Exponential Functions and Their Graphs Digital Lesson

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 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) = = The value of f(x) = 3 x when x = –2 is 9 f(–2) = 3 –2 = Example: Exponential Function

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 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 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 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 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 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The graph of f(x) = e x y x 2 – xf(x)f(x) Graph of Natural Exponential Function f(x) = e x

10 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. The irrational number e, where e  … is used in applications involving growth and decay. Using techniques of calculus, it can be shown that The number e

11 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Financing Formulas for Compound Interest After t years, the balance A in an account with principal P and annual interest rate r (in decimal form) is given by the following formulas. 1.For n compounding per year: 2.For continuous compounding:

12 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Investment A total of $9000 is invested at an annual interest rate of 2.5%, compounded annually. Find the balance in the account after 5 years.

13 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Investment A total of $12,000 is invested at an annual interest rate of 3%. Find the balance after 4 years if the interest is compounded (a)Quarterly (b)continuously

14 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Homework Page all, all, odd, all