Free Response Final Exam Review
Precalculus
Name: ________________________
Date: ________________Hour: ___
1. Find a formula using the sine function for your height above ground after t minutes on
the ferris wheel. The ferris wheel is 20 meters in diameter and boarded in the six o’clock
position from a platform 4 meters above the ground. The wheel completes one full
revolution every 2 minutes. At time t = 0 you are at the 12 o’clock position.
2. A population of animals oscillates between a low of 1300 on January 1 (t = 0) and a high
of 2200 on July 1 (t = 6).
a. Find a formula for the population, P, in terms of time, t, in months.
b. Interpret the amplitude, period, and midline of the function P = f(t).
c. Use a graph to estimate when P = 1500.
3. The pressure, P (in lbs/ft2), in a pipe varies over time. Five times an hour, the pressure
oscillates from a low of 90 to a high of 230 and then back to a low of 90. The pressure at
t = 0 is 90.
a. Graph P = f(t), where t is time in minutes. Label your axes.
b. Find a possible formula for P = f(t).
c. Using the graph of P = f(t) for 0 ≤ t ≤ 2, estimate when the pressure first equals
115 lbs/ft2.
4. Two fire stations are located 56.7 miles apart, at point A and B. There is a forest fire at
point C. The angle at point A is 54° and the angle at point B is 58°.
a. Which station is closer?
b. How much closer?
5. To measure the height of the Eiffel Tower in Paris, a person stands away from the base
and measures the angle of elevation to the top of the tower to be 60°. Moving 210 feet
closer, the angle to the top of the tower is 70°. How tall is the Eiffel Tower?
6. To estimate the width of an archaeological mound, archaeologists place two stakes on
opposite ends of the widest point. (see figure below) They set a third stake 82 feet from
one stake and 97 feet from the other stake. The angle formed is 125°. Find the width of
the mound.
Stake
Mound
Stake
82 ft
97 ft
125°
Stake
7. Use the table to construct values for r(x) = p(q(x)).
x
p(x)
q(x)
r(x)
0
1
5
1
0
2
2
5
3
3
2
1
4
3
4
5
4
8
8. Use the tables for f(x) and g(y) to find g(f(x)).
x
0




6
4
3
2
f(x)
0
1
2
2
2
3
2
1
y
0
g(y)
1


4
2
4
1
2
2
2
3
4
3
2
1
x
0
2

0




3

4
0

6
0
9. From the table and graph calculate:
a. f(g(4)) =
x
f(x)
0
2
1
5
2
3
3
4
4
1
5
0
b. g(f(4)) =
d. g(g(0)) =
g(x)
y
c. f(f(0)) =
5
4
3
2
1
x
1
2
3
4
5
e. Find x for g(g(x)) = 1. x =
10. Let f(x) = ex. Solve each of the following equations exactly for x.
a. (f(x))-1 = 2
b. f-1(x) = 2
c. f(x-1) = 2
6
4
3
2
g(f(x))
11. Let f(x) = x + 1 and g(x) = x2 – 1. In parts a) to e), write a formula in terms of f(x) and
g(x) for the function. Then evaluate the formula for x = 3. Write a formula in terms of x
for each function. Check your formula for x = 3.
a. h(x) is the sum of f(x) and g(x).
b. j(x) is the difference between g(x) and twice f(x).
c. k(x) is the product of f(x) and g(x).
d. m(x) is the ratio of g(x) to f(x).
e. n(x) is defined by n(x) = (f(x))2 – g(x).
For #12, #13 and #14, find possible formulas for the power functions.
12.
x
f(x)
2
12
3
27
4
48
5
75
4
 32
13.
x
g(x)
-6
36
-2
4
3
3
9
x
-2
1
1
h(x)
1
2
3
14.
2
2
-8
4
-32
4
1
8
Using limit notation describe the long run behavior of the function in #15 and #16. What
power function does each resemble?
15. y = 4x4 – 2x2 + 3
16. y = 16x3 – 4023x2 - 2
17. Find:
a.
lim(3x
x 
2
 5x  7)
b.
lim(7 x
x 
2
 9 x3 )
For #18 and #19,
a) find possible polynomial formulas and for each.
b) Describe the behavior of the function for the following:
i) x  
ii) x  
iii) x  0
18.
iv) x  0
19.
For the following rational functions, find the zeros, y-intercept, vertical asymptote(s),
horizontal asymptote, and domain.
2x 2  10x  8
20. f  x  
x 2  16
2x 2  6x
21. f  x   3
x  3x 2  2x