x
function
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.
If a quantity increases by the same proportion r in each unit
of time, then the quantity displays exponential growth and
can be modeled by the equation
y  C (1  r )
t
Where
C = initial amount
r = growth rate (percent written as a decimal)
t = time where t  0
(1+r) = growth factor where 1 + r > 1
You deposit $1500 in an account that pays 2.3% interest
compounded yearly,
1) What was the initial principal (P) invested?
2) What is the growth rate (r)? The growth factor?
3) Using the equation A = P(1+r)t, how much money would
you have after 2 years if you didn’t deposit any more money?
1) The initial principal (P) is $1500.
2) The growth rate (r) is 0.023. The growth factor is 1.023.
3) A  P (1  r ) t
A  1500 (1  0.023) 2
A  $1569.79
Writing an Exponential Growth Model
A population of 20
rabbits is released
into a wildlife region.
The population
triples each year for
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?
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.
Help
GRAPHING EXPONENTIAL GROWTH MODELS
A Model with a Large Growth Factor
Graph the growth of the rabbit population.
Make a table of values, plot the points in a coordinate
plane, and draw a smooth curve through the points.
SOLUTION
t 0
P 20
1
2
3
60
180
540
4
5
1620 4860
6000
Population
5000
4000
P = 20 ( 3 ) t
3000
Here, the large
growth factor of 3
corresponds to a
rapid increase
2000
1000
0
1
2
3
4
Time (years)
5
6
7