MATH 115 FINAL EXAMINATION | Complete Solution

MATH 115 Precalculus Spring, 2016, V4.3
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, and express the answer in interval notation: | 7 – 3x | ≤ 11. 1. ______
A. (–∞, –4/3]
B. (–∞, –4/3] ∪ [6, ∞)
C. [–4/3, 6]
D. (–∞, 6] ∪ [–4/3, ∞)
2. Which of the following polynomials has a graph which exhibits the end behavior of
downward to the left and downward to the right? 2. _______
A. f (x) = –3x
6 + 6x
3
– x
B. f (x) = –4x
5
 – 5x
3 – 1
C. f (x) = 7x
4 + 2x + 6
D. f (x) = 6x
3 + 3x
2
 – x
3. Express as an equivalent expression: 9 log x – log (y + 3) + log 1 3. ______
 A. 9 1 log
3
x
y
  +
    +
 B. log 9 2 ( x y − − )
 C.
9
log
log ( 3)
x
y +
 D.
9
log
3
x
y
      +

vvvv
MATH 115 Precalculus Spring, 2016, V4.3
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4. Determine the interval(s) on which the function is increasing. 4. ______
 A. (–3, 1)
 B. (–3.6, 0) and (6.7, ∞)

C. (–∞, –2) and (4, ∞)

D. (–2, 4)
5. Which of the functions corresponds to the graph? 5. ______

A. ( ) 2
x
f x e = −
B. ( ) 1
x
f x e−
= −
C. ( ) 1
x
f x e−
= +
D. ( )
x
f x e = −
 
MATH 115 Precalculus Spring, 2016, V4.3
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6. Which of the functions corresponds to the graph? 6. ______

 A. f (x) = cos(2x) + 2
B. f (x) = sin x + 3
C. f (x) = 3 – sin x
D. f (x) = 2 cos x + 1
SHORT ANSWER:
7. Points (–5, 8) and (–9, 2) 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: log

    
. Answer: ____________
9. Mimi, a resident of Metropolis, pays Metropolis an annual tax of $35 plus 1.9% of her annual
income. If Mimi paid $1,127.50 in tax, what was Mimi’s income?
 Answer: _____________
 
MATH 115 Precalculus Spring, 2016, V4.3
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10. Water initially at 190° F. is left in a room of temperature 70° F to cool.
After t minutes, the temperature T of the water is given by T(t) = 70 + 120e
– 0.096 t
Find the temperature of the water 10 minutes after it is left to cool. (Round to the nearest degree.)
 Answer: ________

11. Given the function
= 9 − 1
7

, find a formula for the inverse function.
 Answer: __________________

12. (a) State the reference angle associated with 210°. Answer: ________
 (b) Convert 210° to radians. Leave the answer in terms of π. Answer: ________
13. Given y = 6 sin(8x – π), state the
 (a) period Answer: ________
 (b) phase shift Answer: ________
14. Solve the trigonometric equation (cos x)(2cos x + 1) = 0 in the interval [0, 360°).
 Answer: _________________
15. (a) Find the exact value of arcsin sin

 Answer: ________
 (b) Find the exact value of arccos sin

 Answer: ________
16. For the parabola given by (y + 9)2
 = 8(x – 4), find the following:
 (a) direction parabola opens (to the left, right, up, or down) Answer: ___________
 (b) vertex Answer: ___________
 (c) focus Answer: ___________ 
MATH 115 Precalculus Spring, 2016, V4.3
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17. Let 2 18 ( )
5
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 18 ( )
5
x
f x
x
−
=
−
 ? Answer: ______________
GRAPH A. GRAPH B.

GRAPH C. GRAPH D.
 
MATH 115 Precalculus Spring, 2016, V4.3
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SHORT ANSWER, with work required to be shown, as indicated.
18. Find the equation for a line which passes through the points (6, 3) and (2, – 9). Write the
equation in slope-intercept form. Show work.
19. Find the exact solutions and simplify as much as possible: 24x
2
 + 1 = 12x. Show work.
20. Let f (x) = 6x
2
 + 7 and g(x) = x – 4.
(a) Find the composite function ( f o g)(x) and simplify. Show work.
(b) Find )1 ( f o g)(− . Show work.
21. A projectile is launched from a platform 14 feet high with an initial velocity of 80 feet per second,
The height h of the projectile at t seconds after launch is given by h = –16t
2
 + 80t + 14 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
10 42 0
7 49
x
x x
−
+ =
− −
. Show work.
23. Suppose that cos θ = 5/13 and that θ is a Quadrant IV angle.
 (a) Find the exact value of sin θ. Show work.
 (b) Find the exact value of sin 2θ. Show work.
24. Prove the identity (sin x + cos x)
2
− 1 = sin(2x)
 
MATH 115 Precalculus Spring, 2016, V4.3
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25. From a point 48 feet from the base of a redwood tree, the angle of elevation to the top of the
tree is 56.3°. 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 = 46°, B = 64°, and c = 35.0.

Find the length of side a rounded to the nearest tenth. Show work.
27. Let = 〈2, –6〉 and = 〈15, 5〉.
(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
= 1
(a) Is the major axis horizontal or vertical?
(b) Find the exact values of the foci of the ellipse. Show work.