4.2 Exponential Functions and Their Graphs Digital Lesson 4.2 Exponential Functions and Their Graphs

Definition of Exponential Function 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. Definition of Exponential Function

Example: Exponential Function 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 = 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. Example: Exponential Function

Graph of Exponential Function (a > 1) The graph of f(x) = ax, a > 1 y 4 Range: (0, ) (0, 1) x 4 Horizontal Asymptote y = 0 Domain: (–, ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graph of Exponential Function (a > 1)

Graph of Exponential Function (0 < a < 1) 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. Graph of Exponential Function (0 < a < 1)

Example: Sketch the graph of f(x) = 2x. x f(x) (x, f(x)) y x f(x) (x, f(x)) -2 ¼ (-2, ¼) -1 ½ (-1, ½) 1 (0, 1) 2 (1, 2) 4 (2, 4) 4 2 x –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Graph f(x) = 2x

Example: Translation of Graph Example: Sketch the graph of g(x) = 2x – 1. State the domain and range. y f(x) = 2x The graph of this function is a vertical translation of the graph of f(x) = 2x down one unit . 4 2 Domain: (–, ) x y = –1 Range: (–1, ) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Translation of Graph

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

Graph of Natural Exponential Function f(x) = ex The graph of f(x) = ex y x f(x) -2 0.14 -1 0.38 1 2.72 2 7.39 6 4 2 x –2 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Graph of Natural Exponential Function f(x) = ex

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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The number e