Precalculus ~ Review Sheet

Transcription

1 Period: Date: Precalculus ~ Review Sheet Multiple Choice 1. The screen below shows the graph of a sound recorded on an oscilloscope. What is the period and the amplitude? (Each unit on the t-axis equals 0.01 seconds.) A 0.05 seconds, 4.5 C seconds, 9 B 0.05 seconds, 9 D seconds, 4.5 1

2 2. A sound wave has a period of 0.02 seconds and an amplitude of 3 units. Sketch a graph of the sound wave. A B C D 2

3 Use the graph to find the value of y = sin θ for the value of θ π radians A 0.7 B 0.7 C 0.9 D Find the period of the graph shown below. A 2π B 2 3 π 1 C 2 π D 4π 5. A particular sound wave can be graphed using the function y = 3 sin x. Find the amplitude and period of the function. A amplitude = 3, period = 2π C amplitude = 2π, period = 3 B amplitude = 1 2 π, period = 3 D amplitude = 3, period = 1 2 3

4 6. Sketch one cycle of y = 4 sin 4θ. A C B D 4

5 Write a cosine function for the graph. 7. A y = 2 cos x + 3π ˆ Á 4 2 C y = 2 cos x 3π ˆ Á 4 2 B y = 2 cos x 3π ˆ Á 4 2 D y = 2cos x + 3π ˆ Á Use the graph of y = tanx to find the value of y = tan 1 π. Round to the nearest tenth if necessary. If the tangent 8 is undefined at that point, write undefined. A 0 B 1 C 0.4 D 2.4 5

6 9. Sketch the graph of the tangent curve y = tan 2x. A C B D 10. Identify the period for y = tan 3π 4 θ and tell where two asymptotes occur for the function. A period = 3π 4 ; two asymptotes at x = 3π 8 and x = 3π 8 B period = 8 3 ; two asymptotes at x = 4 3 and x = 4 3 C period = 4 3 ; two asymptotes at x = 2 3 and x = 2 D period = 4 3 ; two asymptotes at x = 0 and x = 4 3 6

7 Graph the function in the interval from 0 to 2π. 11. y = 4 cos 1 2 θ A C B D 7

8 12. y = 2 cos x π Á 6 A ˆ + 2 C B D 8

9 13. y = sec x A C B D 14. Write an equation for the translation 3 units down of y = sin x. A y = sin (x + 3) C y = sin x + 3 B y = sin (x 3) D y = sin x 3 9

10 15. The graph below shows height as a function of time for a ride on a Ferris wheel. Find a sine equation for the graph. A y = 25 sin π Á 3 x ˆ + 20 C y = 20 sin π 3 x + 3π ˆ Á B y = 25 sin π Á 3 x ˆ D y = sin π 3 x 3π ˆ Á Find the amplitude of the periodic function. A 3.5 B 3 C 7 D

11 Short Answer 17. Use the graph of the sine function y = 2 sin x shown below. a. How many cycles occur in the graph? b. Find the period of the graph. c. Find the amplitude of the graph. 18. Consider the graph of the cosine function shown below. a. Find the period and amplitude of the cosine function. b. At what values of x for 0 θ 2π do the maximum value(s), minimum values(s), and zeros occur? 19. A particular sound wave is modeled in the laboratory with the function y = 3 sin 4 x π ˆ Á a. Find the amplitude of the function. b. Find the period of the function. c. Describe any phase shift in the graph. d. Describe any vertical shift in the graph. 20. a. Sketch at least one cycle of a sine curve with amplitude 1 and period of 4π. Assume a > 0. b. Write an equation for the graph. 21. An electromagnetic wave is modeled with the function y = 3 2 sin 1 θ. Sketch a graph of this function. 4 11

12 22. Sketch one cycle of y = 2sin 1 8 x + π ˆ and label. Then answer the questions below. Á a. What is the range? b. What is the amplitude? c. What is the period? 23. Sketch one cycle of y = 2csc 1 8 x + π ˆ and label. Then answer the questions below. Á a. What are all the vertical asymptotes? b. What is the range? 24. Graph one cycle of y = cos 1 2 πx ˆ 3 and label. Then answer the questions below. Á a. What is the y-intercept? b. What is the period? c. What is the amplitude? d. What are all the minimum points? 25. Graph one cycle of y = sec 1 2 πx ˆ 3 and label. Then answer the questions below. Á a. What is the domain? b. What is the amplitude? c. What kind of function is it? 26. Graph one cycle of y = tan 1 4 x ˆ and label. Then answer the questions below. Á a. What is the period? b. What are all the x-intercepts? c. What are all the asymptotes? Graph one cycle of y = cot 4 x ˆ and label. Then answer the questions below. Á a. What is the domain? b. What are the extrema? 12

13 Determine whether the function shown below is or is not periodic. If it is, find the period Find the period, range, and amplitude of the cosine function y = 3 2 cos t 2. 13

14 31. Write the equation for the sine function shown below. Essay 32. A sound wave is modeled with cosine function having an equation of y = 1 4 cos 2π 3 θ. a. Find the period of the cosine function. Explain your method. b. Find the amplitude of the cosine function. Explain your method. Other 33. Suppose that you record the average number of cars per month that pass through an intersection in a particular city every month for 10 years. Then you graph the data of the month where x is a number corresponding to the month (January = 1, February = 2, etc.), and y is the number of cars. Is this function periodic? Explain your reasoning. 34. Sketch the graphs of y = sin x, y = sin 2x, and y = 2 sin x. Describe how the graphs are alike and how they are different. 14

15 Precalculus ~ Review Sheet Answer Section MULTIPLE CHOICE 1. A 2. A 3. B 4. B 5. A 6. C 7. B 8. C 9. A 10. C 11. A 12. D 13. A 14. D 15. C 16. A SHORT ANSWER 17. a. 1 cycle b. 2π c a. The period is π and the amplitude is 4. b. maximum values: 0 : π : 2π 19. a. 3 b. 1 2 π minimum values: 1 2 π : 3 2 π zeros: 1 4 π : 3 4 π : 5 4 π : 7 4 π c. phase shift is π units to the right 2 d. vertical shift is 2 units up 1

16 20. a. 21. b. y = sin 1 2 θ 22. È a. ÎÍ 2,2 b. 2 c. 16π 2

17 23. a. x = 8πn and n Ω b. Á, 2 È ÎÍ 2, ˆ 24. a. Á0, 2ˆ b. 4 c. 1 d. Á2 + 4n, 4ˆ and n Ω 3

18 25. Ï a. Ô Ìx: x ò except x 1 + 2n and n Ω Ô ÓÔ Ô b. none c. even 26. a. 4π b. Á2πn,0ˆ and n Ω c. x = 2π + 4πn and n Ω 4

19 27. Ï a. Ô Ìx: x ò except x 2π + 4πn and n Ω Ô ÓÔ Ô b. none 28. not periodic 29. periodic; about period = 4π, range: 3 2 y 3 2, amplitude = y = sin 3θ 5

20 ESSAY 32. [4] a. The period of function y = a cos bθ, with b > 0, is 2π b y = 1 4 cos 2π 3 θ, b = 2π 3.. For the equation 2π b = 2π 2π 3 3 = 2π 2π Substitute. = 3 Simplify. The period of the function is 3. Multiply by the reciprocal to divide. b. The amplitude for a function of the form y = a cos bθ, with b > 0, is a. For the equation y = 1 4 cos 2π 3 θ, a = 1 4. The amplitude of the function is 1 4. [3] one incorrect computation OR an error in reasoning [2] two computational errors OR poor explanation [1] one or more answers missing OR no explanation OTHER 33. Answers may vary. Sample: It will not be a periodic function. The number of cars passing through an intersection at least partially depends upon the population. Population growth is normally exponential and not periodic. 6

21 34. Answers may vary. Sample: All three graphs pass through (0º, 0), (180º, 0) and (360º, 0). The graphs of y = sin x and y = sin 2x have amplitude 1 while the graph of y = 2 sin x has amplitude 2. The graphs of y = sin x and y = 2 sin x have a period of 360º, while the graph of y = sin 2x has a period of 180º. Note: We will be using radians instead of degrees to explain this answer :-) 7