1. Exponential Functions and Their Graphs Digital Lesson

2. 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The exponential function f with base a is defined by f(x) = a x where a > 0, a  1, and x is any real number. For instance, f(x) = 3 x and g(x) = 0.5 x are exponential functions. Definition of Exponential Function

3. 3 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The value of f(x) = 3 x when x = 2 is f(2) = 3 2 = The value of g(x) = 0.5 x when x = 4 is g(4) = 0.5 4 = The value of f(x) = 3 x when x = –2 is 9 f(–2) = 3 –2 = 0.0625 Example: Exponential Function

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

5. 5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = a x, 0 < a < 1 y x (0, 1) Domain: (– ,  ) Range: (0,  ) Horizontal Asymptote y = 0 Graph of Exponential Function (0 < a < 1) 4 4

6. 6 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch the graph of f(x) = 2 x. x xf(x)f(x)(x, f(x)) -2¼(-2, ¼) ½(-1, ½) 01(0, 1) 12(1, 2) 24(2, 4) y 2–2 2 4 Example: Graph f(x) = 2 x

7. 7 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch the graph of g(x) = 2 x – 1. State the domain and range. x y The graph of this function is a vertical translation of the graph of f(x) = 2 x down one unit. f(x) = 2 x y = –1 Domain: (– ,  ) Range: (–1,  ) 2 4 Example: Translation of Graph

8. 8 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example: Sketch the graph of g(x) = 2 -x. State the domain and range. x y The graph of this function is a reflection the graph of f(x) = 2 x in the y- axis. f(x) = 2 x Domain: (– ,  ) Range: (0,  ) 2 –2 4 Example: Reflection of Graph

9. 9 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = e x y x 2 –2 2 4 6 xf(x)f(x) -20.14 0.38 01 12.72 27.39 Graph of Natural Exponential Function f(x) = e x

10. 10 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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 The number e