456 CHAPTER 6 Trigonometric Functions Solution Begin with a scatter plot of the data for one year. See Figure 84.The data will be fitted to a sine function of the form ω φ ( ) = − + y A x B sin Step 1 To find the amplitude A, compute = − = − = Amplitude largest data value smallest data value 2 74.2 30.0 2 22.1 To see the remaining steps in this process, superimpose the graph of the function = y x 22.1sin , where x represents months, on the scatter plot. Figure 85 shows the two graphs. To fit the data, the graph needs to be shifted vertically, shifted horizontally, and stretched horizontally. Step 2 Determine the vertical shift by finding the average of the highest and lowest data values. = + = Vertical shift 74.2 30.0 2 52.1 Now superimpose the graph of = + y x 22.1sin 52.1 on the scatter plot. See Figure 86. We see that the graph needs to be shifted horizontally and stretched horizontally. Step 3 It is easier to find the horizontal stretch factor first. Since the temperatures repeat every 12 months, the period of the function is = T 12. Because π ω = = T 2 12, then ω π π = = 2 12 6 Now superimpose the graph of π( ) = + y x 22.1sin 6 52.1 on the scatter plot. See Figure 87, where the graph still needs to be shifted horizontally. Step 4 To determine the horizontal shift, use the period = T 12 and divide the interval [ ] 0, 12 into four subintervals of length ÷ = 12 4 3: [ ] [ ] [ ] [ ] 0, 3 , 3, 6 , 6, 9 , 9, 12 The sine curve is increasing on the interval [ ] 0, 3 and is decreasing on the interval [ ] 3, 9 , so a local maximum occurs at = x 3. The data indicate that a maximum occurs at = x 7 (corresponding to July’s temperature), so the graph of the function must be shifted 4 units to the right by replacing x by −x 4. Doing this yields π( ) ( ) = − + y x 22.1sin 6 4 52.1 Distributing reveals that a sine function of the form ω φ ( ) = − + y A x B sin that fits the data is π π ( ) = − + y x 22.1sin 6 2 3 52.1 The graph of π π ( ) = − + y x 22.1sin 6 2 3 52.1 and the scatter plot of the data are shown in Figure 88. Figure 88 75 25 0 13 Figure 85 75 225 0 13 Figure 86 75 25 0 13 Figure 84 75 25 0 13 Figure 87 75 25 0 13
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