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
C is the initial amount.
t is the time period.
y = C (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
C 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 = C (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 = C(1 + r) t
Exponential growth model
= 20(1 + 2) 5
Substitute C, r, and t.
= 20 • 3 5
Simplify.
= 4860
Evaluate.
There will be about 4860 rabbits after 5 years.

7. GRAPHING EXPONENTIAL GROWTH MODELS
A Model with a Large Growth Factor
Graph the growth of the rabbit population.
SOLUTION
Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t P
4860 60
180
540
1620 20 5 1 2 3 4
1000
2000
3000
4000
5000
6000 1 7 2 3 4 5 6
Time (years)
Population
Here, the large growth factor of 3 corresponds to a rapid increase
P = 20 ( 3 ) t

8. 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
C is the initial amount.
t is the time period.
y = C (1 – r)t
(1 – r ) is the decay factor, r is the decay rate.
The percent of decrease is 100r.

9. 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 = C (1 – r) t
Exponential decay model
= (1)(1 – 0.035) t
Substitute 1 for C, 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.

10. Purchasing Power (dollars)
GRAPHING EXPONENTIAL DECAY MODELS
Graphing the Decay of Purchasing Power
Help
Graph the exponential decay model in the previous example. Use the graph to estimate the value of a dollar in ten years.
SOLUTION
Make a table of values, plot the points in a coordinate plane, and draw a smooth curve through the points. t y
0.837
0.965
0.931
0.899
0.867
1.00 5 1 2 3 4
0.7
0.808
0.779
0.752
0.726 10 6 7 8 9
0.2
0.4
0.6
0.8
1.0 1 12 3 5 7 9 11
Years From Now
Purchasing Power (dollars) 2 4 6 8 10
y = 0.965t
Your dollar of today
will be worth about 70 cents in ten years.

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