657 6.6 Graphs of the Secant and Cosecant Functions These key points are plotted and joined with a dashed red curve to indicate that this graph is only a guide. An additional period is graphed as shown in Figure 60(a). Step 2 Sketch the vertical asymptotes as shown in Figure 60(a). These occur at x-values for which the guide function equals 0, such as x = -3p, x = -p, x = p, x = 3p. Step 3 Sketch the graph of y = 2 sec 1 2 x by drawing typical U-shaped branches, approaching the asymptotes. See the solid blue graph in Figure 60(b). S Now Try Exercise 11. –2 y 0 2 x y = 2 cos x is used as a guide. 1 2 –4p –3p –2p 2p 3p 4p –p p –2 x y 0 2 y = 2 sec x –4p –3p –2p 2p 3p 4p –p p 1 2 (a) (b) Figure 60 −4 4 4p −4p This is a calculator graph of the function in Example 1. −4 4 11p 4 11p 4 − This is a calculator graph of the function in Example 2. (The use of decimal equivalents when defining y1 eliminates the need for some parentheses.) EXAMPLE 2 Graphing y =a csc 1x −d2 Graph y = 3 2 csc Ax - p 2 B . SOLUTION Step 1 Graph the corresponding reciprocal function y = 3 2 sin ax - p 2b, shown as a red dashed curve in Figure 61. Step 2 Sketch the vertical asymptotes through the x-intercepts of the graph of y = 3 2 sin Ax - p 2 B . These x-values have the form 12n + 12 p 2 , where n is any integer. See the black dashed lines in Figure 61. Step 3 Sketch the graph of y = 3 2 csc Ax - p 2 B by drawing the typical U-shaped branches between adjacent asymptotes. See the solid blue graph in Figure 61. x y 0 1 2 –1 y = sin(x – ) 3 2 p 2 y = csc(x – ) 3 2 p 2 –p p 2p – 3p 2 3p 2 5p 2 p 2 –p 2 Figure 61 S Now Try Exercise 13.
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