428 CHAPTER 6 Trigonometric Functions The graph in Figure 47 is one period, or cycle , of the graph of = y x sin . To obtain a more complete graph of = y x sin , continue the graph in each direction, as shown in Figure 48(a). Figure 48(b) shows the graph using Desmos. Properties of the Sine Function = y x sin • The domain is the set of all real numbers. • The range consists of all real numbers from −1 to 1, inclusive. • The sine function is an odd function, as the symmetry of the graph with respect to the origin indicates. • The sine function is periodic, with period π2 . • The x -intercepts are … π π π π π − − ..., 2 , ,0, ,2 ,3 , ;the y -intercept is 0. • The maximum value is 1 and occurs at π π π π =… − … x , 3 2 , 2 , 5 2 , 9 2 , ; the minimum value is −1 and occurs at … … π π π π = − x , 2 , 3 2 , 7 2 , 11 2 , . Figure 48 = −∞< <∞ y x x sin , x 2 2 p 2 2 p p ( , 21) ( , 1) ( , 21) 3p––– 2 ( , 1) 5p––– 2 y 1 21 p –– 2 p –– 2 p –– 2 p –– 2 3p ––– 2 5p ––– 2 (b) (a) (b) The graph of = y x sin illustrates some properties of the sine function. Now Work PROBLEM 13 Graphing Functions of the Form = y A x sin Using Transformations Graph = y x 3sin using transformations. Use the graph to determine the domain and the range of the function. Solution EXAMPLE 1 Figure 49 illustrates the steps. Figure 49 x 2 2 p p p –– 2 ( , 3) ( , 1) p–– 2 ( , 1) 5p––– 2 y 1 p –– 2 p –– 2 3p ––– 2 5p ––– 2 x y 2p p 23 3 1 p – 2 3p ––– 2 p––– 2 (a) y 5 sin x Multiply by 3; Vertical stretch by a factor of 3 (b) y 5 3 sin x 2 2p (2 , 21) p –– 2 ( , 21) 3p ––– 2 ( , 23) 3p ––– 2 21 The domain of = y x 3sin is the set of all real numbers, or ( ) −∞∞, . The range is { } − ≤ ≤ y y 3 3 , or [ ] −3,3 .
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