Exam 2 Review
Precalculus
Name:_____________________________
1.
x
f(x)
-1
–9.25
0
-6
1.5
– 1.125
2
0.5
a. Define a function that could represent the behavior modeled in the table above.
b. Evaluate f(5).
2.
x
g(x)
0
0.5
1
4.5
1.5
13.5
2
40.5
a. Define a function that could represents the behavior modeled in the table above.
b. Evaluate g(5).
3. The population of Northfield, New Jersey is expected to increase by 1.8% every three years. The
population of Northfield in 2012 was 8,624 people. Let n be the number of years since 2009.
a. What is the 3-unit growth/decay factor?
b. What is the 1-unit percent change?
c. What is the ½ unit growth/decay factor?
d. Approximate the population of Northfield in 2009.
e. Write a function P that models the population of Northfield as a function of the number of years
since 2009.
g. Algebraically determine in what year the population of Northfield will be 9,172 people.
4. Algebraically determine the value of x.
a. log2 2 +log2 (4x) + 1 = 4
b. log4 –2 = x
c. log4 x = –2
5. Joe is pouring water in a pitcher. He pours 3 equal sized cupfuls of water into the pitcher and notices
the height of the water in the pitcher is increasing at a decreasing rate with respect to volume. He
then pours 2 more equal sized cupfuls of water into the pitcher and notices the height of the water in
the pitcher is increasing at a constant rate with respect to the volume of water. Joe adds 5 more equal
sized cupfuls of water and sees that the height of the water in the pitcher is increasing at an increasing
rate with respect to volume.
a. As the first 3 cupfuls of water are poured into the pitcher describe how the height of the water and
the volume of water in the bottle vary together.
b. Construct a graph of the height of the water in the bottle as a function of the volume of water in
the bottle. Be sure to label your axis. Then make a careful sketch of the bottle that would
produce a height-volume graph with that general shape.
6. A ball is kicked up from a platform. The height of the ball, d, above the ground (measured in meters)
can be modeled by the function k(t)=9 + 10t – 4.897t2 where d=k(t) and t represents the amount of
time (in seconds) since the ball was kicked. Approximately how long did it take the ball to fall from
its maximum height to the ground?
Exam 2 Review
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© 2013 Carlson and Oehrtman
Exam 2 Review
Precalculus
7. Let
a.
b.
c.
d.
f (x) = 3x(x − 2)2 (3x + 9)(2x −14) .
Identify the end behavior of the function.
What are the roots of the function f ?
Sketch a graph of the function f .
On what interval(s) is the function increasing?
8. Determine whether the function listed/described is exponential or polynomial. If the function is
exponential determine the 1-unit growth factor AND the initial value. If the function is polynomial
determine the degree of the polynomial function.
a. f(x)=0.2(3.5)7t
b. g(x) = 3x3+1
c.
n
-1
0
1.5
2
t(n)
-9.25
-6
-1.125
0.5
d.
s
0
1
1.5
2
q(s)
0.5
4.5
13.5
40.5
9. Kelsie set p pennies next to a checkerboard. She then placed triple that number of pennies on the first
square. Then she placed on the next square triple the number of pennies that were on the first square.
She continued this pattern of always placing on the next square triple the number of pennies on the
previous square. How many pennies will be on the 5th square?
10. A quadratic function has roots at x = 9 and x = – 3. Determine the axis of symmetry for this function.
How does this relate to the vertex of this function?
11. Define a polynomial function g with the following characteristics:
-g has roots of multiplicity 2 at x=4 and x=1 and a root of multiplicity 1 at x=-3.
-g passes through the point (1, -4).
12. Describe what log 2.1 (24) represents.
13. Rewrite the following statement as an equation: “There are x factors of 4 in 2120.”
Exam 2 Review
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© 2013 Carlson and Oehrtman
Exam 2 Review
Precalculus
14. Use the graph of the function, f, shown below, to answer the following questions.
a.
b.
c.
d.
e.
f.
Given the graph of f as shown above, what are the roots of f, and what are their multiplicities?
f(0) =
Identify the degree of the function.
Identify (by marking on the graph above) the inflection point(s) of the function f.
Approximate on what interval(s) the function is concave down.
Approximate on what interval(s) the function is decreasing.
15. Given the graph of f as shown to the left:
a. What are the roots of f ?
b. What is the y-intercept of f ?
c. Identify (by labeling the points as I on the graph) the
inflection point(s) of the function f .
d. Approximate on what interval(s) the function is concave
up:
e. As x → ∞ , f (x) → ____ . Fill in the blank and then
write a sentence that interprets the meaning of this statement.
f. As x → −∞ , f (x) → ____ . Fill in the blank and then
write a sentence that interprets the meaning of this statement.
g. Is this function of even or odd degree? How do you
know?
h. Write the function definition for f .
Exam 2 Review
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© 2013 Carlson and Oehrtman
Exam 2 Review
Precalculus
16. Given the function h(x) = (x + 1) (x − 2)(x + 3)
a. Determine the roots of h.
2
b. Determine whether h(x) changes sign (from positive to negative or negative to positive) or whether
h(x) keeps the same sign on either side of each root.
c. Examine the leading term of h to determine the end-behavior of h.
d. Determine on what interval(s) of the domain h is increasing.
e. Determine on what interval(s) of the domain h is decreasing.
f. Sketch a graph of h.
Exam 2 Review
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© 2013 Carlson and Oehrtman
Exam 2 Review
Precalculus
17. Consider Segment A and Segment B shown below.
a. The length of Segment A is _______ times as long as the length of Segment B.
b. The length of Segment B is _______ times as long as the length of Segment A.
c. The length of Segment B is ______ % as long as the length of Segment A.
d. The length of Segment B is ______ % longer than the length of Segment A.
18. A customer buys a shirt that is discounted 23%. What percent of the original price will the customer
pay for the shirt?
19. As the value of x increases from x = 3 to x = 5, the value of g(x) increases by 25%.
g(5) is ______ times as large as g(3).
20. Suppose h is an exponential function such that h(-3)=4 and the 3-unit growth factor of h is 2.5.
a. What is the value of h(0)?
b. What is the value of h(3)?
c. What is the value of the 1-unit growth factor?
d. Define the function h.
21. A rock is thrown upward from a bridge that is 20 feet above the surface of the lake. The rock reaches
its maximum height above the surface of the lake 0.25 seconds after it is thrown and reaches the surface
of the lake 1.50 seconds after it was thrown. Define a quadratic function, f, that gives the height of the
rock above the surface of the lake (in feet) in terms of the number of seconds elapsed since the rock was
thrown, t.
22. The domain of the function g is [2, 5] and the range of the function g is [0,7].
a. If the function h is defined as h(x)=g(x)+3, what is:
i. the domain of h?
ii. the range of h?
b. If the function j is defined as j(x)=g(x+1), what is:
i. the domain of j?
ii. the range of j?
c. If the function w is defined as w(x)=g(x-2)-4, what is:
i. the domain of w?
ii. the range of w?
Exam 2 Review
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© 2013 Carlson and Oehrtman