Exponential Functions and Their Graphs

2

The exponential function f with base b is defined by

f(x) = abx

where b > 0, b 1, and x is any real number.

** when b> 1; b is considered a growth factor.

For instance,

f(x) = 3x and g(x) = 0.5x

are exponential functions.

3

The value of f(x) = 3x when x = 2 is

f(2) = 32 =

The value of g(x) = 0.5x when x = 4 is

g(4) = 0.54 =

The value of f(x) = 3x when x = –2 is

9

1

9

f(–2) = 3–2 =

0.0625

4

The graph of f(x) = abx, b > 1

y

x(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4

5

The graph of f(x) = abx, 0 < b < 1

y

x

(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4

Since a< 1; a is decay factor.

6

Example: Sketch the graph of f(x) = 2x.

x

y

2–2

2

4

x f(x)

-2 -1

0

1

2

2-2 = 1/4

2-1 = 1/2

20 = 1

21 = 2

22 = 4

7

Example: Sketch the graph of g(x) = 2x – 1. State the domain and range.

x

yThe graph of this function is a vertical translation of the graph of f(x) = 2x

down one unit .

f(x) = 2x

y = –1 Domain: (–, )

Range: (–1, )

2

4

8

Example: Sketch the graph of g(x) = 2-x. State the domain and range.

x

yThe graph of this function is a reflection the graph of f(x) = 2x in the y-axis.

f(x) = 2x

Domain: (–, )

Range: (0, ) 2–2

4

9

Example: Sketch the graph of g(x) = 4x-3 + 3. State the domain and range.

x

yMake a table.

Domain: (–, )

Range: (3, ) or y > 3

2–2

4 x y

3 4

2 3.25

1 3.0625

4 7

5 19

10

Write an exponential function y = abx for the graph that includes the given points.

(2, 2) and (3, 4)

y = abx Substitute in (2,2) for x and y.2 = ab2

Solve for a2 = ab2

Now substitute (3,4) in for x and y into y = abx

4 = ab3

b3

4 = a b3

Set them equal to each other:2 = 4 = a b3 b2

Now solve for b to get b=2

You must solve for a: 2/2^2 = (1/2) = aSubst you’re a = (1/2) & b = 2 into either one of your 2 equations: y = abx

Y = (1/2)(2)^x

11

Write an exponential function y = abx for the graph that includes the given points.

(2, 4) and (3, 16)

y = abx

12

The irrational number e, where

e 2.718281828…

is used in applications involving growth and decay.

Using techniques of calculus, it can be shown that

ne

n

n

as 1

1

The Natural Base e

13

The graph of f(x) = ex

y

x2 –2

2

4

6

x f(x)

-2 0.14

-1 0.38

0 1

1 2.72

2 7.39

14

Example: Sketch the graph of g(x) = ex-5 + 2. State the domain and range.

x

yMake a table.

Domain: (–, )

Range: (2, ) or y > 2

2–2

4 x y

5 3

6 4.72

7 9.39

4 2.36

3 2.14

15

One to One Property

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

If the bases are the same, then set the exponents equal.

Example—3x = 32

x = ___

Example—3x+ 1 = 9

x = ___

Example—2x - 1 = 1/32

x = ___

Example—e3x + 2 = e3

x = ___

16

Formulas for Compound Interest—

1.) compound per year -- A = P 1 + r nt

n

Interest Applications

Balance in account Principal ($ you invest)

r is the rate

n is the number times you compound your money per year

t is time.

2. Compounded continuously– A = Pert

17

A total of $12000 is invested at an annual interest rate of 9%. Find the balance after 5 years if it is compounded

a. quarterlyb. monthlyc. continuously