430 CHAPTER 6 Trigonometric Functions The graph of = y x cos illustrates some properties of the cosine function. Properties of the Cosine Function • The domain is the set of all real numbers. • The range consists of all real numbers from −1 to 1, inclusive. • The cosine function is an even function, as the symmetry of the graph with respect to the y -axis indicates. • The cosine function is periodic, with period π2 . • The x -intercepts are π π π π π … − − … , 3 2 , 2 , 2 , 3 2 , 5 2 , ; the y -intercept is 1. • The maximum value is 1 and occurs at … … π π π π = − x , 2,0,2,4,6, ; the minimum value is −1 and occurs at … … π π π π = − x , , ,3,5, . Graphing Functions of the Form ω( ) = y A x cos Using Transformations Graph ( ) = y x 2 cos 3 using transformations. Use the graph to determine the domain and the range of the function. Identify the period of the function ( ) = y x 2 cos 3 . Solution EXAMPLE 3 Figure 54 shows the steps. Figure 54 y 1 (a) y 5 cos x 21 x 2p 3p ––– 2 5p ––– 2 (2p, 1) (0, 1) (0, 2) (0, 2) p p –– 2 (p, 21) 2p (2p, 21) 5p ––– 2 3p ––– 2 p –– 2 2 p –– 2 2 p –– 2 x y 2 22 p 2p 2p (b) y 5 2 cos x (2p, 22) (2p, 2) (p, 22) 5p ––– 6 2p ––– 3 2p ––– 3 p –– 2 p –– 6 p –– 3 2 2p –– 6 p –– 3 p –– 3 p –– 3 x y 2 22 (c) y 5 2 cos (3x) ( , 22) (2 , 22) ( , 2) 1 –– 3 Multiply by 2; Vertical stretch by a factor of 2 Replace x by 3x; Horizontal compression by a factor of The domain of ( ) = y x 2cos 3 is the set of all real numbers, or ( ) −∞ ∞, . The range is { } − ≤ ≤ y y 2 2 , or [ ] −2,2 . The period of the function ( ) = y x 2cos3 is π2 3 because of the compression of the original period π2 by the factor of 1 3 . See Figure 54(c). Figure 55 shows the graph on a TI-84 Plus CE graphing calculator, along with the graph of = y x cos . Now Work PROBLEM 45 USING TRANSFORMATIONS 3 Determine the Amplitude and Period of Sinusoidal Functions The sine function and cosine function are referred to as sinusoidal functions . The discussion below provides the rationale for this definition. Begin by shifting the graph of = y x cos to the right π 2 units to obtain the graph of π ( ) = − y x cos 2 . See Figure 56(a) on the next page. Now look at the graph of = y x sin in Figure 56(b). Notice that the graph of = y x sin is the same as the graph of π ( ) = − y x cos 2 . Figure 55 2.5 22.5 2p 5p 2 Y2 5 2cos(3x) Y1 5 cos x
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