Exponential Functions and Their Graphs - Cengage

Digital Lesson Exponential Functions and Their Graphs The exponential function f with base a is defined by f(x) = ax where a > 0, a 1, and x is any real number. For instance, f(x) = 3x and g(x) = 0.5x are exponential functions. Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2 The value of f(x) = 3x when x = 2 is f(2) = 32 = 9 The value of f(x) = 3x when x = 2 is f(2) = 3

2 1 = 9 The value of g(x) = 0.5x when x = 4 is g(4) = 0.54 = 0.0625 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 3 The graph of f(x) = ax, a > 1 y 4 Range: (0, ) (0, 1)

x 4 Domain: (, ) Copyright by Houghton Mifflin Company, Inc. All rights reserved. Horizontal Asymptote y=0 4 The graph of f(x) = ax, 0 < a < 1 y 4 Range: (0, ) Horizontal Asymptote y=0 (0, 1)

x 4 Domain: (, ) Copyright by Houghton Mifflin Company, Inc. All rights reserved. 5 Example: Sketch the graph of f(x) = 2x. x -2 -1 0 1 2 f(x) (x, f(x)) (-2, ) (-1, )

1 (0, 1) 2 (1, 2) 4 (2, 4) y 4 2 x 2 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2 6 Example: Sketch the graph of g(x) = 2x 1. State the

domain and range. The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . y f(x) = 2x 4 2 Domain: (, ) x Range: (1, ) y = 1

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 7 Example: Sketch the graph of g(x) = 2-x. State the domain and range. y The graph of this function is a reflection the graph of f(x) = 2x in the yaxis. f(x) = 2x 4 Domain: (, ) Range: (0, )

x 2 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2 8 The graph of f(x) = ex y x -2 -1 0 1 2 6 4

2 f(x) 0.14 0.38 1 2.72 7.39 x 2 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2 9 The irrational number e, where e 2.718281828 is used in applications involving growth and

decay. Using techniques of calculus, it can be shown that n 1 1 e as n n Copyright by Houghton Mifflin Company, Inc. All rights reserved. 10