306 CHAPTER 5 Exponential and Logarithmic Functions SUMMARY Properties of the Exponential Function ( ) = > f x a a , 1 x • Domain: the interval ( ) −∞ ∞, ; range: the interval ( )∞ 0, • x -intercepts: none; y -intercept: 1 • Horizontal asymptote: x -axis ( ) = y 0 as →−∞ x • Increasing; one-to-one; smooth; continuous • See Figure 23 on page 300 for a typical graph. ( ) = < < f x a a , 0 1 x • Domain: the interval ( ) −∞ ∞, ; range: the interval ( )∞ 0, • x -intercepts: none; y -intercept: 1 • Horizontal asymptote: x -axis ( ) = y 0 as →∞ x • Decreasing; one-to-one; smooth; continuous • See Figure 28 on page 301 for a typical graph. If = a a , u v then = u v. ‘Are You Prepared?’ Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red. 5.3 Assess Your Understanding 1. = 43 ; = 82 3 ; = −3 2 . (pp. A7–A9 and pp. A91–A92) 2. Solve: + = x x3 4 2 (pp. A47–A54) 3. True or False To graph ( ) = − y x 2 , 3 shift the graph of = y x3 to the left 2 units. (pp. 112–120) 4. Find the average rate of change of ( ) = − f x x3 5 from = x 0 to = x 4 . (pp. 93–95) 5. True or False The graph of the function ( ) = − f x x x 2 3 has = y 2 as a horizontal asymptote. (pp. 241–244) 6. If ( ) = − + f x x3 10, then the graph of f is a with slope and y -intercept . (pp. 139–145) 7. Where is the function ( ) = − + f x x x4 3 2 increasing? Where is it decreasing? (pp. 157–166) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure Concepts and Vocabulary 8. Interactive Figure Exercise Exploring Exponential Functions Open the “Exponential Functions” interactive figure, which is available in the Video & Resource Library of MyLab Math (under Sullivan Interactive Figures). (a) In the interactive figure, the graph of ( ) = ⋅ + − f x c a k x h is drawn. Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to increase the value of a from 2 to 4. Note the points on the graph. (i) The graph of ( ) = f x 2x contains the points 1, , 0, , ( ) ( ) − and 1, ( ) (ii) The graph of ( ) = f x 3x contains the points 1, , 0, , ( ) ( ) − and 1, ( ) (b) Check the “Show Reciprocal of Base a ” box. The graph of ( ) ( ) = ⋅ + − g x c a k 1 x h is drawn. Use the sliders to set the value of c to 1, a to 2, h to 0, and k to 0. Now, use the slider to increase the value of a from 2 to 4. Note the points on the graph. (i) The graph of ( ) ( ) = g x 1 2 x contains the points 1, , 0, , ( ) ( ) − and 1, ( ) (ii) The graph of ( ) ( ) = g x 1 3 x contains the points 1, , 0, , ( ) ( ) − and 1, ( ) (c) The graph of ( ) ( ) = g x a 1 x is a reflection about the ( x -axis/ y -axis/origin) of the graph of ( ) = > f x a a , 1. x (d) In the graph of ( ) = f x ax with < < a 0 1, the graph is (increasing/decreasing) over its domain. In the graph of ( ) = f x ax with > a 1, the graph is (increasing/decreasing) over its domain. (e) Uncheck the box “Show Reciprocal of Base a.” Use the sliders to set the value of c to 1, a to 1, h to 0, and k to 0. What type of graph is ( ) = f x 1 ?x Explain why ≠ a 1 for an exponential function.
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