656 CHAPTER 6 The Circular Functions and Their Graphs Calculators do not have keys for the cosecant and secant functions. To graph them with a graphing calculator, use csc x = 1 sin x and sec x = 1 cos x . Reciprocal identities 7 Techniques for Graphing Guidelines for Sketching Graphs of Cosecant and Secant Functions To graph y =a csc bx or y =a sec bx, with b 70, follow these steps. Step 1 Graph the corresponding reciprocal function as a guide, using a dashed curve. To Graph Use as a Guide y = a csc bx y = a sin bx y = a sec bx y = a cos bx Step 2 Sketch the vertical asymptotes. They will have equations of the form x = k, where k corresponds to an x-intercept of the graph of the guide function. Step 3 Sketch the graph of the desired function by drawing the typical U-shaped branches between the adjacent asymptotes. The branches will be above the graph of the guide function when the guide function values are positive and below the graph of the guide function when the guide function values are negative. The graph will resemble those in Figures 56 and 59 in the function boxes given earlier in this section. Like graphs of the sine and cosine functions, graphs of the secant and cosecant functions may be translated vertically and horizontally. The period of both basic functions is 2p. EXAMPLE 1 Graphing y =a sec bx Graph y = 2 sec 1 2 x. SOLUTION Step 1 This function involves the secant, so the corresponding reciprocal function will involve the cosine. The guide function to graph is y = 2 cos 1 2 x. Using the guidelines given earlier, we find that this guide function has amplitude 2 and that one period of the graph lies along the interval that satisfies the following inequality. 0 … 1 2 x … 2p 0 … x … 4p Multiply each part by 2. Dividing the interval 30, 4p into four equal parts gives these key points. 10, 22, 1p, 02, 12p, -22, 13p, 02, 14p, 22 Key points
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