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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