654 CHAPTER 6 The Circular Functions and Their Graphs Secant Function ƒ1x2 =sec x Domain: Ex x ≠12n + 12 p 2 , where n is any integerF Range: 1-∞, -14 ´31, ∞2 x y - p 2 undefined - p 4 22 0 1 p 4 22 p 2 undefined 3p 4 - 22 p -1 3p 2 undefined x y =sec x 0 1 { p 6 223 3 ≈1.2 { p 4 22 ≈1.4 { p 3 2 { 2p 3 -2 { 3p 4 - 22 ≈ -1.4 { 5p 6 - 223 3 ≈ -1.2 {p -1 x y –2 –1 0 2 y = sec x P p 2 –P 2 –p Figure 54 Because secant values are reciprocals of corresponding cosine values, the period of the secant function is 2p, the same as for y = cos x. When cos x = 1, the value of sec x is also 1. Likewise, when cos x = -1, sec x = -1. For all x, -1 … cos x … 1, and thus, sec x Ú 1 for all x in its domain. Figure 55 shows how the graphs of y = cos x and y = sec x are related. 0 y = cos x y y = sec x Period: 2p –2p 2p –p p 1 –1 x Figure 55 x 0 y –1 1 f(x) = sec x –2p 2p –p p −4 4 11p 4 f(x) = sec x 11p 4 − Figure 56 • The graph is discontinuous at values of x of the form x = 12n + 12 p 2 and has vertical asymptotes at these values. • There are no x-intercepts. • Its period is 2p. • There are no minimum or maximum values, so its graph has no amplitude. • The graph is symmetric with respect to the y-axis, so the function is an even function. For all x in the domain, sec1-x2 = sec x. As we shall see, locating the vertical asymptotes for the graph of a function involving the secant (as well as the cosecant) is helpful in sketching its graph.
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