SECTION 6.6 Phase Shift; Sinusoidal Curve Fitting 455 SUMMARY Steps for Graphing Sinusoidal Functions y A x B sin ω φ ( ) = − + or y A x B cos ω φ ( ) = − + Step 1 Find the amplitude A , 8 period π ω = T 2 , and phase shift φ ω . Step 2 Determine the starting point of one cycle of the graph, φ ω . Determine the ending point of one cycle of the graph, φ ω π ω + 2 . Divide the interval φ ω φ ω π ω + ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ , 2 into four subintervals, each of length π ω ÷ 2 4. Step 3 Use the endpoints of the subintervals to find the five key points on the graph. Step 4 Plot the five key points, and connect them with a sinusoidal graph to obtain one cycle of the graph. Extend the graph in each direction to make it complete. Step 5 If ≠ B 0, apply a vertical shift. 2 Build Sinusoidal Models from Data Scatter plots of data sometimes resemble the graph of a sinusoidal function. For example, the data given in Table 11 represent the average monthly temperatures in Denver, Colorado. Since the data represent average monthly temperatures collected over many years, the data will not vary much from year to year and so will essentially repeat each year. In other words, the data are periodic. Figure 83 shows a scatter plot of the data, where = x 1 represents January, = x 2 represents February, and so on. Notice that the scatter plot looks like the graph of a sinusoidal function. We choose to fit the data to a sine function of the form ω φ ( ) = − + y A x B sin where ω A B , , ,and φ are constants. January, 1 February, 2 March, 3 April, 4 May, 5 June, 6 July, 7 August, 8 September, 9 October, 10 November, 11 December, 12 30.7 32.5 40.4 47.4 57.1 67.4 74.2 72.5 63.4 50.9 38.3 30.0 Source: U.S. National Oceanic & Atmospheric Administration Month, x Average Monthly Temperature, °F Table 11 Figure 83 Denver average monthly temperature 0 12 2 4 6 8 10 80 30 y x Finding a Sinusoidal Function from Temperature Data Fit a sine function to the data in Table 11. EXAMPLE 3 (continued)
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