5.4 Take Some Time to Reflect

5.4
Take Some Time
to Reflect
Reflections of Linear
and Exponential Functions
LEARNING GOALS
KEY TERMS
In this lesson, you will:
t Reflect linear and exponential
t reflection
t line of reflection
functions vertically.
t Reflect linear and exponential
functions horizontally.
t Determine characteristics of graphs
after transformations.
Y
ou are already familiar with many different types of “reflections” in mathematics.
When a negative sign is present, this is a good indication of a reflection.
For example, 25 is a “reflection” of 5 over 0 on the number line.
2
1
The power 21 5 __, but 221 5 __.
1
2
In this lesson, you will learn about reflecting functions. Watch out for those
negative signs!
© 2012 Carnegie Learning
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5.4
Reflections of Linear and Exponential Functions
327
Problem 1
Basic linear and exponential
functions are used to explore
vertical and horizontal
reflections. Students will write
given functions in terms of
basic functions, describe the
operation performed on the
basic function, and then a
graphing calculator to compare
the graph of the basic function
to the graph of the reflected
functions. Students then
complete a table of values
comparing the ordered pairs
of the graphs of the basic
function to the ordered pairs
of the graphs of the reflected
functions. They will generalize
about the constraints involved
when linear or exponential
functions are reflected and are
able to recognize the graphical
implications of the movements
using only the equations of the
functions.
PROBLEM 1
Reflections
Consider the three exponential functions shown, where h(x) 5 2x is the basic function.
r h(x) 5 2x
r m(x) 5 2(2x)
r n(x) 5 2(2x)
1. Write the functions m(x) and n(x) in terms of the basic function h(x).
2h(x)
m(x) 5
n(x) 5
h(2x)
2. Compare m(x) to h(x). Does an operation performed on h(x) or on the argument of h(x)
result in the equation for m(x)? What is the operation?
The function h(x) is multiplied by 21 to result in the equation for m(x).
3. Compare n(x) to h(x). Does an operation performed on h(x) or on the argument of h(x)
result in the equation for n(x)? What is the operation?
The argument of the function h(x) is multiplied by 21 to result in the equation for n(x).
4. Use a graphing calculator to graph each function with the
bounds [210, 10] 3 [210, 10]. Then, sketch the graph of each
function. Label each graph.
y
h(x)
x 5 2x
n(x)
x 5
Before you
press the
GRAPH key, make a
prediction about the
shapes of m(x)
and n(x).
x
2(2x)
x
m(x)
x 5 2(2x)
Grouping
5
Guiding Questions for
Share Phase,
Questions 1 through 4
t Is there a difference between
the function 2h(x) and the
function h(2x)? What is the
difference?
t Does the function 2h(x) have
an asymptote? Where?
t Does the function h(2x) have
t Is there a difference between the function m(x) and the function n(x)? What is
the difference?
t Does the function m(x) have an asymptote? Where?
t Does the function n(x) have an asymptote? Where?
t Which function is reflected over the y-axis?
t Which function is reflected over the x-axis?
an asymptote? Where?
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Chapter 5 Exponential Functions
© 2012 Carnegie Learning
Have students complete
Questions 1 through 7 with
a partner. Then share the
responses as a class.
Guiding Questions
for Share Phase,
Questions 5 through 7
t Looking at only the equation
of an exponential function,
how can you determine if
the graph of the reflected
function will be reflected over
the y-axis?
t Looking at only the equation
of an exponential function,
how can you determine if
the graph of the reflected
function will be reflected over
the x-axis?
5. Compare the graphs of m(x) and n(x) to the graph of the basic function h(x).
What do you notice?
The graph of m(x) is a mirror image of the graph of h(x), reflected over the horizontal
line y 5 0. The graph of n(x) is a mirror image of the graph of h(x), reflected over the
vertical line x 5 0.
6. Complete the table of ordered pairs for the three given functions.
h(x) 5 2x
m(x) 5 2(2x)
__1
1)
(22, __
4
(22,
22,
2,
1)
(21, __
2
21,
1,
(21,
2
4 )
__1
2
2 )
n(x) 5 2(2x)
( 2
1)
, __
4
( 1
1)
, __
2
, 1)
(0, 1)
(0, 21 )
( 0
(1, 2)
(1, 22 )
( 21 , 2)
(2, 4)
(2, 24 )
( 22 , 4)
Grouping
Ask a student to read the
information and definition
following Question 7. Discuss
as a class.
7. Use the table to compare the ordered pairs of the graphs of m(x) and n(x) to the ordered
pairs of the graph of the basic function h(x). What do you notice?
For the same x-coordinate, the y-coordinate of m(x) is the opposite of the
y-coordinate of h(x). For the same y-coordinate, the x-coordinate of n(x) is the
opposite of the x-coordinate of h(x).
A reflection of a graph is a mirror image of the graph about
a line of reflection. A line of reflection is the line that the
graph is reflected about. A horizontal line of reflection
affects the y-coordinates, and a vertical line of reflection
affects the x-coordinates.
You can use the coordinate notation shown to indicate a
reflection about a horizontal line of reflection.
When the
negative is on the
outside of the function, like
–g(x), all the y-values become
the opposite of the y-values of
g(x). The x-values remain
unchanged.
5
© 2012 Carnegie Learning
(x, y) → (x, 2y)
5.4
Reflections of Linear and Exponential Functions
329
Grouping
t Have students complete
You can also use the coordinate notation shown to indicate a
reflection about a vertical line of reflection.
Questions 8 and 9 with a
partner. Then share the
responses as a class.
When the
negative is on the
inside of the function, like
g(–x), all the x-values become
the opposite of the x-values of
g(x). The y-values remain
unchanged.
(x, y) → (2x, y)
t Have students complete
8. Which function represents a reflection of h(x) over a
horizontal line? Which function represents a reflection of
h(x) over a vertical line?
Questions 10 and 11 with
a partner. Then share the
responses as a class.
The function m(x) is a reflection of h(x) over a horizontal
line.
The function n(x) is a reflection of h(x) over a vertical line.
9. Describe each graph in relation to its basic function.
Guiding Questions
for Share Phase,
Questions 8 and 9
t When coordinate notation
a. Compare f(x) 5 2(bx) to the basic function h(x) 5 bx.
The graph of f(x) is a reflection of the graph of h(x) over
the horizontal line y 5 0.
is used to represent a
horizontal reflection, is the
negation associated with
the x-coordinate or is the
negation associated with the
y-coordinate?
b. Compare f(x) 5 b(2x) to the basic function h(x) 5 bx.
The graph of f(x) is a reflection of the graph of h(x)
over the vertical line x 5 0.
10. The graph of a function w(x) is shown. Sketch the graphs of w9(x) and w0(x).
a. w9(x) 5 2w(x)
b. w0(x) 5 w(2x)
t Does a vertical reflection
y
affect the x-coordinate or
the y-coordinate of the basic
function?
4
3
2
1
w(x)
w(2x)
Guiding Questions
for Share Phase,
Questions 10 and 11
t Is there a difference between
t Do you need to create a
table of values to graph the
reflections? Why or why not?
330
4
x
2w(x)
23
24
11. Write the equation of each function after a reflection about the horizontal line y 5 0.
Then, write the equation after a reflection about the vertical line x 5 0.
a. a(x) 5 5x
Reflection about y 5 0: a9(x) 5
over the y-axis?
describing the function w(x)
to graph the reflections? Why
or why not?
3
22
t Which function is reflected
t Do you need the equation
2
21
the function 2w(x) and the
function w(2x)? What is the
difference?
t Which function is reflected
over the x-axis?
1
Reflection about x 5 0: a0(x) 5
25
x
2x
5
t What points did you use to graphically reflect the function w(x)?
t How does the graph of a(x) 5 5x compare to the graph of a(x) 5 25x?
t How does the graph of a(x) 5 5x compare to the graph of a(x) 5 52x?
t How does the graph of b(x) 5 22x2 compare to the graph of b(x) 5 2(2x)2?
t How does the graph of c(x) 5 __45x3 compare to the graph of c(x) 5 2__45 x3?
t How does the graph of c(x) 5 __45 x3 compare to the graph of c(x) 5 __45 (2x)3 ?
Chapter 5 Exponential Functions
© 2012 Carnegie Learning
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0
24 23 22 21
t A horizontal reflection of a
function involves negating
what part of the basic
function?
b. b(x) 5 22x2
t A vertical reflection of a
function involves negating
what part of the basic
function?
Reflection about y 5 0: b9(x) 5
2x2
Reflection about x 5 0: b0(x) 5
22x2
5x3
c. c(x) 5 __
4
Reflection about y 5 0: c9(x) 5
__5
2 x
4
__5(2x)
3
Reflection about x 5 0: c0(x) 5
Problem 2
A horizontal reflection, s(x) 5 (2x)
and a vertical reflection,
r(x) 5 2(x) are performed on
the basic function g(x) 5 x
resulting in the same expression
and graphically, the same line.
Students will complete a table
of values used to compare the
ordered pairs of each function
to conclude that the reflection
of the basic function is the
same over the x- and y-axes.
3
PROBLEM 2
4
Linear Functions . . . Another Curious Case!
Consider the three linear functions shown, where g(x) 5 x is the basic function.
r g(x) 5 x
r r(x) 5 2(x)
r s(x) 5 (2x)
1. Write the functions r (x) and s(x) in terms of the basic function g(x).
2g(x)
r(x) 5
s(x) 5
g(2x)
2. Use a graphing calculator to graph each function with the bounds [210, 10] 3 [210, 10].
Then, sketch the graph of each function. Label each graph.
y
r( x) 5 2(x) and
g(x)
g(
( xx)) 5 x
s(x)
x 5 (2x)
x
Grouping
x
© 2012 Carnegie Learning
Have students complete
Questions 1 through 6 with
a partner. Then share the
responses as a class.
Guiding Questions
for Share Phase,
Questions 1 through 6
t Which function represents
5
3. Compare the graphs of r(x) and s(x) to the graph of the basic function g(x).
What do you notice?
The graphs of r(x) and s(x) are the same. They are both mirror images of the graph
of g(x).
the vertical reflection of g(x)?
t Which function represents
the horizontal reflection
of g(x)?
t What is the argument of g(x)?
t If the functions simplify to
t Can you think of any other basic function in which the graph of the horizontal
and the graph of the vertical reflection would result in the same line?
the same expression, what
graphical implications might
this have?
5.4
Reflections of Linear and Exponential Functions
331
4. Complete the table of ordered pairs for the three given functions.
g(x) 5 x
r(x) 5 2(x)
s(x) 5 (2x)
(22, 22)
(22,
22,
2,
2 )
( 2 , 22)
(21, 21)
(21,
21,
1,
1 )
( 1 , 21)
(0, 0)
(0,
(1, 1)
(1, 21 )
( 21 , 1)
(2, 2)
(2, 22 )
( 22 , 2)
0 )
(
0 , 0)
5. Use the table to compare the ordered pairs of the graphs of r(x) and s(x) to the ordered
pairs of the graph of the basic function g(x). What do you notice?
For the same x-coordinate, the y-coordinate of r(x) is the opposite of the
y-coordinate of g(x). For the same y-coordinate, the x-coordinate of s(x) is the
opposite of the x-coordinate of g(x).
6. Which function represents a reflection of the basic function g(x) over a vertical line?
Which function represents a reflection of the basic function g(x) over a horizontal line?
Why do you think they produce the same graph?
The function r(x) is a reflection of the basic function g(x) over a horizontal line.
The function s(x) is a reflection of the basic function g(x) over a vertical line.
They produce the same graph because the reflection of the basic function g(x) 5 x
is the same over the y- and x-axes.
© 2012 Carnegie Learning
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332
Chapter 5 Exponential Functions
Problem 3
Students are given several
descriptions of functions and
will write an equation and
sketch a graph of the function.
A Who’s Correct? question
provides two different methods
of performing transformations
that result in the same graph.
In the last activity, students
will name all possible
transformations performed on
a basic function that would
produce a graph of a described
function.
PROBLEM 3
Characteristics of Graphs after Transformations
1. Use the given characteristics to write an equation and then sketch a graph of g(x).
a. Write an equation and sketch a graph that:
r is an exponential function,
r is continuous,
r is increasing, and
r is translated 2 units to the left of f(x) 5 2x.
x12
2
Equation: g(x) 5
y
x
Grouping
Have students complete
Questions 1 through 3 with
a partner. Then share the
responses as a class.
b. Write an equation and sketch a graph that:
r is an exponential function,
r is continuous,
Guiding Questions
for Share Phase,
Question 1
t Do all exponential functions
r is increasing, and
r is translated 5 units down from f(x) 5 2x.
Equation: g(x) 5
have an asymptote? Why or
why not?
x
2 25
y
t If the function is exponential,
what is the basic function?
5
x
© 2012 Carnegie Learning
t What feature of the
exponential function
ensures that the function is
increasing?
t What feature of the
exponential function ensures
that the function has been
translated 2 units to the left?
t What feature of the
exponential function ensures
that the function has been
translated 5 units down?
t What feature of the exponential function ensures that the function is
decreasing?
t What feature of the exponential function ensures that the function has been
reflected over the y-axis?
5.4
Reflections of Linear and Exponential Functions
333
Guiding Questions
for Share Phase,
Question 2
t What is the basic function
Jacob and Kate are using?
t Who used a vertical
reflection?
t Who used a horizontal
reflection?
c. Write an equation and sketch a graph that:
r is an exponential function,
r is continuous,
r is decreasing, and
r is a reflection of f(x) 5 2x over the line x 5 0.
22x
Equation: g(x) 5
y
t Who used a upward
translation?
t Who used a downward
x
translation?
t Could Jacob and Kate both
be correct?
?
2. Jacob and Kate are comparing the two graphs shown.
y
y
f(x)
x
x
g(x)
Jacob says that to get the graph of g(x), first translate f(x) down 3 units, and then
reflect over the line y 5 0. Kate says that to get the graph of f(x), first reflect g(x) over
the line y 5 0, and then translate up 3 units.
Who is correct? Explain your reasoning.
Both Jacob and Kate are correct. To transform f(x) to g(x), you can translate and
then reflect. To transform g(x) to f(x), you can reflect and then translate.
© 2012 Carnegie Learning
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334
Chapter 5 Exponential Functions
Guiding Questions
for Share Phase,
Question 3
t If the graph of g(x) and
the graph of f(x) have a
different y-intercept, what
does this imply about the
transformation that was
used?
3. Choose the transformations in the box performed on f(x) that would produce the
graph of g(x). Then sketch the graph of g(x) on the coordinate plane.
vertical translation
horizontal translation
reflection over the line y 5 0
reflection over the line x 5 0
a. The graph of f(x) is shown.
y
t If the graph of g(x) and
the graph of f(x) have a
different y-intercept, what
does this imply about the
transformation that was not
used?
t If the graph of g(x) and
the graph of f(x) have the
same asymptote, what
does this imply about the
transformation that was
used?
x
The graph of g(x) has a different y-intercept than the graph of f(x), but the same
asymptote as f(x).
Possible transformation(s) on
f(x) to produce g(x):
Sketch of g(x)
y
Sample answers.
horizontal translation
reflection over the line y 5 0
x
t If the graph of g(x) and
the graph of f(x) have the
same asymptote, what
does this imply about the
transformation that was not
used?
b. The graph of f(x) is shown.
y
x
t If the graph of g(x) and
© 2012 Carnegie Learning
the graph of f(x) have the
same y-intercept, what
does this imply about the
transformation that was
used?
The graph of g(x) has the same y-intercept as f(x), but g(x) is decreasing.
Possible transformation(s) on
f(x) to produce g(x):
Sketch of g(x)
y
t If the graph of g(x) and
the graph of f(x) have the
same y-intercept, what
does this imply about the
transformation that was not
used?
t If the graph of g(x) is
decreasing and the graph
of f(x) is increasing, what
does this imply about the
transformation that was
used?
5
reflection over the line x 5 0
x
t If the graph of g(x) is decreasing and the graph of f(x) is increasing, what does
this imply about the transformation that was not used?
t If the graph of g(x) has a different asymptote than the graph of f(x), what does
this imply about the transformation that was used?
t If the graph of g(x) has a different asymptote than the graph of f(x), what does
this imply about the transformation that was used?
5.4
Reflections of Linear and Exponential Functions
335
c. The graph of f(x) is shown.
y
x
The graph of g(x) has a different asymptote than the graph of f(x), and g(x) is
increasing.
Possible transformation(s) on
f(x) to produce g(x):
Sketch of g(x)
y
vertical translation
x
Be prepared to share your solutions and methods.
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Chapter 5 Exponential Functions
© 2012 Carnegie Learning
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