639 6.4 Translations of the Graphs of the Sine and Cosine Functions SOLUTION (a) We use the maximum and minimum average monthly temperatures to find the amplitude a. a = 83 - 53 2 = 15 Amplitude a For c, we find the average of the maximum and minimum temperatures. c = 83 + 53 2 = 68 Vertical translation c Because temperatures repeat every 12 months, b can be found as follows. 12 = 2p b , so b = p 6 Solve for b. The coldest month is January, when x = 1, and the hottest month is July, when x = 7. A good choice for d is 4 because April, when x = 4, is located at the midpoint between January and July. Also, notice that the average monthly temperature in April is 69°F, which is very close to the value of the vertical translation, c. The average monthly temperature in New Orleans is modeled closely by the following equation. ƒ1x2 = a sin 3b1x - d24 + c ƒ1x2 = 15 sin c p 6 1x - 42d + 68 Substitute for a, b, c, and d. (b) Figure 39 shows two iterations of the data points from the table, along with the graph of y = 15 sinC p 6 1x - 42D + 68. The graph of y = 15 sin p 6 x + 68 is shown for comparison. (c) We used the given data for a two-year period and the sine regression capability of a graphing calculator to produce the model described in Figure 40(a). ƒ1x2 = 15.14 sin10.52x - 2.112 + 69.67 Its graph, along with the data points, is shown in Figure 40(b). S Now Try Exercise 63. 25 0 85 50 Figure 39 50 0 25 85 (b) Figure 40 (a) 6.4 Exercises CONCEPT PREVIEW Fill in the blank(s) to correctly complete each sentence. 1. The graph of y = sin Ax + p 4 B is obtained by shifting the graph of y = sin x unit(s) to the . (right / left) 2. The graph of y = cos Ax - p 6 B is obtained by shifting the graph of y = cos x unit(s) to the . (right / left)
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