638 CHAPTER 6 The Circular Functions and Their Graphs Step 3 Make a table of values. y = –1 y x 0 1 –2 –3 y = –1 + 2 sin(4x + P) –1 –p 4 –p 2 p 4 p 2 p 8 3p 8 – 3p 8 p 8 – Figure 37 x - p 4 - p 8 0 p 8 p 4 x +P 4 0 p 8 p 4 3p 8 p 2 41x +P 4 2 0 p 2 p 3p 2 2p sin 341x +P 4 2 4 0 1 0 -1 0 2 sin 341x +P 4 2 4 0 2 0 -2 0 −1 +2 sin 14 x +P2 -1 1 -1 -3 -1 Steps 4 and 5 Plot the points found in the table and join them with a sinusoidal curve. Figure 37 shows the graph, extended to the right and left to include two full periods. S Now Try Exercise 59. A Trigonometric Model For natural phenomena that occur in periodic patterns (such as seasonal temperatures, phases of the moon, or heights of tides), a sinusoidal function will provide a good approximation of a set of data points. EXAMPLE 6 ModelingTemperature with a Sine Function The maximum average monthly temperature in New Orleans, Louisiana, is 83°F, and the minimum is 53°F. The table shows the average monthly temperatures. The scatter diagram for a two-year interval in Figure 38 strongly suggests that the temperatures can be modeled with a sine curve. 50 0 25 85 Figure 38 Month °F Month °F Jan 53 July 83 Feb 57 Aug 83 Mar 63 Sept 80 Apr 69 Oct 71 May 77 Nov 63 June 82 Dec 56 Data from National Climatic Data Center. (a) Using only the maximum and minimum temperatures, determine a function of the form ƒ1x2 = a sin 3b1x - d24 + c, where a, b, c, and d are constants, that models the average monthly temperature in New Orleans. Let x represent the month, with January corresponding to x = 1. (b) On the same coordinate axes, graph ƒ for a two-year period together with the actual data values found in the table. (c) Use the sine regression feature of a graphing calculator to determine a second model for these data.
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