Math 1330 Precalculus Electronic Homework (EHW 6) Section 5.2 Work the following problems and choose the correct answer. The problems that refer to the Textbook may be found at www.casa.uh.edu in the Online Textbook under the Exercises link in the appropriate section. Record your answers by logging into your CASA account, go to the EMCF link in the top menu bar, selecting the appropriate assignment in the EMCF menu and recording your choices on the EMCF form. You MUST record your answers by the deadline given. No late homework will be accepted. Problem 5.2.2 refers to problem 2 in Chapter 5, Section 2 in the online text. 1. Problem 5.2.2 A. period is 2 units, amplitude is 8 units B. period is 8 units, amplitude is 5 units C. period is 5 units, amplitude is 8 units D. period is 5 units, amplitude is 4 units E. None of the above 2. Problem 5.2.4 A. period is 2 units, amplitude is 6 units B. period is 2 units, amplitude is 3 units C. period is 2 units, amplitude is -6 units D. period is 6 units, amplitude is 2 units E. None of the above 3. Problem 5.2.6 A. period is 2 units, amplitude is 5 units 3 Page 1 of 8 2 units, amplitude is 10 units 3 B. period is C. period is 3 units, amplitude is 10 units D. period is 10 units, amplitude is 2 units 3 E. None of the above 4. Problem 5.2.10 A. A B. A C. A D. None of the above 5. 4 , 1 , 1 2 , 1 B 3 , 1 4 C B 3 , 1 B 3 , 1 2 b. x D 2 ,0 ,0 C 4 ,0 D 8 ,0 C 2 ,0 D 4 ,0 Problem 5.2.14 A. a. x 0, B. a. x 3 2 2 b. x C. a. x 0, b. D. a. x 3 2 2 b. E. None of these , , c. x 0 c. x 0 x c. x 0 x c. x 0 2 3 2 Page 2 of 8 6. Problem 5.2.16 A. a. x B. a. x C. a. x D. a. x E. None of these 7. 2 , 2 3 2 , 2 b. x b. x b. x b. x 3 2 2 2 3 2 c. x c. x c. x c. x , 2 3 2 , 2 2 Problem 5.2.28 - use the following list rather than the one in the text A. reflect about the y-axis, shift left 2 and up 3 B. reflect about the x-axis, shift right 2 and up 3 C. reflect about the x-axis, shift left 2 and up 3 D. reflect about the y-axis, shift right 2 and up 3 E. None of the above 8. Problem 5.2.32 - use the following list rather than the one in the text A. shrink (compress) horizontally, shift 8 units right B. shrink (compress) vertically shift 8 units right C. shrink (compress) vertically, shift 4 units right D. shrink (compress) horizontally, shift 4 units right E. None of the above. Page 3 of 8 9 & 10 You are strongly urged to: Work these problems completely before looking at any of the answer choices. You will probably need to produce a graph of a trigonometric function on your test. 9. A. Problem 5.2.46 1 2 Period: Amplitude: 3 5 Phase Shift: None C. Period: 6 Phase Shift: 1 right 3 B. Vertical Shift: None Amplitude: 2 5 Vertical Shift: None Period: 2 5 Amplitude: 6 Phase Shift: None Vertical Shift: None D. Period: 6 Phase Shift: None Amplitude: 2 5 Vertical Shift: None E. None of these Page 4 of 8 10. A. Problem 5.2.58 Period: 4 Phase Shift: 3 2 Right Amplitude: 5 Vertical Shift: None C. Phase Shift: Period: Amplitude: 5 3 4 right Vertical Shift: None B. Period: 4 Amplitude: 5 D. Period: 4 Amplitude:5 3 2 Phase Shift: Left Vertical Shift: None Phase Shift: 3 4 Left Vertical Shift: None E. None of these Page 5 of 8 11. Which of the following is a sine function which has amplitude 2 and period 4? A. y 2 sin 4 x B. y 2 sin C. y 4 sin 2 x D. y 4 sin E. None of these 12. 2 4 x x Given the following, determine the maximum, minimum, and x-intercepts in the interval 2 ,2 . y 3 cos x 2 A. Maximum: 3/2 Minimum: -3/2 x-intercepts: B. Maximum: 3 Minimum: -3 x-intercepts: 0, C. Maximum: 3 Minimum: -3 x-intercepts: D. Maximum: 3 Minimum: -3 x-intercepts: 2 ,0 , 2 2 3 ,0 2 , 0, ,0 , 3 2 3 ,0 2 ,0 E. None of these Page 6 of 8 13. Find the range of the following function: f x 3 sin x 5 2 3 A. [-5,5] B. [2,8] C. [-3,3] D. [-8,8] E. [-2,-8] F. None of these 14. Scientists believe that the average temperatures at various places on Earth vary from cooler to warmer over thousands of years. At one place on Earth, the highest average temperature is 800 and the lowest is 600. Suppose it takes 20,000 years to go from the high to low average and the average temperature was at a high point of 800 in year 2000. Set up a sinusoidal function T (t ) (where t is time in years) to model this phenomenon. A. T (t ) 20 cos B. T (t ) 20 cos C. T (t ) 10 cos D. T (t ) 10 cos E. T (t ) 10 cos t 2000 70 t 2000 70 t 2000 70 t 2000 70 20, 000 10, 000 20, 000 10, 000 20, 000 t 70 F. None of these Page 7 of 8 15. The voltage V produced by an alternating current generator is V t 229 sin 120 t . What are the amplitude and period of V(t)? A. amplitude: 114.5 period: 1/60 B. amplitude: 229 period: 60 C. amplitude: 229 period: 120 D. amplitude: 229 period: 1/60 E. amplitude: 458 period: 1/60 F. None of these Page 8 of 8 CHAPTER 5 Graphing Trigonometric Functions Section 5.2: Graphs of the Sine and Cosine Functions Basic Graphs Transformations Basic Graphs Graphing the Sine Function: Graphing the Cosine Function: 446 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Definition of a Periodic Function and Its Period: Sine Function: MATH 1330 Precalculus 447 CHAPTER 5 Graphing Trigonometric Functions Cosine Function: Definition of Amplitude: Sine Function: 448 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Cosine Function: Example: Solution: MATH 1330 Precalculus 449 CHAPTER 5 Graphing Trigonometric Functions Additional Example 1: Solution: Additional Example 2: Solution: 450 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Additional Example 3: Solution: Additional Example 4: Solution: MATH 1330 Precalculus 451 CHAPTER 5 Graphing Trigonometric Functions Transformations Vertical Scaling: Graph of y = Asin(x): 452 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Graph of y = Acos(x): MATH 1330 Precalculus 453 CHAPTER 5 Graphing Trigonometric Functions Vertical Translations: Graph of y = sin(x) + D: Graph of y = cos(x) + D: 454 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Example: Solution: MATH 1330 Precalculus 455 CHAPTER 5 Graphing Trigonometric Functions Horizontal Scaling: Graph of y = sin(Bx): 456 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Example: Solution: Graph of y = cos(Bx): MATH 1330 Precalculus 457 CHAPTER 5 Graphing Trigonometric Functions Example: Solution: 458 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Horizontal Translations: Graph of y = sin(x - C): Graph of y = cos(x - C): MATH 1330 Precalculus 459 CHAPTER 5 Graphing Trigonometric Functions Example: Solution: Graphs of y = Asin(Bx - C) and y = Acos(Bx - C): 460 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Example: Solution: MATH 1330 Precalculus 461 CHAPTER 5 Graphing Trigonometric Functions Graphs of y = Asin(Bx - C) + D and y = Acos(Bx - C) + D: 462 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Example: Solution: MATH 1330 Precalculus 463 CHAPTER 5 Graphing Trigonometric Functions Additional Example 1: Solution: 464 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Additional Example 2: Solution: MATH 1330 Precalculus 465 CHAPTER 5 Graphing Trigonometric Functions Additional Example 3: 466 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Solution: Part (a): MATH 1330 Precalculus 467 CHAPTER 5 Graphing Trigonometric Functions Part (b): 468 University of Houston Department of Mathematics SECTION 5.2 Graphs of the Sine and Cosine Functions Additional Example 4: Solution: Part (a): MATH 1330 Precalculus 469 CHAPTER 5 Graphing Trigonometric Functions Part (b): 470 University of Houston Department of Mathematics Exercise Set 5.2: Graphs of the Sine and Cosine Functions For each of the following functions, (a) Find the period. (b) Find the amplitude. 1. 6. y x y x Answer the following. 7. 2. y (a) Use your calculator to complete the following chart. Round to the nearest hundredth. x x y sin x x y sin x 0 3. y x 6 7 6 4 5 4 3 4 3 2 3 2 2 3 5 3 3 4 7 4 5 6 11 6 2 4. y x (b) Plot the points from part (a) to discover the graph of f x sin x . y x 5. y x (c) Use the graph from part (b) to sketch an extended graph of f x sin x , where 4 x 4 . Be sure to show the intercepts as well as the maximum and minimum values of the function. MATH 1330 Precalculus 471 Exercise Set 5.2: Graphs of the Sine and Cosine Functions (d) State the domain and range of f x sin x . Do not base this answer (c) Use the graph from part (b) to sketch an extended graph of g x cos x , where simply on the limited domain from part (c), but the entire graph of f x sin x . 4 x 4 . Be sure to show the intercepts as well as the maximum and minimum values of the function. (e) State the amplitude and period of f x sin x . (f) Give an interval on which f x (d) State the domain and range of g x cos x . Do not base this answer simply on the limited domain from part (c), but the entire graph of g x cos x . sin x is increasing. 8. (e) State the amplitude and period of g x cos x . (a) Use your calculator to complete the following chart. Round to the nearest hundredth. x y cos x x y (f) Give an interval on which g x cos x is decreasing. cos x 0 6 7 6 4 5 4 3 4 3 2 3 2 2 3 5 3 3 4 7 4 5 6 11 6 Answer the following. 9. Use the graph of f x cos x below to determine the coordinates of points A, B, C, and D. y f C D A x B 2 10. Use the graph of f x (b) Plot the points from part (a) to discover the graph of g x cos x . sin x below to determine the coordinates of points A, B, C, and D. y y f x C D x A 472 B University of Houston Department of Mathematics Exercise Set 5.2: Graphs of the Sine and Cosine Functions Use the graphs of f x sin x and g x cos x to answer the following. 11. Use the appropriate graph to find the following function values. which: (a) sin x 1 (b) sin x (a) cos 0 (c) sin 5 2 (d) sin 3 1 (c) sin x 3 2 (b) cos x 0 , use the graph of 16. For 2 f x sin x to find the value(s) of x for 0 Graph each of the following functions over the interval 2 x 2 . 12. Use the appropriate graph to find the following function values. 17. f x cos x 18. g x sin x (a) sin 0 (b) sin Matching. The left-hand column contains equations that represent transformations of f x sin x . 2 Match the equations on the left with the description on the right of how to obtain the graph of y g x from (c) cos 7 2 (d) cos 13. For 0 x the graph of f . 2 , use the graph of f x sin x to find the value(s) of x for which: 19. y sin x 3 20. y sin x A. Shrink horizontally by a 1 factor of 3 . 3 (a) sin x 1 21. y sin 3x B. Shift left 3 units, then reflect in the y-axis. (b) sin x 0 22. y 1 sin x 3 C. Reflect in the x-axis, then shift downward 3 units. 23. y 3sin x D. Shift right 3 units. 24. y sin x 3 25. y sin E. Shift right 2 units, then reflect in the x-axis, then shift upward 3 units. 26. y 1 sin x 3 27. y sin x 3 (c) sin x 14. For 0 x 1 2 , use the graph of g x cos x to find the value(s) of x for which: (a) cos x 0 (b) cos x (c) cos x 1 1 x 0 , use the graph of 15. For 2 g x cos x to find the value(s) of x for which: (a) cos x 1 (b) cos x 1 (c) cos x 0 MATH 1330 Precalculus x 3 F. Shift upward 3 units. G. Stretch horizontally by a factor of 3. 28. y sin x 2 29. y sin x 30. y sin x 3 3 H. Shift left 3 units, then shift upward 2 units. 2 3 I. Shift left 3 units. J. Shift downward 3 units. K. Stretch vertically by a factor of 3. L. Shrink vertically by a 1 factor of 3 . 473 Exercise Set 5.2: Graphs of the Sine and Cosine Functions Matching. The left-hand column contains equations that represent transformations of f x cos x . 37. f x 4sin x Match the equations on the left with the description on the right of how to obtain the graph of y g x from 38. f x 3cos x the graph of f . 39. f x 5cos x 40. f x 2sin x 31. y cos 2 x 8 32. y cos 2 x 8 33. y cos 2 x 34. y cos 35. y 36. y 1 2 8 x 8 cos 1 2 x 8 cos 1 2 x 8 A. Stretch horizontally by a factor of 2, then shift right 16 units. 41. f x 6sin x 2 42. f x 4cos x 3 43. f x cos 2 x B. Shrink horizontally by a 1 factor of 2 , then shift right 4 units. C. Shrink horizontally by a 1 44. f x sin 3x D. Stretch horizontally by a factor of 2, then shift right 8 units. 45. f x cos factor of 2 , then shift downward 8 units. E. Stretch horizontally by a factor of 2, then shift downward 8 units. 46. f x 2 x cos 5 3 47. f x 4 sin 3 x 48. f x cos x 3 F. Shrink horizontally by a 1 factor of 2 , then shift right 8 units. sin State the period. State the amplitude. State the phase shift. State the vertical shift. Use transformations to sketch the graph of the function over one period. For consistency in solutions, perform the appropriate transformations on the function g x sin x or h x cos x where 50. f x 4 cos 51. f x cos x 52. f x sin x 53. f x 3cos 0 x 2 . (Note: The resulting graph may not fall within this same interval.) The x-axis should be labeled to show the transformations of the intercepts of g x sin x or 54. f x 5sin 4 x 55. f x sin 2 x 3 cos x , as well as any x-values where 56. f x cos 3x h x a maximum or minimum occurs. The y-axis should be labeled to reflect the maximum and minimum values of the function. 57. f x 58. f x 474 x 2 49. f x For each of the following functions, (a) (b) (c) (d) (e) x 4 4 1 x 3 2 2 x 7 cos 4 x 5sin 1 3 x 2 4 University of Houston Department of Mathematics Exercise Set 5.2: Graphs of the Sine and Cosine Functions 59. f x 3sin 2 x 60. f x 5cos 61. f x 4sin 62. f x 3 x 2 7sin 64. f x 10sin 65. f x 66. f x 2 2 2sin 2 x Omit part (c) 6sin 3x 68. f x 5sin y Omit part (c) x 2 71. 4 2 Omit part (c) 4 cos y x x 67. f x 70. 5 2 x 3cos 2 x 63. f x 2 3 4 x Omit part (c) y 72. 3 Omit part(c) x x Omit part(c) x For each of the following graphs, (a) Give an equation of the form f x A sin Bx C D which could be 73. y x used to represent the graph. (Note: C or D may be zero. Answers vary.) (b) Give an equation of the form f x A cos Bx C D which could be used to represent the graph. (Note: C or D may be zero. Answers vary.) 69. y 74. y x x MATH 1330 Precalculus 475 Exercise Set 5.2: Graphs of the Sine and Cosine Functions 75. y x y 76. x Answer the following. 77. The depth of the water at a certain point near the shore varies with the ocean tide. Suppose that the high tide occurs today at 2AM with a depth of 6 meters, and the low tide occurs at 8:30AM with a depth of 2 meters. (a) Write a trigonometric equation that models the depth, D, of the water t hours after midnight. (b) Use a calculator to find the depth of the water at 11:15AM. (Round to the nearest hundredth.) 78. The depth of the water at a certain point near the shore varies with the ocean tide. Suppose that the high tide occurs today at 4AM with a depth of 9 meters, and the low tide occurs at 11:45AM with a depth of 3 meters. (a) Write a trigonometric equation that models the depth, D, of the water t hours after midnight. (b) Use a calculator to find the depth of the water at 2:30PM. (Round to the nearest hundredth.) 476 University of Houston Department of Mathematics