3.1 Exponential Functions and their Graphs

Students will recognize and evaluate exponential functions with base a.

Students will graph exponential functions.

Students will recognize, evaluate, and graph exponential functions with base e.

Students will use exponential functions to model and solve real-life problems.

Example 1

Use a calculator to evaluate each function at the indicated value of x.

a. b. c.f x x( ) 2 f x x( ) 2 f x x( ) .6

Example 2

In the same coordinate plane, sketch the graph of each function by hand.

a. b. f x x( ) 2 g x x( ) 4

y

x–2

2

Example 3

In the same coordinate plane, sketch the graph of each function by hand.

a. b. f x x( ) 2 g x x( ) 4

y

x–2

2

Example 4

Each of the following graphs is a transformation of

a. b.

c. d.

h x x( ) 3 1 g x x( ) 3 2y

x–2

2

f x x( ) 3

k x x( ) 3 j x x( ) 3

Compound Interest

Annually Compounded Interest:

= Accumulated amount after n years

P = principle (invested amount)

r = the interest rate in decimal form (5% = .05)

n = time in years

Compounded Interest m times per year:

m = # of times per year interest is compounded (i.e. m = 1 is yearly, m = 12 is monthly, m = 365 is daily)

Compounded Interest continuously:

e is the natural number

A P rnn ( )1

An

A Pr

mnmn ( )1

A Penrn

e2 718281828.

Example: Invest $1 for 1 year at 100%Consider how m affects the accumulated amount:

Yearly m = 1

Bi-annually m = 2

Monthly m = 12

Daily m = 365

Hourly m = 8760

Example 5

A Pr

mnmn ( )1 A

mm

111 1

1 ( ) A

mm ( )1

1

Continuously CompoundedInterest:

A Penrn

Example 6Use a calculator to evaluate the function

where:a. b. c.

f x ex( )

y

x–2

2

x 2 x .25 x .4

Copyright © Houghton Mifflin Company. All rights reserved. Digital Figures, 3–2

Section 3.1, Figure 3.9, The Natural Exponential Function, pg. 180

Example 7

A total of $12000 is invested at 7% interest. Find the balance after 10 years if the interest is compounded a) quarterly

b) continuously

a)

b)

A Pr

mnmn ( )1

A Penrn