1. EXPONENTIAL GROWTH MODEL
WRITING EXPONENTIAL GROWTH MODELS
A quantity is growing exponentially if it increases by the same percent in each time period.
EXPONENTIAL GROWTH MODEL
A is the initial amount.
t is the time period.
y = A (1 + r)t
(1 + r) is the growth factor, r is the growth rate.
The percent of increase is 100r.

2. METHOD 1 SOLVE A SIMPLER PROBLEM
Finding the Balance in an Account
COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years?
SOLUTION
METHOD 1 SOLVE A SIMPLER PROBLEM
Find the account balance A1 after 1 year and multiply by the growth factor to find the balance for each of the following years. The growth rate is 0.08, so the growth factor is = 1.08.
A1 = 500(1.08) = 540
Balance after one year
A2 = 500(1.08)(1.08) =
Balance after two years
A3 = 500(1.08)(1.08)(1.08) =
Balance after three years • •
A6 = 500(1.08) 6 
Balance after six years

3. EXPONENTIAL GROWTH MODEL EXPONENTIAL GROWTH MODEL
Finding the Balance in an Account
COMPOUND INTEREST You deposit $500 in an account that pays 8% annual interest compounded yearly. What is the account balance after 6 years?
SOLUTION
METHOD 2 USE A FORMULA
Use the exponential growth model to find the account balance A. The growth rate is The initial value is 500.
EXPONENTIAL GROWTH MODEL
500 is the initial amount.
6 is the time period.
( ) is the growth factor, 0.08 is the growth rate.
A6 = 500(1.08) 6  Balance after 6 years
A6 = 500 ( ) 6
EXPONENTIAL GROWTH MODEL
A is the initial amount.
t is the time period.
(1 + r) is the growth factor, r is the growth rate.
The percent of increase is 100r.
y = A (1 + r)t

4. Writing an Exponential Growth Model
A population of 20 rabbits is released into a wildlife region. The population triples each year for 5 years.

5. Reminder: percent increase is 100r.
Writing an Exponential Growth Model
A population of 20 rabbits is released into a wildlife region.
The population triples each year for 5 years.
a. What is the percent of increase each year?
SOLUTION
The population triples each year, so the growth factor is 3.
The population triples each year, so the growth factor is 3.
1 + r = 3
1 + r = 3
1 + r = 3
So, the growth rate r is 2 and the percent of increase each year is 200%.
So, the growth rate r is 2 and the percent of increase each year is 200%.
So, the growth rate r is 2 and the percent of increase each year is 200%.
Reminder: percent increase is 100r.

6. There will be about 4860 rabbits after 5 years.
Writing an Exponential Growth Model
A population of 20 rabbits is released into a wildlife region.
The population triples each year for 5 years.
b. What is the population after 5 years?
Help
SOLUTION
After 5 years, the population is
P = A(1 + r) t
Exponential growth model
= 20(1 + 2) 5
Substitute A, r, and t.
= 20 • 3 5
Simplify.
= 4860
Evaluate.
There will be about 4860 rabbits after 5 years.

7. Practice Problems:
There are 20 bears in a zoo. What will be their population after 3 years, if the population doubles each year?

8. Practice Problems:
2. The cost of a movie ticket is $8, which increases by 3% each year. Find the cost of the ticket after 8 years.

9. Practice Problems:
3. Rents is a particular area are increasing by 3% every year. Predict what the rent of an apartment would be after 2 years, if its rent is $400 per month now.

10. Practice Problems:
4. The population of the United States was about 250 million in 2003, and is growing exponentially at a rate of 0.7% each year. What was the population in 2013?

11. EXPONENTIAL DECAY MODEL
WRITING EXPONENTIAL DECAY MODELS
A quantity is decreasing exponentially if it decreases by the same percent in each time period.
EXPONENTIAL DECAY MODEL
A is the initial amount.
t is the time period.
y = A (1 – r)t
(1 – r ) is the decay factor, r is the decay rate.
The percent of decrease is 100r.

12. The purchasing power of a dollar in 1997 compared to 1982 was $0.59.
Writing an Exponential Decay Model
COMPOUND INTEREST From 1982 through 1997, the purchasing power of a dollar decreased by about 3.5% per year. Using 1982 as the base for comparison, what was the purchasing power of a dollar in 1997?
SOLUTION
Let y represent the purchasing power and let t = 0 represent the year The initial amount is $1. Use an exponential decay model.
y = A (1 – r) t
Exponential decay model
= (1)(1 – 0.035) t
Substitute 1 for A, for r.
= t
Simplify.
Because 1997 is 15 years after 1982, substitute 15 for t.
y =
Substitute 15 for t.
0.59
The purchasing power of a dollar in 1997 compared to 1982 was $0.59.

13. EXPONENTIAL GROWTH AND DECAY MODELS
GRAPHING EXPONENTIAL DECAY MODELS
EXPONENTIAL GROWTH AND DECAY MODELS
CONCEPT
SUMMARY
EXPONENTIAL GROWTH MODEL
EXPONENTIAL DECAY MODEL
y = A (1 + r)t
y = A (1 – r)t
An exponential model y = a • b t represents exponential growth if b > 1 and exponential decay if 0 < b < 1.
(1 – r) is the decay factor, r is the decay rate.
(1 + r) is the growth factor, r is the growth rate.
A is the initial amount.
𝒕 𝒊𝒔 𝒕𝒊𝒎𝒆
1 + r > 1
0 < 1 – r < 1

14. EXPONENTIAL GROWTH MODEL
Back
EXPONENTIAL GROWTH MODEL
A is the initial amount.
t is the time period.
(1 + r) is the growth factor, r is the growth rate.
The percent of increase is 100r.
y = A (1 + r) t

15. EXPONENTIAL DECAY MODEL
Back
EXPONENTIAL DECAY MODEL
A is the initial amount.
t is the time period.
(1 – r) is the decay factor, r is the decay rate.
The percent of decrease is 100r.
y = A (1 – r) t