625 6.3 Graphs of the Sine and Cosine Functions The interval 30, p4 is divided into four equal parts using these x-values. 0, p 4 , p 2 , 3p 4 , p We plot the points from the table of values for y = sin 2x given at the top of the previous page and join them with a smooth sinusoidal curve. More of the graph can be sketched by repeating this cycle. See Figure 26. The amplitude is not changed. Left endpoint First-quarter point Midpoint Third-quarter point Right endpoint y x 1 0 –1 y = sin x y = sin 2x 2p –p p – – 3p 2 3p 2 3p 4 p 2 p 2 p 4 Figure 26 We can think of the graph of y =sin bx as a horizontal stretching of the graph of y =sin x when 0 *b *1 and as a horizontal shrinking when b +1. S Now Try Exercise 27. Period For b 70, the graph of y =sin bx will resemble that of y = sin x, but with period 2p b . Also, the graph of y =cos bx will resemble that of y = cos x, but with period 2p b . −2 2 3p 0 This screen shows a graph of the function in Example 3. When we choose Xscl = 3p 4 , the x-intercepts, maxima, and minima coincide with tick marks on the x-axis. EXAMPLE 3 Graphing y =cos bx Graph y = cos 2 3 x over one period. SOLUTION Divide 2p by b = 2 3 to determine the period. 2p 2 3 = 2p , 2 3 = 2p # 3 2 = 3p. We divide the interval 30, 3p4 into four equal parts to obtain the x-values 0, 3p 4 , 3p 2 , 9p 4 , and 3p that yield minimum points, maximum points, and x-intercepts. We use these values to obtain a table of key points for one period. To divide by a fraction, multiply by its reciprocal. x 0 3p 4 3p 2 9p 4 3p 2 3 x 0 p 2 p 3p 2 2p cos 2 3 x 1 0 -1 0 1 x y 0 –1 –2 1 2 y = cos x2 3 3p 3p 2 3p 4 9p 4 Figure 27 The amplitude is 1 because the maximum value is 1, the minimum value is -1, and 1 2 31 - 1-124 = 1 2 122 = 1. We plot these points and join them with a smooth curve. The graph is shown in Figure 27. S Now Try Exercise 25.
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