# Exponential Functions and Their Graphs Digital Lesson Exponential Functions and Their Graphs Copyright by Houghton Mifflin Company, Inc. All rights reserved. 2 Definition of the Exponential Function The exponential function f with base b is defined by f (x) = bxx or y = bxx Where b is a positive constant other than and x is any real number. Here are some examples of exponential functions. f (x) = 2x g(x) = 10x h(x) = 3x+1 Base is 2. Base is 10.

Copyright by Houghton Mifflin Company, Inc. All rights reserved. Base is 3. 3 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.

Senteo Question To set the properties right click and select Senteo Question Object->Properties... 4 Characteristics of Exponential Functions The domain of f (x) = bxx consists of all real numbers. The range of f (x) = bxx consists of all positive real numbers. The graphs of all exponential functions pass through the point (0, 1) because f (0) = b00 = 1. If b > 1, f (x) = bxx has a graph that goes up to the right and is an increasing function.

If 0 < b < 1, f (x) = bxx has a graph that goes down to the right and is a decreasing function. f (x) = bxx is a one-to-one function and has an inverse that is a function. The graph of f (x) = bxx approaches but does not cross the x-axis. The xaxis is a horizontal asymptote. f (x) = bx 01 5 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

6 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: r nt 1. For n compoundings per year : A p (1 ) n 2. For continuous compounding : A Pe rt Copyright by Houghton Mifflin Company, Inc. All rights reserved. 7 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= Pert A = 2000e(.07)(8) A = 2000 e(.56) Senteo Question A = 2000 * 1.75 A = \$3500 To set the properties right click and select Senteo Question Object->Properties... Copyright by Houghton Mifflin Company, Inc. All rights reserved. 8 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. r A P 1 n

nt .0685 A 8000 1 12 126 72 A 8000 1.0057 Senteo Question To set the properties right click and select Senteo Question Object->Properties... A \$12,052.51

Copyright by Houghton Mifflin Company, Inc. All rights reserved. 9 Exponential Growth and Decay The function y = kax, k > 0 is a model for exponential growth if a > 1, and a model for exponential decay if 0 < a < 1. Copyright by Houghton Mifflin Company, Inc. All rights reserved. 10 Example The population of Silver Run in the year 1980 was 6250. Assume the population increased at a rate of 2.75% per year. About when did the population reach 50,000? x x

y ka x y 6250a x y 6250(1.0275) y 6250(1.0275) x Copyright by Houghton Mifflin Company, Inc. All rights reserved. 50000 6250(1.0275) x 8 (1.0275) ln 8 x ln(1.0275) ln 8 x 76.65 ln 1.0275 11 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? y ka y 5a 1 5 x x / 20 1 x / 20 2 Copyright by Houghton Mifflin Company, Inc. All rights reserved. 1 1 x / 20

2 5 1 ln ( x / 20) ln 12 5 46.4 x 12