642 CHAPTER 6 The Circular Functions and Their Graphs Graph each function over a two-period interval. See Examples 1 and 2. 39. y = cos ax - p 2b 40. y = sin ax - p 4b 41. y = sin ax + p 4b 42. y = cos ax + p 3b 43. y = 2 cos ax - p 3b 44. y = 3 sin ax - 3p 2 b Graph each function over a one-period interval. See Example 3. 45. y = 3 2 sinc 2 ax + p 4b d 46. y = - 1 2 cos c 4 ax + p 2b d 47. y = -4 sin12x - p2 48. y = 3 cos14x + p2 49. y = 1 2 cos a 1 2 x - p 4b 50. y = - 1 4 sin a 3 4 x + p 8b (Modeling) Solve each problem. See Example 6. 63. Average Monthly Temperature The average monthly temperature (in °F) in Seattle, Washington, is shown in the table. (a) Plot the average monthly temperature over a two-year period, letting x = 1 correspond to January of the first year. Do the data seem to indicate a translated sine graph? (b) The highest average monthly temperature is 66°F, and the lowest average monthly temperature is 41°F. Their average is 53.5°F. Graph the data together with the line y = 53.5. What does this line represent with regard to temperature in Seattle? (c) Approximate the amplitude, period, and phase shift of the translated sine wave. (d) Determine a function of the form ƒ1x2 = asin3b1x - d24 + c, where a, b, c, and d are constants, that models the data. (e) Graph ƒ together with the data on the same coordinate axes. How well does ƒ model the given data? (f) Use the sine regression capability of a graphing calculator to find the equation of a sine curve that fits these data (over a two-year interval). Month °F Month °F Jan 42 July 66 Feb 43 Aug 66 Mar 47 Sept 61 Apr 50 Oct 53 May 56 Nov 45 June 61 Dec 41 Data from National Climatic Data Center. 51. y = -3 + 2 sinx 52. y = 2 - 3 cos x 53. y = -1 - 2 cos 5x 54. y = 1 - 2 3 sin 3 4 x 55. y = 1 - 2 cos 1 2 x 56. y = -3 + 3 sin 1 2 x 57. y = -2 + 1 2 sin3x 58. y = 1 + 2 3 cos 1 2 x Graph each function over a two-period interval. See Example 4. Graph each function over a one-period interval. See Example 5. 59. y = -3 + 2 sinax + p 2b 60. y = 4 - 3 cos1x - p2 61. y = sinc 2ax + p 4b d + 1 2 62. y = cos c 3ax - p 6b d - 5 2
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