Review: exponential growth and Decay

functions

In this lesson, you will review how to write an exponential growth and decay function modeling a percent of increase situation or decrease situation by interpreting the percent of increase as a growth factor or percent of decrease as a decay factor.

Review: exponential growth functions

Exponential growth happens by repeated multiplication by the same number.

Review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

Review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

(𝟎𝐰𝐞𝐞𝐤𝐬 ,𝟏𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚 )

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

(𝟏𝐰𝐞𝐞𝐤𝐬 ,𝟐𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟒𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

(𝟑𝐰𝐞𝐞𝐤𝐬 ,𝟖𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚 )

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

(𝟒𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟔𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

(𝟓𝐰𝐞𝐞𝐤𝐬 ,𝟑𝟐𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

A certain species of bacteria will double in population each week in a laboratory.

review: exponential growth functions

(𝟔𝐰𝐞𝐞𝐤𝐬 ,𝟔𝟒𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

Review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory.

The growth factor is 2.

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟎𝐰𝐞𝐞𝐤𝐬 ,𝟐𝟓𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚 )

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟏𝐰𝐞𝐞𝐤𝐬 ,𝟓𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚 )

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

The initial value is 25.

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)𝑥 𝑦

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

Let x = number of time periods since growth began

𝒙 𝑦

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚𝑥

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐𝐰𝐞𝐞𝐤𝐬 ,𝟏𝟎𝟎𝐛𝐚𝐜𝐭𝐞𝐫𝐢𝐚)

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝑦𝑥

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

(𝟐 ,𝟏𝟎𝟎)

Let x = number of time periods since growth beganLet y = amount of the growing quantity

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚=()()𝒙growth factorinitial value (𝟐 ,𝟏𝟎𝟎)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚=()()𝒙growth factorinitial value (𝟐 ,𝟏𝟎𝟎)

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚=()()𝒙initial value (𝟐 ,𝟏𝟎𝟎)

2

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚=()()𝒙initial value (𝟐 ,𝟏𝟎𝟎)

2

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚=()()𝒙(𝟐 ,𝟏𝟎𝟎)

225

review: exponential growth functions

A certain species of bacteria will double in population each week in a laboratory. Suppose you initially have 25 specimens of this bacteria.

Let x = number of time periods since growth beganLet y = amount of the growing quantity

𝒚=(𝟐𝟓)(𝟐)𝒙(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝒙(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝒙

(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝒙

(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝟐

(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝟐

(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝟐

(𝟐 ,𝟏𝟎𝟎)

𝒚=(𝟐𝟓)(𝟐)𝟐

(𝟐 ,𝟏𝟎𝟎)

interpreting a percent of increase as a growth factor

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

1

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

1 2 3

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

1

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

3%=0.03

1

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.033%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.03

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.03

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.03

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.03

13%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.031

3%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.0313%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

0.0313%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

1+0.033%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

1 .033%=0.03

interpreting a percent of increase as a growth factor The amount of money in your savings account increased by 3% last year.

What does it mean to grow by 3%?

is the growth factor!3%=0.03

an example

an exampleA certain savings account pays 2% interest per year. You open one of these accounts and put $300 into the account. Assume you do not take out or put in any money from the account after the initial $300. Write a two-variable equation modeling the situation. Be sure to define the variables.

an exampleA certain savings account pays 2% interest per year. You open one of these accounts and put $300 into the account. Assume you do not take out or put in any money from the account after the initial $300. Write a two-variable equation modeling the situation. Be sure to define the variables.

an exampleA certain savings account pays 2% interest per year. You open one of these accounts and put $300 into the account. Assume you do not take out or put in any money from the account after the initial $300. Write a two-variable equation modeling the situation. Be sure to define the variables.

an exampleA certain savings account pays 2% interest per year. You open one of these accounts and put $300 into the account. Assume you do not take out or put in any money from the account after the initial $300. Write a two-variable equation modeling the situation. Be sure to define the variables.

Growth factor:

an exampleA certain savings account pays 2% interest per year. You open one of these accounts and put $300 into the account. Assume you do not take out or put in any money from the account after the initial $300. Write a two-variable equation modeling the situation. Be sure to define the variables.

Growth factor:

In this lesson, you will learn to write an exponential decay function modeling a percent of decrease situation by interpreting the percent of decrease as a decay factor.

Review: exponential growth and exponential decay

𝑦=¿

Review: exponential growth and exponential decay

𝑦=¿

If the change factor is greater than 1:

Review: exponential growth and exponential decay

𝑦=¿

If the change factor is greater than 1:

exponential growth

Review: exponential growth and exponential decay

𝑦=¿

If the change factor is greater than 1:

exponential growth(growth factor)

Review: exponential growth and exponential decay

𝑦=¿

If the change factor is greater than 1:

If the change factor is between 0 and 1:

exponential growth(growth factor)

Review: exponential growth and exponential decay

𝑦=¿

If the change factor is greater than 1:

If the change factor is between 0 and 1:

exponential growth(growth factor)

exponential decay

Review: exponential growth and exponential decay

𝑦=¿

If the change factor is greater than 1:

If the change factor is between 0 and 1:

exponential growth(growth factor)

exponential decay(decay factor)

Review: exponential growth and exponential decay

growth factor (greater than 1)

decay factor (less than 1)

Review: exponential growth and exponential decay

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

review: exponential growth and exponential decay

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Review: exponential growth and exponential decay

1

growth factor (greater than 1)

decay factor (less than 1)

Example: 2 becomes

Review: exponential growth and exponential decay

1

growth factor (greater than 1)

decay factor (less than 1)

Example: 2 becomes

Review: exponential growth and exponential

decay

1 2

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Example:

becomes

Review: exponential growth and exponential

decay

1 2

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Example:

becomes

Review: exponential growth and exponential

decay

1 2

1

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Example:

becomes

becomes

Review: exponential growth and exponential

decay

1 2

1

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Example:

becomes

becomes

Review: exponential growth and exponential

decay

1 2

1

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Example:

becomes

becomes

Review: exponential growth and exponential

decay

1 2

1

growth factor (greater than 1)

decay factor (less than 1)

Example: 2

Example:

becomes

becomes

Review: exponential growth and exponential

decay

1 2

1 34

percent of decrease as a decay factor

percent of decrease as a decay factor

The population of a country decreases by 3%.

percent of decrease as a decay factor

The population of a country decreases by 3%.

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.031

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.031

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.031

0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.031

0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.031

0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

1−0.03=0.971

0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

1−0.03=0.971

0.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

1−0.03=0.9710.03

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

1−0.03=0.97

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

1−0.03=0.97

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

becomes1 0.97

percent of decrease as a decay factor

The population of a country decreases by 3%.

3%=0.03

The decay factor is 0.97!becomes1 0.97

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿

15%=0.15

1

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿

15%=0.15

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿

15%=0.15

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿

1−0.15=0.85

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿1

0.85

becomes

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿1

0.85

becomes

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿1

0.85

becomes

0.85

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿0.85

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿0.85

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿0.85

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿0.8520000

an example The value of a car depreciates 15% per year after it is purchased. The value of the car at the time of its purchase was $20,000. Write a two-variable equation to model this situation.

𝑦=¿0.8520000

Now you know how to write an exponential decay function modeling a percent of decrease situation!