Exponential Functions and Their Graphs

Digital Lesson

2

The exponential function f with base a is defined by

f(x) = ax

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

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

91

9

f(–2) = 3–2 =

0.0625

4

The graph of f(x) = ax, a > 1y

x(0, 1)

Domain: (–, )

Range: (0, )

Horizontal Asymptote y = 0

4

4

5

The graph of f(x) = ax, 0 < a < 1y

x(0, 1)

Domain: (–, )

Range: (0, )Horizontal Asymptote

y = 0

4

4

6

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

x

x f(x) (x, f(x))-2 ¼ (-2, ¼)-1 ½ (-1, ½)0 1 (0, 1)1 2 (1, 2)2 4 (2, 4)

y

2–2

2

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

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 11

The Natural Base e

11

The graph of f(x) = ex

y

x2 –2

2

4

6

x f(x)-2 0.14-1 0.380 11 2.722 7.39

12

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

13

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 raten is the number times you compound your money per yeart is time.

2. Compounded continuously– A = Pert

14

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