Galen Kaspar College Algebra — sample problems

Galen Kaspar
College Algebra — sample problems
This sampler of College Algebra problems has items in several categories:
1, 2) review simple arithmetic and algebra
3, 4, 5) write an expression to fit a prose description
6, 7) coordinate plane and distance formula
8) inequalities and interval notation
9) average rate-of-change
10, 11) lines and slopes
12) examine 5 tables, detect linear functions
13, 14, 15, 16) quadratic polynomials
17) translation of a graph
18) power functions
19) box without a lid (cubic model)
20) polynomial division
Problems 6, 9, 10, 15, 17, 18 create and display a graph.
3. (1 pt) (a) Write an expression for the total cost of buying
7 apples at $a each and 5 pears at $p each. Your expression
should be in terms of a and p.
$
help (formulas)
(b) Find the total cost if apples cost $0.45 each and pears cost
$0.70 each.
help (numbers)
$
4. (1 pt)
You buy a pot and its lid for a total of $ 11. The sales person tells
you that the pot by itself costs $ 10 more than the lid. The price
of the pot is $
and the price of the lid is $
5. (1 pt) The selling price of a textbook is $125.35. If the
markup is 15% of the bookstore’s cost, what is the bookstore’s
cost of the textbook?
a) Write an equation to model the problem. Use x to represent the number.
Answer:
b) Solve the equation to find the bookstore’s cost. (Note:
Your answer should be in the form $ddd.cc. )
Answer:
1. (1 pt) REVIEW: Use the order of operations to simplify:
7(2x − 1) − (4x − 9) =
2. (1 pt)
For each statement below enter a T (true) the statement is
true and an F (false) otherwise. In this problem you need to get
everything correct before receiving credit. Whenever there is a
division below,we assume that the divisor is non-zero.
For all real numbers a, b, and x
a − b(x − 1) = a − bx − b.
For all real numbers a, b, and x
a − b(x − 1) = a − bx + b.
For all real numbers a, b, and x
3 + bx
= 1 + bx.
3
6. (1 pt)
For all real numbers a, b, and x
2
2
REVIEW: Give the coordinates of the points on the graph.
a) A :
b) B :
c) C :
d) D :
2
(x − r) = x − r .
For all real numbers a, b, and x
(x − r)2 = x2 − 2rx + r2 .
7. (1 pt) Find the perimeter of the triangle with the vertices
at
(2, 0), (-3, 6), and (-3, -6).
For all real numbers a, b, and x
(x − r)(x + r) = x2 − r2 .
1
• A. It is the average velocity of the car over the first two
hours.
• B. It is how far the car will travel in a half-hour.
• C. It is the slope of the line.
• D. It is the total distance the car travels in five hours.
• E. It represents the car’s velocity.
• F. It is the acceleration of the car over the five hour time
interval.
• G. None of the above
8. (1 pt) Match the statements defined below with the letters
labeling their equivalent intervals.
You must get all of the answers correct to receive credit.
1.
2.
3.
4.
5.
A.
B.
C.
D.
E.
x ∈ [1, 7)
x ∈ (−∞, 1)
x ∈ [1, ∞)
x ∈ (−∞, 1]
x ∈ [1, 7]
1≤x<7
x≤1
x<1
1≤x≤7
1≤x
9. (2 pts) Question 10:
The graph below shows the distance traveled, D (in miles)
as a function of time, t (in hours).
10. (1 pt)
Use the graphs given above and list the slopes m1 , m2 , m3 , m4 in
order of decreasing size, i.e. biggest slope first. (Type m1 for
m1 and so on. Separate the slopes with commas.)
Answer:
11. (1 pt) For the line given by, 3y − 2x + 4 = 0, find the slope
of a line that is:
a) Parallel to the given line: m parallel =
b) Perpendicular to the given line: m perpendicular =
(Click on the graph to get a larger version.)
12. (1 pt) (a) Could the table represent a linear function? ?
a) For each of the intervals, find the values of 4D and 4t between the indicated start and end times. Enter your answers in
their respective columns in the table below.
x=
y=
7
30
12
60
17
90
22
120
27
150
(b) Could the table represent a linear function? ?
Time Interval
t = 1.5 to t = 4.5
t = 1 to t = 3
t = 2 to t = 4.5
4D
4t
x=
y=
2
6
4
8
8
12
16
20
32
36
(c) Could the table represent a linear function? ?
x=
y=
b) Based on your results from (a) it follows that the average
rate of change of D is constant, it does not depend over which
interval of time you choose. What is the constant rate of change
of D ? 4D
4t =
-6
21
-3
12
0
3
3
-6
6
-15
(d) Could the table represent a linear function? ?
c) Which of the statements below CORRECTLY explains the
significance of your answer to part (b)? Select ALL that apply
(more than one may apply).
x=
y=
2
-2
3
0
2
4
0
10
-3
18
-7
(e) Could the table represent a linear function? ?
x=
y=
2
5
4
10
8
15
16
20
17. (1 pt) The graph of y = x2 is given below. (To look at the
graph in a separate window, you can click on it).
32
25
13. (1 pt) The expression (3 − 3x)2 equals Ax2 + Bx +C
where A equals:
and B equals:
and C equals:
14. (1 pt) Factor the trinomial 10x2 − 39x + 35.
Find a formula for the function whose graph is given below.
Note: your answer should be in the form (Ax − B)(Cx − D).
.
15. (1 pt)
Find a possible formula for the quadratic function in
the graph.
f (x) =
y=
help (formulas)
18. (1 pt)
16. (1 pt) (a) Complete the square by writing 4x2 + 56x + 1
in the form a(x − h)2 + k. Note: the numbers a, h and k can be
positive or negative.
4x2 + 56x + 1 =
·
2
+
help (formulas)
Each graph in the figure is the graph of a power function f (x) = kx p . Match each graph with its corresponding value (or range of values) for p.
(b) Solve the equation 4x2 + 56x + 1 = 0 by completing the
square or using the quadratic formula. If there is more than one
correct answer, enter your answers as a comma separated list. If
there are no solutions, enter NONE.
x=
?
?
?
?
help (numbers)
3
1.
2.
3.
4.
p>1
0< p<1
p=1
p<0
A
19. (1 pt) A box without a lid is constructed from a 28 inch
by 28 inch piece of cardboard by cutting x in. squares from each
corner and folding up the sides.
a) Determine the volume of the box as a function of the variable
x.
V (x) =
b) Use a graphing calculator to approximate the values of x that
produce a volume of 1543.5.
Note: There are 3 values of x that produce the given value but
only two of them are acceptable in the context of the problem.
List the two answers, to at least one decimal place, separated by
commas.
x=
B
20. (1 pt) Find the quotient and remainder using long division for
x3 − 5x2 + 9x − 21
.
x−4
C
D
The quotient is
The remainder is
(Click on a graph to enlarge it)
c
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Team, Department of Mathematics, University of Rochester
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