Trigonometry Homework july 12

MATH 115 Section 6381 Summer 2020 OL1 Page 1 of 6

MATH 115 Summer 2020 Section 6381 Final Exam

The final exam is due at 11:59 pm EDT on July 14, 2020.

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. It is a violation of the UMGC Academic Dishonesty and

Plagiarism policy to use unauthorized materials or work from others.

Answer all 20 questions. Make sure your answers are as complete as possible.

Show all of your supporting work and reasoning. Answers that come straight

from calculators, programs or software packages without any explanation will

not be accepted. If you need to use technology (for example, Excel, online or

hand-held calculators, software packages) to aid in your calculation, you must

cite the sources and explain how you get the results.

Record your answers and work on the separate answer sheet provided.

This exam has 20 questions; 5 points for each question.

You must include the Honor Pledge on the title page of your submitted final

exam. Exams submitted without the Honor Pledge will not be accepted.

MATH 115 Section 6381 Summer 2020 OL1 Page 2 of 6

Questions 1 through 5 are multiple choice questions. Show the justification and / or work in the

answer sheet for full credit.

1. If 𝑓(𝑥) = 𝑥2−25

2𝑥−10 , and 𝑔(𝑥) = 𝑥 + 5, then

𝑓(5)

𝑔(5) =

(a) 0

(b) 1

(c) 1 2⁄

(d) Undefined

(e) None of the above

2. Given 𝑓(𝑥) = −𝑥2 + 4𝑥 + 3. The range of 𝑓(𝑥) is:

(a) (−∞, 3]

(b) (−∞, −1]

(c) [−1, ∞)

(d) (−∞, 7]

(e) [7, ∞)

3. Assume 𝑓(𝑥) = 3𝑥2−𝑥+2

−2𝑥 . What is the end behavior of 𝑓(𝑥) when 𝑥 → −∞?

(a) 𝑓(𝑥) → ∞

(b) 𝑓(𝑥) → −∞

(c) 𝑓(𝑥) → − 3

2

(d) 𝑓(𝑥) → 1

2

(e) None of the above

4. Let 𝑓 and 𝑔 be two functions, and 𝑔(𝑥) = 𝑓(𝑥 + 2) . Given 𝑓(−2) = 2,

𝑓(0) = 3, 𝑓(2) = 5, 𝑎𝑛𝑑 𝑓(4) = 4. What is 𝑔(0)?

(a) 2

(b) 3

(c) 5

(d) 4

(e) Cannot be determined

MATH 115 Section 6381 Summer 2020 OL1 Page 3 of 6

5. The graph of 𝑦 = 𝑓(𝑥) is shown below.

Which of the following graphs is 𝑦 = −𝑓(𝑥) ?

(c) Neither

(a) (b)

MATH 115 Section 6381 Summer 2020 OL1 Page 4 of 6

Questions 6 through 10 are True or False questions. Show the justification and / or work in the

answer sheet for full credit.

6. (True or False) The relation 𝑔(𝑥) is a function if

𝑔(𝑥) = { 𝑥 + 3 if 𝑥 < −1 2 if − 1 ≤ 𝑥 ≤ 1 𝑥2 + 1 if 𝑥 ≥ 1

7. (True or False) Since cos ( 5𝜋

3 ) =

1

2 , arccos (

1

2 ) =

5𝜋

3 .

8. (True or False) If an ellipse has the center at (2,5), one vertex at (2,1), and one focus at (2, 7),

then its major axis is parallel to the y-axis.

9. (True or False) If the equation of a parabola is 𝑦2 + 2𝑦 = −𝑥 + 6, then the directrix of the

parabola is parallel to y-axis

10. (True or False) If cos(𝜃) = 3

5 , then cos(−𝜃) = −

3

5 .

11. Solve the equation: 2𝑥 − 6 = √12 + 4𝑥. Be sure to check for extraneous solution(s).

Show work and rationale, and simplify your answer for full credit. Answers based on

technology (calculators, applets, software packages, etc.) are not accepted.

12. Solve the following equations analytically:

(a) 25−2𝑥 = 1

625

(b) log3(𝑥 − 4) + log3(𝑥 + 4) = 4

Show work and rationale for full credit. Answers based on technology (calculators, applets,

software packages, etc.) are not accepted.

MATH 115 Section 6381 Summer 2020 OL1 Page 5 of 6

13. Let 𝑓(𝑥) = 4𝑥−1

𝑥−2 when 𝑥 ≠ 2 .

(a) Find the inverse function 𝑓−1(𝑥). (3 points)

(b) What is the domain of 𝑓−1(𝑥)? Write your answer using interval notation. (1 point)

(c) What is the range of 𝑓−1(𝑥)? Write your answer using interval notation. (1 point)

Show work and rationale for full credit.

14. Suppose −2𝜋 ≤ 𝜃 ≤ −𝜋 with cos(𝜃) = − 12

13 .

(a) What Quadrant does the half angle 𝜃

2 lie in? (2 point)

(b) Find sin ( 𝜃

2 ). (3 points)

Show work and rationale, and simplify your answer for full credit. Answers based on

technology (calculators, applets, software packages, etc.) are not accepted.

15. Given 𝑦 = −3 sin(2𝑥 + 𝜋) − 1.

(a) Graph at least one cycle of the given function using graphing calculator or other software.

(3 points)

(b) State the period, amplitude, phase shift and vertical shift of the function. (2 points)

Show work and rationale, and label the x-axis and y-axis clearly for full credit.

16. Solve the equation, giving the exact solutions which lie in [−𝜋, 𝜋):

cos(2𝑥) + sin(𝑥) = 0

Express your solutions in radian measure. Show work and rationale, and simplify your

answer for full credit. Answers based on technology (calculators, applets, software packages,

etc.) are not accepted.

17. Write the standard equation of the circle (𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟2 which has (-1, 3) and (2, -1) as the endpoints of a diameter.

Show work and rationale, and simplify your answer for full credit. Answers based on

technology (calculators, applets, software packages, etc.) are not accepted.

MATH 115 Section 6381 Summer 2020 OL1 Page 6 of 6

18. Assume 𝛽 = 30°, 𝑏 = 7 𝑎𝑛𝑑 𝑐 = 14 in a triangle. (As in the text, (𝛼, 𝑎), (𝛽, 𝑏) and (𝛾, 𝑐) are

angle-side opposite pairs.)

(a) Use the Law of Sines or Law of Cosines to find the remaining side a and angles α and γ.

Give the exact answers and decimal approximations rounded to hundredths. (3 points)

(b) Find the area of the triangle. Give the exact answers and decimal approximations rounded to

hundredths. (2 points)

Show work and rationale, and simplify your answer for full credit. Answers based on

technology (calculators, applets, software packages, etc.) are not accepted.

19. Answer the following questions:

(a) ‖�⃗� ‖ = 4; when drawn in standard position 𝑢 ⃗⃗ ⃗ lies in Quadrant II and makes a 30° angle

with the positive y-axis. Find the component form of 𝑢 ⃗⃗ ⃗. (2 points)

(b) ‖𝑣 ‖ = 3; when drawn in standard position 𝑣 ⃗⃗⃗ lies in Quadrant III. In addition, 𝑣 ⃗⃗⃗ and 𝑢 ⃗⃗ ⃗

are orthogonal. Find the component form of 𝑣 ⃗⃗⃗ . (3 points)

Show work and rationale, and simplify your answer for full credit. Answers based on

technology (calculators, applets, software packages, etc.) are not accepted.

20. Given the equation of a hyperbola as: 25𝑥2 − 9𝑦2 − 150𝑥 + 36𝑦 + 414 = 0.

(a) Put the equation of the hyperbola in standard form (𝑥−ℎ)2

𝑎2 −

(𝑦−𝑘)2

𝑏2 = 1 or

(𝑦−𝑘)2

𝑏2 −

(𝑥−ℎ)2

𝑎2 = 1. (3 points)

(b) What are the slopes of the asymptotes? (1 point)

(c) Express the equations of the asymptotes in point-slope form. (1 point)

Show work and rationale, and simplify your answer for full credit. Answers based on

technology (calculators, applets, software packages, etc.) are not accepted.