Exponential Growth and Exponential Decay

Section 8.1 and 8.2

WHAT YOU WILL LEARN:

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

W 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17

1 .01

.02

.04

.08

.16

.32

.64

1.28

2.56

5.12

10.24

20.48

40.96

81.92

163.84

327.68

655.36

1310.72

2 $1 $2 $3 $4 $5 $6 $7 $8 $9 $10 $11 $12 $13

$14

$15

$16

$17

$18

• 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:

Why!

Exponential Growth!

The equation for option 1 is: y = 2n 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 bx 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

x-10 -5 5 10

y

-10

-5

5

10

Question: Will the graph ever pass below y of 0?

𝑦=2𝑥

Let’s Graph an Example

x-10 -5 5 10

y

-10

-5

5

10

Question: Will the graph ever pass below y of 0?

We say that there is an asymptote at y = 0.

Let’s Graph an Example

x-10 -5 5 10

y

-10

-5

5

10

Question: Will the graph ever pass below y of 0?

We say that there is an asymptote at y = 0.

An asymptote is a line that a graph approaches as you move away from the origin.

Try the following on your graphing calculator

x

x

x

y

y

y

2

23

23

1

x

x

x

y

y

y

2

25

25

1

xaby

Group 1: Group 2:

How does “a” in the function affect the graph?

A Definition

y = abx is an exponential growth function. When a is greater than 0 and b is greater than 1.

Graphing Examplesxy 3

2

1• Graph

x-10 -5 5 10

y

-10

-5

5

10

Another Example

xy )2

3(Graph

x-10 -5 5 10

y

-10

-5

5

10

Graphing by Translation

The generic form of an exponential function is:

y = abx-h + k

Where h is movement along the x axis and k is movement along the y axis.

An Example of Graphing by Translation

423 1 xyGraph

x-10 -5 5 10

y

-10

-5

5

10

You Try

132 2 xy• Graph

x-10 -5 5 10

y

-10

-5

5

10

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 = abx where a is greater than zero and b is between 0 and 1.

19

Example 1

x

x

x

xf

xf

xf

)3(10)(.3

)2

3(8)(.2

)3

2(5)(.1

20

• State whether the function is an exponential growth or exponential decay function.

You Try

x

x

xf

xf

)8

5(4)(.2

)2(3

1)(.1

• State whether the function is an exponential decay or growth function.

A Basic Graphx

y

2

1• A graph of

x-10 -5 5 10

y

-10

-5

5

10

Graphing Exponential Functions…again

x

y

4

13• Graph:

x-10 -5 5 10

y

-10

-5

5

10

Another Examplex

y

3

25• Graph:

x-10 -5 5 10

y

-10

-5

5

10

Graphing by Translation

The generic form of an exponential function is:

y = abx-h + k

Where h is movement along the x axis and k is movement along the y axis.

Graphing by Translation

12

13

2

x

y• Graph:

x-10 -5 5 10

y

-10

-5

5

10

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

:

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