SECTION 6.4 Graphs of the Sine and Cosine Functions 431 Based on Figure 56, we conjecture that Figure 56 (b) y 5 sin x x y 1 21 2p p 2p 2p –– 2 p –– 2 3p ––– 2 5p ––– 2 2p –– 2 p –– 2 3p ––– 2 5p ––– 2 x y 1 21 (a) y 5 cos x; y 5 cos (x 2 ) p – 2 2p p 2p π ( ) = − x x sin cos 2 We prove this in Chapter 7. Because of this relationship, the graphs of functions of the form ω( ) = y A x sin or ω( ) = y A x cos are referred to as sinusoidal graphs . Figure 57 uses transformations to obtain the graph of = y x 2cos . Note that the values of = y x 2 cos lie between −2 and 2, inclusive. Figure 57 x y 1 21 2p 2 p 2p (a) y 5 cos x (b) y 5 2 cos x p –– 2 p –– 2 3p ––– 2 5p ––– 2 (2p, 1) (p, 21) (2p, 21) Multiply by 2; Vertical stretch by a factor of 2 x y 1 2 21 22 2p p 2p 2p –– 2 p –– 2 3p ––– 2 5p ––– 2 (2p, 22) (p, 22) (2p, 2) The values of the functions = y A x sin and = y A x cos , where ≠ A 0, satisfy the inequalities − ≤ ≤ − ≤ ≤ A A x A A A x A sin and cos respectively. The number A is called the amplitude of = y A x sin or = y A x cos . See Figure 58. Figure 58 (a) π = > = y A x A sin , 0;period 2 x y A 2A p 2p 2p –– 2 p –– 2 3p ––– 2 5p ––– 2 (b) π = > = y A x A cos , 0;period 2 x y A 2A p 2p 2p –– 2 p –– 2 3p ––– 2 5p ––– 2 If ω > 0, the functions ω( ) = y x sin and ω( ) = y x cos have period π ω = T 2 . To see why, recall that the graph of ω( ) = y x sin is obtained from the graph of = y x sin by performing a horizontal compression or stretch by a factor ω 1 . This horizontal compression replaces the interval π [ ] 0,2 , which contains one period of the graph of = y x sin , by the interval π ω ⎡ ⎢ ⎣ ⎤ ⎦ ⎥ 0, 2 , which contains one period of the graph of ω( ) = y x sin . This is why the function ( ) = y x 2cos 3 , graphed in Figure 54(c) on the previous page, with ω = 3, has period π ω π = 2 2 3 . In Words The amplitude determines the vertical stretch or compression of the graph of = y x sin or = y x cos . Seeing the Concept Using a graphing utility, graph = Y x sin 1 and π ( ) = − Y x cos 2 . 2 How many graphs do you see?
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