Section 8.1 and 8.2

1. How to graph exponential growth functions.
2. How to graph exponential decay functions.

Exponential Growth
• This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at
$1 and increases by $1 each week.

Exponential Growth
• This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by $1 each week.
Although the second option, growing at a constant rate of
$1/week, pays more in the short run, the first option eventually grows much larger:
W 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
1
2
.01
$1
.02
$2
.04
$3
.08
$4
.16
$5
.32
$6
.64
$7
1.28
$8
2.56
$9
5.12
10.2
4
20.4
8
40.
96
81.
92
163
.84
327
.68
655
.36
131
0.7
2
$10 $11 $12 $13 $14 $15 $16 $17 $18

Why!
Exponential Growth!
The equation for option 1 is: y = 2 n where n is the number of weeks.
The equation for option 2 is y = 1 + n where n is the number of weeks.

Oh Boy! Vocabulary
An exponential function involves the expression b x where the base “ b ” is a positive number other than
1.
The variable is going to be in the “ position ” of the exponent.

Let ’ s Graph an Example y 𝑦 = 2 𝑥
10
5
-10 -5 5
Question: Will the graph ever pass below y of 0?
10 x
-5
-10

Let ’ s Graph an Example y
10
5
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.
10 x
-10 -5
5
-5
-10

Let ’ s Graph an Example y
10
5
5
Question: Will the graph ever pass below y of 0?
We say that there is an asymptote at y = 0.
10 x
-10 -5
An asymptote is a line that a graph approaches as you move away from the origin.
-5
-10

Try the following on your graphing calculator
Group 2: Group 1: y y


1
3

2 x
3

2 x y

2 x y
 y


1

2 x
5

5

2 x y

2 x y
 ab

A Definition y = ab x is an exponential growth function . When a is greater than 0 and b is greater than 1.

Graphing Examples
• Graph y

1
2

3 x y
10
5
-10 -5
-5
-10
5 10 x

Another Example
Graph y
 
(
3
2
) x
10 y
5
-10 -5
-5
-10
5 10 x

Graphing by Translation
The generic form of an exponential function is: y = ab x-h + k
Where h is movement along the x axis and k is movement along the y axis.

An Example of Graphing by Translation
Graph y

3

2 x

1 
4
10 y
5
10 x
-10 -5 5
-5
-10

You Try
• Graph y

2

3 x

2 
1 y
10
5
-10 -5
-5
-10
5 10 x

Exponential Growth Model
• We will use the formula: y = a(1 + r) t a is the initial amount, r is the percent increase expressed as a decimal and t is the number of years.
The term 1 + r is called the growth factor.

An Example Problem
• In January 1993, there were about 1,313,000
Internet hosts. During the next five years, the number of hosts increased by about 100% per year.
• Write a model.
• How many hosts were there in 1996?
• Graph the model.
• When will there be 30 million hosts?

Section 8.2 – Exponential Decay
• These functions will have the form y = ab x where a is greater than zero and b is between 0 and 1.
19

Example 1
• State whether the function is an exponential growth or exponential decay function.
1 .
f ( x )

5 (
2
3
) x
2 .
f
3 .
f (
( x ) x )

8 (
3
2
) x

10 ( 3 )
 x
20

You Try
• State whether the function is an exponential decay or growth function.
1 .
f ( x )

1
3
( 2 )
 x
2 .
f ( x )

4 (
5
8
) x

A Basic Graph
• A graph of y

1
2 x
10 y
5
-10 -5
-5
-10
5 10 x

Graphing Exponential Functions … again x
• Graph: y

3

1
4 y
10
5
10 x
-10 -5 5
-5
-10

Another Example x
• Graph: y
 
5

2
3
10 y
5
-10 -5
-5
-10
5 10 x

Graphing by Translation
The generic form of an exponential function is: y = ab x-h + k
Where h is movement along the x axis and k is movement along the y axis.

Graphing by Translation
• Graph: y
 

3

1
2 x

2

1 y
10
5
-10 -5 5
-5
-10
10 x

An Exponential Decay Word Problem
• We will use the formula: y = a(1 - r) t
(1-r) is called the decay factor.

The Word Problem
• You buy a new car for $24,000. The value y of the car decreases by 16% each year.
1. Write an exponential decay model for the value of the car.
2. Use the model to estimate the value after 2 years.
3. Graph the model.
4. When will the car have a value of $12,000.

Homework
:
Page 469, 14-18 even, 19-24 all, 34, 36, 38,
43-45 all
Page 477, 12, 16, 18, 19-24 all, 36, 40, 42,
47-49 all