SECTION 6.4 Graphs of the Sine and Cosine Functions 429 Now Work PROBLEM 41 USING TRANSFORMATIONS 2 Graph the Cosine Function = y x cos and Functions of the Form ω = ( ) y A x cos The cosine function also has period π2 . To graph = y x cos , begin by constructing Table 7, which lists some points on the graph of = y x cos , π ≤ ≤ x 0 2 . As the table shows, the graph of π = ≤ ≤ y x x cos ,0 2 , begins at the point ( ) 0, 1 . As x increases from 0 to π 2 to π, the value of y decreases from 1 to 0 to −1; as x increases from π to π3 2 to π2 , the value of y increases from −1 to 0 to 1. As before, plot the points in Table 7 to get one period or cycle of the graph. See Figure 52. Graphing Functions of the Form ω = ( ) y A x sin Using Transformations Graph ( ) = − y x sin 2 using transformations. Use the graph to determine the domain and the range of the function. Identify the period of the function ( ) = − y x sin 2 . EXAMPLE 2 Solution Figure 50 illustrates the steps. x = cos y x ( ) , x y 0 1 ( ) 0 1, π 3 1 2 π( ) 3 1 2 , π 2 0 π( ) 2 0, π2 3 − 1 2 π ( ) − 2 3 1 2 , π −1 π( ) −1 , π4 3 − 1 2 π ( ) − 4 3 1 2 , π3 2 0 π ( ) 3 2 0, π5 3 1 2 π ( ) 5 3 1 2 , π2 1 π ( ) 2 1, Table 7 The domain of ( ) = − y x sin 2 is the set of all real numbers, or ( ) −∞ ∞, . The range is { } − ≤ ≤ y y 1 1 , or [ ] −1, 1 . The period of the function ( ) = − y x sin2 is π because of the horizontal compression of the original period π2 by a factor of 1 2 . See Figure 50(c). Figure 51 shows the graph using Desmos, along with the graph of = y x sin . Figure 51 ( ) = =− y x y x sin ; sin 2 1 2 Figure 50 p –– 4 2 p –– 4 3p ––– 4 x y 21 1 p ( , 21) p –– 2 p –– 2 2 (c) y 5 2 sin (2x) ( , 1) 3 p––– 4 p –– 4 (2 , 1) p –– 4 p –– 2 p –– 2 3p ––– 2 x y 21 1 p ( , 21) p–– 2 2 (b) y 5 2 sin x p –– 2 (2 , 1) p –– 2 ( , 1) 3p ––– 2 ( , 1) 2p 2p Replace x by 2x; Horizontal compression by a factor of 1 – 2 Multiply by 21; Reflect about the x2axis (a) y 5 sin x x 2 p y 1 p –– 2 p –– 2 3p ––– 2 21 2p 2p ( , 21) 3 p––– 2 p –– 2 (2 , 21) Figure 52 π = ≤ ≤ y x x cos ,0 2 p–– 3 x p 2p ( , )1 – 2 ( , ) 1 – 2 (0, 1) (2p, 1) (p, 21) 5p––– 3 ( , 2 )1 – 2 2p––– 3 ( , 2 ) 1 – 2 4p––– 3 p –– 2 3p ––– 2 y 1 21 A more complete graph of = y x cos is obtained by continuing the graph in each direction, as shown in Figure 53(a). Figure 53(b) shows the graph on a TI-84 Plus CE graphing calculator. Figure 53 = −∞< <∞ y x x cos , (a) x 2p p 2 2 p p (2 , 21) p( , 21) p (2 , 1) y 1 21 p –– 2 p –– 2 3p ––– 2 5p ––– 2 (b) 21.5 2p 3p 1.5
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