SECTION 5.3 Exponential Functions 301 The graph of ( ) = y 1 2 x also can be obtained from the graph of = y 2x using transformations.The graph of ( ) = = − y 1 2 2 x x is a reflection about the y -axis of the graph of = y 2x (replace x by −x ). See Figures 25(a) and (b). Need to Review? Reflections about the y -axis are discussed in Section 2.5, p. 118. Seeing the Concept Using a graphing utility, simultaneously graph: (a) ( ) = = Y Y 3 , 1 3 x x 1 2 (b) ( ) = = Y Y 6 , 1 6 x x 1 2 Conclude that the graph of ( ) = Y a 1 , x 2 for > a 0, is the reflection about the y -axis of the graph of = Y a . x 1 Figure 25 x y 3 (2, 4) (1, 2) (0, 1) ( ) –2, 6 3 1 – 4 ( ) –1, 1 – 2 ( ) –3, 1 – 8 x y –3 3 (–2, 4) (–1, 2) (0, 1) 1, 3, ( ) ( ) ( ) 2, 6 3 1 – 2 ( )1 – 2 1 – 4 1 – 8 (a) y 5 2x (b) y 5 22x 5( ) 1 – 2 y 5 0 y 5 0 y 5 y 5 2x x Replace x by 2x; reflect about the y-axis x The graph of ( ) ( ) = f x 1 2 x in Figure 24 is typical of all exponential functions of the form ( ) = f x ax with < < a 0 1. Such functions are decreasing and one-to-one. Their graphs lie above the x -axis and contain the point ( ) 0, 1 . The graphs rise rapidly as →−∞ x . As →∞ x , the x -axis ( ) = y 0 is a horizontal asymptote. There are no vertical asymptotes. Finally, the graphs are smooth and continuous, with no corners or gaps. Figures 26 and 27 illustrate the graphs of two other exponential functions whose bases are between 0 and 1. Notice that the smaller base results in a graph that is steeper when < x 0. When > x 0, the graph of the equation with the smaller base is closer to the x -axis. Figure 26 x ( ) ( ) y = x y –3 3 (–1, 3) (0, 1) 1, 6 3 1 – 3 1 – 3 y 5 0 y = x ( ) ( ) (–1, 6) 1, 1 – 6 1 – 6 (b) 1 0 0 4 Y1 5( ) 1 3 x Y2 5( ) 1 6 x Figure 27 10 21 24 4 Y1 5( ) 1 3 x Y 2 5( ) 1 6 x (a) Properties of the Exponential Function f x , 0 1 x ( ) = < < a a • The domain is the set of all real numbers, or ( ) −∞ ∞, using interval notation; the range is the set of positive real numbers, or ( )∞ 0, using interval notation. • There are no x -intercepts; the y -intercept is 1. • The x -axis ( ) = y 0 is a horizontal asymptote of the graph of f as →∞ x . • ( ) = < < f x a a , 0 1, x is a decreasing function and is one-to-one. • The graph of f contains the points ( ) ( ) − a 1, 1 , 0, 1 , and ( )a 1, . • The graph of f is smooth and continuous, with no corners or gaps. See Figure 28. Figure 28 ( ) = < < f x a a , 0 1 x x y (0, 1) (1, a) ( ) –1, 1 –a y 5 0
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