623 6.3 Graphs of the Sine and Cosine Functions The calculator graphs of ƒ1x2 = sin x in Figure 22 and ƒ1x2 = cos x in Figure 24 are shown in the ZTrig viewing window c - 11p 4 , 11p 4 d by 3-4, 44 A11p 4 ≈8.639379797B of the TI-84 Plus calculator, with Xscl = p 2 and Yscl = 1. (Other models have different trigonometry viewing windows.) 7 Techniques for Graphing, Amplitude, and Period The examples that follow show graphs that are “stretched” or “compressed” (shrunk) either vertically, horizontally, or both when compared with the graphs of y = sin x or y = cos x. EXAMPLE 1 Graphing y =a sin x Graph y = 2 sin x, and compare to the graph of y = sin x. SOLUTION For a given value of x, the value of y is twice what it would be for y = sin x. See the table of values. The change in the graph is the range, which becomes 3-2, 24. See Figure 25, which also includes a graph of y = sin x. x 0 p 2 p 3p 2 2p sin x 0 1 0 -1 0 2 sin x 0 2 0 -2 0 x y 2 1 0 –1 –2 y = sin x Period: 2p y = 2 sin x –2p 2p –p p – – 3p 2 3p 2 p 2 p 2 Figure 25 The amplitude of a periodic function is half the difference between the maximum and minimum values. It describes the height of the graph both above and below a horizontal line passing through the “middle” of the graph. Thus, for the basic sine function y = sin x (and also for the basic cosine function y = cos x), the amplitude is computed as follows. 1 2 31 - 1-124 = 1 2 122 = 1 Amplitude of y = sin x For y = 2 sin x, the amplitude is 1 2 32 - 1-224 = 1 2 142 = 2. Amplitude of y = 2 sin x We can think of the graph of y =a sin x as a vertical stretching of the graph of y =sin x when a +1 and as a vertical shrinking when 0 *a *1. S Now Try Exercise 15. −4 4 11p 4 11p 4 − The graph of y = 2 sin x is shown in blue, and that of y = sin x is shown in red. Compare to Figure 25. Amplitude The graph of y =a sin x or y =a cos x, with a≠0, will have the same shape as the graph of y = sin x or y = cos x, respectively, except with range 3- a , a 4. The amplitude is a .
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