# MATH 115 Precalculus Final Exam Fall, 2016

MATH 115 Precalculus

MATH 115 FINAL EXAMINATION

Fall, 2016, V4.3

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: | 3 – 8x | ≤ 21.

A. (–∞, −9/4] ∪ [3, ∞)

B. (–∞, –9/4]

C. [–9/4, 3]

D. [–9/4, ∞)

1. ______

2. Which of the following polynomials has a graph which exhibits the end behavior of

downward to the left and downward to the right?

A. f (x) = –6x5– x3– x2– 5

B. f (x) = 8x3+ 6x2– x + 3

2. _______

C.

D.

f (x) = 5x6+ 7x2+ x

f (x) = –3x4 + 9x2– x – 1

3. Express as an equivalent expression: 8 log y + log 1 – log (x – 3)

3. ________

A.

log

y

8

B.

y +

C.

log

x

(

)

-13D.

-13log 8y + 4 − x

MATH 115 Precalculus

4. Determine the interval(s) on which the function is increasing.

A. (− 4 / 3, 1)

B. (−1,1) and (3,∞)

Fall, 2016, V4.3

4. ______

C. (

)

11/ 3, ∞)

C. f ( x) = −ex

D. ( ) x 1

-40f x = e− +

5. _____

Page 2 of 7

MATH 115 Precalculus

6. Which of the functions corresponds to the graph?

A. f (x) = cos(2x) + 2

B. f (x) = 3 – sin x

C. f (x) = sin x + 3

D. f (x) = 2 cos x + 1

SHORT ANSWER:

7. Points (–7, 2) and (5, 6) are endpoints of the diameter of a circle.

Fall, 2016, V4.3

6. ______

(a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________

(b) What is the center of the circle?

(c) What is the equation of the circle?

Answer: ____________

Answer: ___________________________

8. Find the value of the logarithm: log

Answer: ____________

9. Bill, a resident of Metropolis, pays Metropolis an annual tax of $55 plus 1.8% of his annual

income. If Bill paid $1,441 in tax, what was Bill’s income?

Answer: _____________

Page 3 of 7

MATH 115 Precalculus

Fall, 2016, V4.3

10. A can of soda at 71° F. is placed in a refrigerator that maintains a constant temperature of 39°

F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by

T(t) = 39 + 32 e – 0.058 t

Find the temperature of the soda 20 minutes after it is placed in the refrigerator. (Round to the

nearest tenth of a degree.)

Answer: ________

11. Given the function

= 3 − 18 , find a formula for the inverse function.

Answer: __________________

12. (a) State the reference angle associated with 300°.

(b) Convert 300° to radians. Leave the answer in terms of ð.

13. Given y = 9 sin(8x – ð), state the

(a) period

(b) phase shift

Answer: ________

Answer: ________

Answer: ________

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 arccos sin

(b) Find the exact value of arcsin tan

16. For the parabola given by (y + 5)2= 8(x – 2), find the following:

(a) direction parabola opens (to the left, right, up, or down)

(b) vertex

(c) focus

Answer: ________

Answer: ________

Answer: ___________

Answer: ___________

Answer: ___________

Page 4 of 7

3x + 1

Fall, 2016, V4.3

17. Let

f (x) =

x − 2

(a) State the domain.

(b) State the vertical asymptote(s).

(c) State the horizontal asymptote.

(d) Which of the following represents the graph of

f x

( )

=

3x + 1

x − 2

Answer: _________________

Answer: _________________

Answer: _________________

Answer: ______________

GRAPH A.

GRAPH C.

GRAPH B.

GRAPH D.

MATH 115 Precalculus

SHORT ANSWER, with work required to be shown, as indicated.

Fall, 2016, V4.3

(a) Find the composite function ( f o g)(x) and simplify. Show work.

(b) Find (

f o g − . Show work.

21. A projectile is launched from a platform 15 feet high with an initial velocity of 48 feet per second.

The height h of the projectile at t seconds after launch is given by h = –16t2 + 48t + 15 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:

x − 6

x − 4

+

16

x2−16

= 0 . Show work.

23. Suppose that cos è = 5/13 and that è is a Quadrant IV angle.

(a) Find the exact value of sin è.

(b) Find the exact value of sin(2è ).

Show work.

Show work.

24. Prove the identity (sin x + cos x)2− sin(2x) = 1

MATH 115 Precalculus

Fall, 2016, V4.3

25. From a point 48 feet from the base of a redwood tree, the angle of elevation to the top of the

tree is 52.3°. Find the height of the tree to the nearest tenth of a 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 = ⟨10, –5⟩ and = ⟨2, 4⟩.

(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.

MATH 115 FINAL EXAMINATION

Fall, 2016, V4.3

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: | 3 – 8x | ≤ 21.

A. (–∞, −9/4] ∪ [3, ∞)

B. (–∞, –9/4]

C. [–9/4, 3]

D. [–9/4, ∞)

1. ______

2. Which of the following polynomials has a graph which exhibits the end behavior of

downward to the left and downward to the right?

A. f (x) = –6x5– x3– x2– 5

B. f (x) = 8x3+ 6x2– x + 3

2. _______

C.

D.

f (x) = 5x6+ 7x2+ x

f (x) = –3x4 + 9x2– x – 1

3. Express as an equivalent expression: 8 log y + log 1 – log (x – 3)

3. ________

A.

log

y

8

B.

y +

C.

log

x

(

)

-13D.

-13log 8y + 4 − x

MATH 115 Precalculus

4. Determine the interval(s) on which the function is increasing.

A. (− 4 / 3, 1)

B. (−1,1) and (3,∞)

Fall, 2016, V4.3

4. ______

C. (

)

11/ 3, ∞)

C. f ( x) = −ex

D. ( ) x 1

-40f x = e− +

5. _____

Page 2 of 7

MATH 115 Precalculus

6. Which of the functions corresponds to the graph?

A. f (x) = cos(2x) + 2

B. f (x) = 3 – sin x

C. f (x) = sin x + 3

D. f (x) = 2 cos x + 1

SHORT ANSWER:

7. Points (–7, 2) and (5, 6) are endpoints of the diameter of a circle.

Fall, 2016, V4.3

6. ______

(a) What is the exact length of the diameter? (Simplify as much as possible) Answer: ________

(b) What is the center of the circle?

(c) What is the equation of the circle?

Answer: ____________

Answer: ___________________________

8. Find the value of the logarithm: log

Answer: ____________

9. Bill, a resident of Metropolis, pays Metropolis an annual tax of $55 plus 1.8% of his annual

income. If Bill paid $1,441 in tax, what was Bill’s income?

Answer: _____________

Page 3 of 7

MATH 115 Precalculus

Fall, 2016, V4.3

10. A can of soda at 71° F. is placed in a refrigerator that maintains a constant temperature of 39°

F. The temperature T of the soda t minutes after it is placed in the refrigerator is given by

T(t) = 39 + 32 e – 0.058 t

Find the temperature of the soda 20 minutes after it is placed in the refrigerator. (Round to the

nearest tenth of a degree.)

Answer: ________

11. Given the function

= 3 − 18 , find a formula for the inverse function.

Answer: __________________

12. (a) State the reference angle associated with 300°.

(b) Convert 300° to radians. Leave the answer in terms of ð.

13. Given y = 9 sin(8x – ð), state the

(a) period

(b) phase shift

Answer: ________

Answer: ________

Answer: ________

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 arccos sin

(b) Find the exact value of arcsin tan

16. For the parabola given by (y + 5)2= 8(x – 2), find the following:

(a) direction parabola opens (to the left, right, up, or down)

(b) vertex

(c) focus

Answer: ________

Answer: ________

Answer: ___________

Answer: ___________

Answer: ___________

Page 4 of 7

3x + 1

Fall, 2016, V4.3

17. Let

f (x) =

x − 2

(a) State the domain.

(b) State the vertical asymptote(s).

(c) State the horizontal asymptote.

(d) Which of the following represents the graph of

f x

( )

=

3x + 1

x − 2

Answer: _________________

Answer: _________________

Answer: _________________

Answer: ______________

GRAPH A.

GRAPH C.

GRAPH B.

GRAPH D.

MATH 115 Precalculus

SHORT ANSWER, with work required to be shown, as indicated.

Fall, 2016, V4.3

(a) Find the composite function ( f o g)(x) and simplify. Show work.

(b) Find (

f o g − . Show work.

21. A projectile is launched from a platform 15 feet high with an initial velocity of 48 feet per second.

The height h of the projectile at t seconds after launch is given by h = –16t2 + 48t + 15 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:

x − 6

x − 4

+

16

x2−16

= 0 . Show work.

23. Suppose that cos è = 5/13 and that è is a Quadrant IV angle.

(a) Find the exact value of sin è.

(b) Find the exact value of sin(2è ).

Show work.

Show work.

24. Prove the identity (sin x + cos x)2− sin(2x) = 1

MATH 115 Precalculus

Fall, 2016, V4.3

25. From a point 48 feet from the base of a redwood tree, the angle of elevation to the top of the

tree is 52.3°. Find the height of the tree to the nearest tenth of a 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 = ⟨10, –5⟩ and = ⟨2, 4⟩.

(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.

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