log3 x — 2 log3 y
37. i log2 a + log2 (b + 1)
38. 5 In (x + 3) + 3 In 2 x - 4 In ( x - 1)
Exponential and Logarithmic Equations
1
(pp. 1 90 -1 9S )
39. 3 * + 3 = 27* “ 2
Solve 7 In 2x = 28.
40. 253x+ 2 = 125
7 In 2x = 28
41. e2* - 8 e * + 15 = 0
42.
e* -
4e-x = 0
Original equation
In 2x = 4
e ln2* = e 4
2x = e 4
43. log2 x + lo g 2 3 = log2 18
44. log6 x + log6 ( x - 5)
=
Inverse Property
x = 0.5e4 or about 27.299
Solve and simplify.
2
Modeling With Nonlinear Regress;ion (pp. 200- 210)
Example 6
Complete each step.
Linearize the data shown assuming a logarithmic model, and
calculate the equation for the line of best fit. Use this equation to
find a logarithmic model for the original data.
a. Linearize the data according to the given model.
b. Graph the linearized data, and find the linear regression
equation.
c. Use the linear model to find a model for the original data
'
and graph it.
45.
exponential
E
46.
»
k W ill
1
2
3
4
5
6
5
17
53
166
517
1614
2
3
4
5
6
7
4
8
10
12
14
15
logarithmic
'
-
3
5
7
9
10
-7
-1 5
-21
-2 5
-2 7
To linearize y = a In x + b, graph (In x, y).
0
1.1
1.6
1.9
2.2
2.3
12
-7
-1 5
-21
-2 5
-2 7
W W 1U
The line of best fit is y = - 1 6 .9 4 x + 11.86.
PTFflfH
y = —16.94 In x + 11.86
x=lnx
Continued
Applications and Problem Solving
47. INFLATION Prices of consumer goods generally increase each year
due to inflation. From 2000 to 2008, the average rate of inflation in
the United States was 4.5%. At this rate, the price of milk fyears
after January 2000 can be modeled with M(t) = 2.75(1.045)f.
(Lesson 3-1)
52. SOUND The intensity level of a sound, measured in decibels,
can be modeled by d(w) = 10 log
where w is the
intensity of the sound in w atts per square meter and w0 is
the constant 1 x 1 0 ~ 12 w atts per square meter. (Lesson 3-4)
a. Determine the intensity of the sound at a concert that reaches
a. What was the price of milk in 2000? 2005?
b. If inflation continues at 4.5%, approximately what will the price
100 decibels.
b. Tory compares the concert w ith the music she plays at home.
of milk be in 2015?
She plays her music at 50 decibels, so the intensity of her music
is half the intensity of the concert. Is her reasoning correct?
Justify your answer mathematically.
c. In what year will the price of milk reach $4?
c. Soft music is playing w ith an intensity of 1
x 10 - 8 w atts per
square meters. By how much do the decimals increase if the
intensity is doubled?
48. CARS The value of a new vehicle depreciates the moment the car
is driven off the dealership’s lot. The value of the car will continue to
depreciate every year. The value of one car fyears after being
bought is f(x) = 18,000(0.8)f. (Lesson 3-1)
a. What is the rate of depreciation for the car?
b. How many years after the car is bought will it be worth
half of its original value?
49. CHEMISTRY A radioactive substance has a half-life of
53. FINANCIAL LITERACY Delsin has $8000 and wants to put it
into an interest-bearing account that compounds continuously.
His goal is to have $12,000 in 5 years. (Lesson 3-4)
a. Delsin found a bank that is offering 6% on investments.
How long would it take his investment to reach
$12,000 at 6%?
b. What rate does Delsin need to invest at to reach his goal of
16 years. The number of years t it takes to decay from
an initial amount W0 to /Vcan be determined using
,
t=
16l09i
— . (Lesson 3-2)
lo g i
$12,000 after 5 years?
54. INTERNET The number of people to visit a popular
Web site is given below. (Lesson 3-5)
a. Approximately how many years will it take 100 grams to
Total Number of Visitors
decay to 30 grams?
b. Approximately what percentage of 100 grams will there be
in 40 years?
100
-9 4 -
98
101
■
_82-
50. EARTHQUAKES The Richter scale is a number system for
determining the strength of earthquakes. The number R is
dependent on energy E released by the earthquake
in kilowatt-hours. The value of R is determined by
R = 0.67 • log (0.37 E) + 1.46. (Lesson 3-2)
0
JJ-
80
-6 5 -
1
60
40
a. Find flfor an earthquake that releases
50
“33“
20
1,000,000 kilowatt-hours.
b. Estimate the energy released by an earthquake that
registers 7.5 on the Richter scale.
^
^
^
^
Year
51. BIOLOGY The time it takes for a species of animal to
double is defined as its generation time and is given by
a. Make a scatterplot of the data. Let 1990 = 0.
G = - — t— 3, where b is the initial number of animals,
b.
2.5 log,, d
d is the final number of animals, t is the time period, and G is the
generation time. If the generation time 6 of a species is 6 years,
how much time t will it take for 5 animals to grow into a population
of 3125 animals? (Lesson 3-3)
214
C h a p te r 3
Linearize the data w ith a logarithm ic model.
c. Graph the linearized data, and find the linear regression
equation.
d.
Use the linear model to find a model for the original data and
graph it.
Practice Test
range, intercepts, asymptotes, end behavior, and where the function is
2. f(x) = 2 | | j
1. f(x) = - e x+1
I I '- 3
18. 3 X+ 8 = 9 2x
19. e2x - 3 e x + 2 = 0
20. log x + log (x - 3) = 1
g(x) = - 5 ~ x - 2
4. f(x) = 5x
16. 2 log4 m + 6 log4 n - 3(log4 3 + log4 y)
17. 1 + In 3 - 4 In x
* —4
Use the graph of f(x) to describe the transformation that results in the
graph of g(x). Then sketch the graphs of f(x) and g(x).
3 .
21. log 2 (x — 1) + 1 = log 2 (x + 3)
5. MULTIPLE CHOICE For which function is lim f(x) =
X— >oo
-o o ?
C f(x-) = —log8 (x — 5)
A f(x) = —2 • 3 _x
22. MULTIPLE CHOICE Which equation has no solution?
D f(x) = log3 ( - x ) - 6
6- iog3 ^ -
7. log32 2
8.
9_ g lo g 9 5.3
log 1012
f(x) = —log4 (x + 3)
H log5 x = l o g 9 x
G 2X~ 1 = 3 X + 1
J log2 ( x + 1) = log2 x
For Exercises 23 and 24, complete each step.
a.
Find an exponential or logarithm ic function to model the data.
b.
Find the value of each model at x = 20.
23.
Sketch the graph of each function.
10.
F ex = e~x
11. g(x)
= log
(—x) + 5
12. FINANCIAL LITERACY You invest $1500 in an account with an
interest rate of 8% for 12 years, making no other deposits or
withdrawals.
24.
■
3
3
5
7
9
11
13
3
0
-2
-3
-4
-5
3
5
7
9
11
13
4
5
6
7
9
10
a. What will be your account balance if the interest is compounded
monthly?
b.
What will be your account balance if the interest is compounded
continuously?
25. CENSUS The table gives the U.S. population between 1790 and
1940. Let 1780 = 0.
c. If your investment is compounded daily, about how long will it
take for it to be worth double the initial amount?
4. i
3\fb
14- l09 3 ^
13. log6 36xy2
15. GEOLOGY Richter scale magnitude of an earthquake can be
calculated using R = \ log -J- where Eis the energy produced and
E0 is a constant.
3
tg
a. An earthquake with a magnitude of 7.1 hit San Francisco in
1989. Find the scale of an earthquake that produces 10 times
the energy of the 1989 earthquake.
b.
In 1906, San Francisco had an earthquake registering 8.25. How
many times as much energy did the 1906 earthquake produce
as the 1989 earthquake?
Year
Population
(m illions)
1790
4
1820
10
1850
23
1880
50
1910
92
1940
132
a. Linearize the data, assuming a quadratic model. Graph the data,
and write an equation for a line of best fit.
b.
Use the linear model to find a model for the original data. Is a
quadratic model a good representation of population growth?
Explain.
$
:
~1
connectEO.m cgraw-hill.com |
215
Connect to AP Calculus
'Approximating Rates of .Change
Objective
Use secant lines and the
difference quotient to
approximate rates of
change.
f
In Chapter 1, we explored the rate of change of a function
at a point using secant lines and the difference quotient.
You learned that the rate of change of a function at a point
can be represented by the slope of the line tangent to the
function at that point. This is called the instantaneous rate
of change at that point.
The constant e is used in applications of continuous growth
and decay. This constant also has many applications in
differential and integral calculus. The rate of change of
f(x) = e *a t any of its points is unique, which makes it a
useful function for exploration and application in calculus.
Activity 1 Approximate Rate of Change
A pproxim ate the rate o f change o f f i x ) = e x at x — 1.
CflSffn
Graph/(x) = ex, and plot the points P (l,/ (1))
and Q(2,/(2)).
PT71TTW
Draw a secant line of f( x ) through P and Q.
U se)
f(x 2) - f ( x x)
to calculate the average rate of change m for/(x) using P and Q.
Repeat Steps 1 -3 two more times. First use P (l,/ (1 )) and Q(1.5,/(1.5)) and then use
P (l,/ (1 )) and Q(1.25,/(1.25)).
w A nalyze the Results
1. As the x-coordinate of Q approaches 1, what does the average rate of change m appear to
approach?
2. Evaluate and describe the overall efficiency and the overall effectiveness of using secant lines
to approximate the instantaneous rate of change of a function at a given point.
In Chapter 1, you developed an expression, the difference quotient, to calculate the slope of secant lines for different
values of h.
Difference Quotient
f(x + h) - f(x)
h
m= ■
As h decreases, the secant line moves closer and closer to a line tangent to the function. Substituting decreasing
values for h into the difference quotient produces secant-line slopes that approach a limit. This limit represents the
slope of the tangent line and the instantaneous rate of change of the function at that point.
216
C h a p te r 3
Activity 2 A p p ro x im a te R ate o f C hange
Approxim ate the rate of change of f(x ) = ex at several points.
EflTW
Substitute / ( x ) = e x into the difference quotient.
m =
Approxim ate the rate of change of/(x) at x = 1
using values of h that approach 0. Let h = 0.1, 0.01,
0.001, and 0.0001.
m ■
f( x + h) - f ( x )
Repeat Steps 1 and 2 for x = 2 and for x = 3.
^ A nalyze the Results
3. As h —>0, what does the average rate of change appear to approach for each value of x?
4. Write an expression for the rate of change of f ( x ) = ex at any point x.
5. Find the rate of change of g(x) = 3ex at x = 1. H ow did m ultiplying ex by a constant affect the
rate of change at x = 1?
6. W rite an expression for the rate of change of g(x) = aex at any point x.
In this chapter, you learned that f(x) = In x is the inverse of g(x) = ex,
and you also learned about some of its uses in exponential growth
and decay applications. Similar to e, the rate of change of f(x) = In x
at any of its points is unique, thus also making it another useful
function for calculus applications.
Activity 3 A p p ro x im a te Rate o f C hange
Approxim ate the rate of change of f(x ) = In x at several points.
f(x + h) - f ( x )
Substitute/(x) = In x into the difference quotient.
Approxim ate the rate of change o f f (x) at x = 2
using values of h that approach 0. Let h = 0.1, 0.01,
0.001, and 0.0001.
EflSjiFI
In (x + h) - In x
h
Repeat Steps 1 and 2 for x = 3 and for x = 4.
p A nalyze the Results
7. As h —>0, what does the average rate of change appear to approach for each value of x?
8. Write an expression for the rate of change of the function/(x) = In x at any point x.
Model and Apply
9. In this problem , you will investigate the rate of change of the
function g(x) = —3 In x at any point x.
a. Approxim ate the rates of change of g(x) at x = 2 and then at x = 3.
b. How do these rates of change com pare to the rates of change for
f( x ) = In x at these points?
c. Write an expression for the rate of change of the function
g(x) = a In x for any point x.
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rn
1
217

Chapters 2-3

Transcription

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