Exponential Functions and Their Graphs Digital Lesson

2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved.

3 Definition of the Exponential Function Here are some examples of exponential functions. f (x) = 2 x g(x) = 10 x h(x) = 3 x+1 Base is 2.Base is 10.Base is 3. The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than and x is any real number. The exponential function f with base b is defined by f (x) = b x or y = b x Where b is a positive constant other than and x is any real number.

4 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

5 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Characteristics of Exponential Functions The domain of f (x) = b x consists of all real numbers. The range of f (x) = b x consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. If b > 1, f (x) = b x has a graph that goes up to the right and is an increasing function. If 0 < b < 1, f (x) = b x has a graph that goes down to the right and is a decreasing function. f (x) = b x is a one-to-one function and has an inverse that is a function. The graph of f (x) = b x approaches but does not cross the x-axis. The x- axis is a horizontal asymptote. The domain of f (x) = b x consists of all real numbers. The range of f (x) = b x consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b 0 = 1. If b > 1, f (x) = b x has a graph that goes up to the right and is an increasing function. If 0 < b < 1, f (x) = b x has a graph that goes down to the right and is a decreasing function. f (x) = b x is a one-to-one function and has an inverse that is a function. The graph of f (x) = b x approaches but does not cross the x-axis. The x- axis is a horizontal asymptote. f (x) = b x b > 1 f (x) = b x 0 < b < 1

6 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

7 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 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:

8 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example Find the accumulated value of an investment of $2000 for 8 years at an interest rate of 7% if the money is compounded continuously Solution: A= Pe rt A = 2000e (.07)(8) A = 2000 e (.56) A = 2000 * 1.75 A = $3500

9 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example Find the accumulated value of an investment of $8000 for 6 years at an interest rate of 6.85% if the money is compounded monthly.

10 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Exponential Growth and Decay The function y = ka x, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1.

11 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example The population of Silver Run in the year 1980 was Assume the population increased at a rate of 2.75% per year. About when did the population reach 50,000?

12 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example Suppose the half-life of a radioactive substance is 20 days and that there are 5 grams present initially. When will there be only 1 gram left?