7-7 Vocabulary
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Exponential growth
Growth factor
Compound interest
Exponential decay
Decay factor
7-7 Exponential Growth
Functions
Algebra 1
Exponential Growth
y  a  b x , where a  0 and b  1
• b is the growth factor = 1 + r
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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.
Ex. 1
• Suppose the population of a town was
25,000 people in the year 2000. If the
population grows about 1.5% each year,
then what will the approximate population
be in 2025?
Compound Interest Formula
 r
A  P 1  
 n
nt
A = amount earned after investing
P= principal invested or deposited
r=rate (%)
n=# of times compounded
t=time in years
Example 2
• A savings certificate of $1000 pays 6.5%
annual interest compounded yearly. What
is the balance when the certificate matures
after 5 years?
Ex. 3
• Suppose you invest $2500 with interest rate
of 3% compounded monthly for 5 years.
What will be your account balance after 5
years?
Assignment
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Example 2
• A company starts with 50 employees and
after one year has 75. It increases the
employees at the same rate every year for 4
years. What is the percent of increase each
year? Find the number of employees after 4
years.
Example 3
• A stock with initial cost of $20 per share
doubled its price per share every year for 3
years. What is the percent of increase each
year? What is the price per share after 3
years?
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
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 + 0.08 = 1.08.
A1 = 500(1.08) = 540
Balance after one year
A2 = 500(1.08)(1.08) = 583.20
Balance after two years
A3 = 500(1.08)(1.08)(1.08) = 629.856
Balance after three years
A6 =
Balance after six years
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500(1.08) 6 793.437
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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 0.08. The initial value is 500.
EXPONENTIAL GROWTH MODEL
is the
initial
amount.
C 500
is the
initial
amount.
6tisisthe
period.
thetime
time
period.
A6y==500
C (1
(1++0.08)
r)t 6
(1(1++0.08)
thegrowth
growth factor,
factor, 0.08
is the
growthrate.
rate.
r) isisthe
r is the
growth
6  793.437
The)percent
of increase
is 100r.
A6 = 500(1.08
Balance
after 6 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.
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.
1+r = 3
So, the growth rate r is 2 and the percent of increase each
year is 200%.
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.
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
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.
Writing an Exponential Decay Model
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?
COMPOUND INTEREST
SOLUTION
Let y represent the purchasing power and let t = 0 represent the year
1982. 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, 0.035 for r.
= 0.965 t
Simplify.
Because 1997 is 15 years after 1982, substitute 15 for t.
y = 0.96515
Substitute 15 for t.
0.59
The purchasing power of a dollar in 1997 compared to 1982 was $0.59.
GRAPHING EXPONENTIAL DECAY MODELS
Graphing the Decay of Purchasing Power
Graph the exponential decay model in the previous example.
Use the graph to estimate the value of a dollar in ten years.
SOLUTION
Help
Make a table of values, plot the points in a coordinate
plane, and draw a smooth curve through the points.
t 0
1
2
3
4
5
6
7
8
9
10
y 1.00 0.965 0.931 0.899 0.867 0.837 0.808 0.779 0.752 0.726 0.7
y = 0.965t
0.8
(dollars)
Purchasing Power
1.0
0.6
0.4
0.2
0
1
2
3
Your dollar of today
will be worth about 70
cents in ten years.
4
5
6
7
8
Years From Now
9
10
11
12
GRAPHING EXPONENTIAL DECAY MODELS
CONCEPT
EXPONENTIAL GROWTH AND DECAY MODELS
SUMMARY
EXPONENTIAL GROWTH MODEL
y = C (1 + r)t
EXPONENTIAL DECAY MODEL
y = C (1 – r)t
An exponential model y = a • b t represents exponential
b > 1 and
exponential (1
decay
0 <decay
b < 1.factor,
(1 growth
+ r) is theifgrowth
factor,
– r) isifthe
Ct is
is the
the initial
time period.
amount.
C)
(0, C) rate.
r is the(0,growth
rate.
r is the decay
1+r>1
0<1–r<1