Pre-Calculus Final
MATH 115 Precalculus Fall, 2018, V1.4
Page 1 of 7
MATH 115 FINAL EXAMINATION
This is an open-book exam. You may refer to your text and other course materials as you work
on the exam, and you may use a calculator. You must complete the exam individually.
Neither collaboration nor consultation with others is allowed.
Record your answers and work on the separate answer sheet provided.
There are 28 problems.
Problems #1–6 are Multiple Choice.
Problems #7–17 are Short Answer. (Work not required to be shown)
Problems #18–28 are Short Answer with work required to be shown.
MULTIPLE CHOICE
1. Solve | 7 – 5x | 8 and write interval notation for the solution set. 1. _______
A. (−∞, − 1
5 ] ∪ [3, ∞)
B. (−∞, 3] ∪ [− 1
5 , ∞)
C. [− 1
5 , ∞)
D. [− 1
5 , 3]
2. Which of the following polynomials has a graph which exhibits the end behavior of
upward to the left and upward to the right? 2. ______
A. f (x) = x3 – 3x2 – x
B. f (x) = 3x6 + x2 – x
C. f (x) = –7x4 – x3 – 5
D. f (x) = –6x5 + x3 + 8
3. Write as an equivalent expression: 9 log x – log (y + 3) + log 1 3. _______
A. 9
log
log ( 3)
x
y +
B. ( )log 9 2x y− −
C. 9 1
log 3
x
y
+
+
D. 9
log 3
x
y
+
vvvv
MATH 115 Precalculus Fall, 2018, V1.4
Page 2 of 7
4. Determine the interval(s) on which the function is decreasing. 4. ______
A. (–, –3.5) (0, 3.5)
B. (– 5.3, 5.3)
C. (– 3.5, 3.5) D. (– 2, 2)
5. Which of the functions corresponds to the graph? 5. ______
A. ( ) 1xf x e−= +
B. ( ) 2xf x e−= +
C. ( ) 2xf x e−=
D. ( ) 2 xf x e−=
MATH 115 Precalculus Fall, 2018, V1.4
Page 3 of 7
6. Which of the functions corresponds to the graph? 6. ______
A. f (x) = 3 – sin x
B. f (x) = sin x + 3
C. f (x) = 2 cos x + 1
D. f (x) = cos(2x) + 2
SHORT ANSWER:
7. Points (–9, 2) and (–5, 8) are endpoints of the diameter of a circle.
(a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________
(b) What is the center of the circle? Answer: ____________
(c) What is the equation of the circle? Answer: ___________________________
8. Find the value of the logarithm: log8 ( 1
64 ). Answer: ____________
9. A salesperson earns a base salary of $1,020 per month and a commission of 7.8% on the
amount of sales. If the salesperson has a paycheck of $4,725 for one month, what was the
amount of sales for the month?
Answer: _____________
MATH 115 Precalculus Fall, 2018, V1.4
Page 4 of 7
10. A bowl of soup at 170 F. is placed in a room of constant temperature of 60 F. The
temperature T of the soup t minutes after it is placed in the room is given by
T(t) = 60 + 110 e – 0.075 t
Find the temperature of the soup 8 minutes after it is placed in the room. (Round to the nearest degree.) Answer: _________
11. Given the function 𝑓(𝑥) = 3
8 −
1
6 𝑥, find a formula for the inverse function.
Answer: __________________
12. (a) State the reference angle associated with 300°. Answer: ________
(b) Convert 300° to radians. Leave the answer in terms of . Answer: ________
13. Given y = 7 sin(4x – ), state the
(a) period Answer: ________
(b) phase shift Answer: ________
14. Solve the trigonometric equation (√2 sin(𝑥) + 1)(2 cos(𝑥) + 1) = 0 in the interval [0, 360°). Find the exact values of the solutions in degrees.
Answer: _________________
15. (a) Find the exact value of arccos (tan 5𝜋
4 ) Answer: ________
(b) Find the exact value of arcsin (sin 2𝜋
3 ) Answer: ________
16. For the parabola given by (y – 2)2 = –8(x + 7), find the following:
(a) direction parabola opens (to the left, right, up, or down) Answer: ___________
(b) vertex Answer: ___________
(c) focus Answer: ___________
MATH 115 Precalculus Fall, 2018, V1.4
Page 5 of 7
17. Let 2 4
( ) 3
x f x
x
− =
− .
(a) State the domain. Answer: _________________
(b) State the horizontal asymptote. Answer: _________________
(c) State the vertical asymptote(s). Answer: _________________
(d) Which of the following represents the graph of 2 4
( ) 3
x f x
x
− =
− ? Answer: ______________
GRAPH A. GRAPH B.
GRAPH C. GRAPH D.
MATH 115 Precalculus Fall, 2018, V1.4
Page 6 of 7
SHORT ANSWER, with work required to be shown, as indicated.
18. Find the equation for a line which passes through the points (–1, 9) and (3, 1). Write the
equation in slope-intercept form. Show work.
19. Find the exact solutions and simplify as much as possible: 8x2 = 12x + 5. Show work.
20. Let f (x) = 5x2 + 8 and g(x) = x − 3.
(a) Find the composite function ))(( xgf o and simplify. Show work.
(b) Find )1)(( −gf . Show work.
21. A projectile is launched from a platform 25 feet high with an initial velocity of 112 feet per second,
The height h of the projectile at t seconds after launch is given by h = –16t2 + 112t + 25 feet.
(a) How many seconds after launch does the projectile attain maximum height? Show work.
(b) What is the maximum height? Show work.
22. Solve: 2
0 8 30
5 25
x
x x =
− +
− − . Show work.
23. Suppose that sin = 1/5 and that is a Quadrant II angle.
(a) Find the exact value of cos . Show work.
(b) Find the exact value of sin(2 ). Show work.
24. Prove the identity (sin x – cos x)2 = 1 – sin(2x)
MATH 115 Precalculus Fall, 2018, V1.4
Page 7 of 7
25. From a point 63 feet from the base of a redwood tree, the angle of elevation to the top of the
tree is 51.8°. Find the height of the tree to the nearest foot. Show work. (sketch is not to scale)
26. For the triangle ABC, we are given that A = 42, B = 58, and c = 16.0.
Find the length of side b, rounded to the nearest tenth. Show work.
27. Let �⃗⃗� = 15, −3 and �⃗⃗� = 4, 20.
(b) Calculate the dot product �⃗⃗� ∙ �⃗⃗� . Show work.
(c) Determine the angle between �⃗⃗� and �⃗⃗� . Round the result to the nearest degree. Show work.
28. An ellipse has the equation (𝑥 − 5)2
25 +
(𝑦 + 1)2
49 = 1
(a) Is the major axis horizontal or vertical?
(b) Find the exact values of the foci of the ellipse. Show work.