627 6.3 Graphs of the Sine and Cosine Functions Step 4 Plot the points 10, -32, A1 2 , 0B, 11, 32, A 3 2 , 0B, and 12, -32, and join them with a sinusoidal curve having amplitude -3 = 3. See Figure 29. Step 5 The graph can be extended by repeating the cycle. Notice that when b is an integer multiple of P, the first coordinates of the x-intercepts of the graph are rational numbers. S Now Try Exercise 37. EXAMPLE 5 Graphing y =a cos bx (Where b Is a Multiple of P) Graph y = -3 cos px over one period. SOLUTION Step 1 Here b = p and the period is 2p p = 2, so we will graph the function over the interval 30, 24. Step 2 Dividing 30, 24 into four equal parts yields the x-values 0, 1 2 , 1, 3 2 , and 2. Step 3 Make a table using these x-values. x 0 1 2 1 3 2 2 Px 0 p 2 p 3p 2 2p cos Px 1 0 -1 0 1 −3 cos Px -3 0 3 0 -3 x y –3 –2 –1 1 0 2 3 2 1 y = –3 cos Px 1 2 3 2 Figure 29 Connecting Graphs with Equations EXAMPLE 6 Determining an Equation for a Graph Determine an equation of the form y = a cos bx or y = a sin bx, where b 70, for the given graph. SOLUTION This graph is that of a cosine function that is reflected across its horizontal axis, the x-axis. The amplitude is half the distance between the maximum and minimum values. 1 2 32 - 1-224 = 1 2 142 = 2 The amplitude a is 2. Because the graph completes a cycle on the interval 30, 4p4, the period is 4p. We use this fact to solve for b. 4p = 2p b Period = 2p b 4pb = 2p Multiply each side by b. b = 1 2 Divide each side by 4p. An equation for the graph is y = -2 cos 1 2 x. Horizontal stretch x-axis reflection S Now Try Exercise 41. x y –3 –2 –1 1 0 2 3 4p 3p 2 p p
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