622 CHAPTER 6 The Circular Functions and Their Graphs Graph of the Cosine Function The graph of y =cos x in Figure 24 is the graph of the sine function shifted, or translated, to the left P 2 units. x y 0 0 1 2 1 (1, 0) (0, –1) (0, 1) –1 1 2 y = sin x p 3p 2 2p 3 3p 2 2p 3 7p 6 p 2 p 2 p 6 p 6 7p 6 – p 2 , 1 ( ) 3p 2 , –1 ( ) – , ( ) 1 2 √3 2 , ( ) 1 2 √3 2 , – –( ) 1 2 √3 2 Figure 23 LOOKING AHEAD TO CALCULUS The discussion of the derivative of a function in calculus shows that for the sine function, the slope of the tangent line at any point x is given by cos x. For example, look at the graph of y = sin x and notice that a tangent line at x = { p 2 , { 3p 2 , { 5p 2 , . . . will be horizontal and thus have slope 0. Now look at the graph of y = cos x and see that for these values, cos x = 0. Cosine Function ƒ 1x2 =cos x Domain: 1-∞, ∞2 Range: 3-1, 14 x y 0 1 p 6 23 2 p 4 22 2 p 3 1 2 p 2 0 p -1 3p 2 0 2p 1 x y 2 1 0 –1 –2 f(x) = cos x, –2p " x " 2p –2p 2p –p p – – 3p 2 3p 2 p 2 p 2 f(x) = cos x −4 4 11p 4 11p 4 − Figure 24 • The graph is continuous over its entire domain, 1-∞, ∞2. • Its x-intercepts have x-values of the form 12n + 12p 2 , where n is an integer. • Its period is 2p. • The graph is symmetric with respect to the y-axis, so the function is an even function. For all x in the domain, cos1-x2 = cos x. NOTE A function ƒ is an even function if for all x in the domain of ƒ, ƒ1 −x2 =ƒ1x2. The graph of an even function is symmetric with respect to the y-axis. This means that if 1x, y2 belongs to the function, then 1-x, y2 also belongs to the function. For example, Ap 2 , 0B and A - p 2 , 0B are points on the graph of y = cos x, illustrating the property cos1-x2 = cos x.
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