Did you know that if you put a group of
fruit flies together, their number will double each day?
Suppose we start with an
initial population of 4
How many
after
1 day
8
Initial population 4
How many
after
2 days
16
Initial population 4
How many
after
3 days
32
Initial population 4
How many
after
4 days
64
Initial population:
4 fruit flies
After 1 day:
4x2 fruit flies = 8
After 2 days
4x2x2 fruit flies = 4x22 = 16
After 3 days
4x2x2x2 fruit flies = 4x23 = 32
After 4 days
4x2x2x2x2 fruit flies = 4x24 = 64
This is an example of
an exponential function.
After t days, the number N, of fruit flies is
N  N(t)  4x2
t
How many fruit flies will be present after 10 days (assuming none die)?
N(10) = 4x210 = 4,096
A function N(t) is an exponential function with base a if N changes by
constant multiples of a. That is, if t is increased by 1, the new value of N
is formed by multiplying by a.
1. The formula for an exponential function with base a and initial value P is
N = Pat.
In the fruit fly illustration, P = 4 (the initial population),
and a = 2 (daily growth factor),
and N = N(t) = 4x2t.
What if we began with 1,000 fruit flies and we killed half each day?
What would the exponential function look like?
N = 1000(.5t)
This is an example of exponential decay (as opposed to growth)
1,000 is the initial value and .5 is the daily decay factor.
What if we began with 1,000 fruit flies and we killed half each day?
What would the exponential function look like?
N = 1000(.5t)
This is an example of exponential decay (as opposed to growth)
1,000 is the initial value and .5 is the daily decay factor.
What if we only killed ¼ of the flies each day?
What would the exponential function look like?
N = 1000(.75t)
A function N(t) is an exponential function with base a if N changes by
constant multiples of a. That is, if t is increased by 1, the new value of N
is formed by multiplying by a.
1. The formula for an exponential function with base a and initial value P is
N = Pat.
2. If a > 1, then N shows exponential growth with growth factor a.
3. If 0 < a < 1, the N shows exponential decay with decay factor a.
The limiting value of such a function is 0.
a>1
0<a<1
A common situation where an exponential
function would apply is a situation where there is
a constant percent increase or decrease.
A common situation where an exponential function would apply is a
situation where there is a constant percent increase or decrease.
For example: Suppose you find an investment that allows you to earn
8% per year on your investment.
You decide to invest $5,000.
How much will your investment be worth after 9 years
(assuming you make no withdrawals)?
$9,995.02
Year 1 - Begin with $5,000. Interest first year = .08 x 5,000 = $400
After 1 year, your investment is worth: 5,000 + 400 = $5,400
Year 2 - Begin with $5,400. Interest second year = .08 x 5,400 = $432
After 2 years, your investment is worth: 5,400 + 432 = $5,832
This can be modeled with the exponential function:
W = 5,000(1.08t)
8% is the growth rate
1.08 is the growth factor
Where W is the worth of the investment and t is the number of years.
A function is exponential if it shows constant percentage growth or decay.
1. For an exponential function with (yearly, monthly, etc.) percentage
growth rate r (as a decimal), the growth factor is a = 1 + r
In our investment example, we added .08 to 1 to obtain the growth factor
W = 5,000(1.08t) where W = worth of the investment and t = number of years invested
8% is the percentage growth rate (or just growth rate).
1 + .08 = 1.08 is the yearly growth factor (or just growth factor).
In 1971, the average selling price for a house in
Cobb County was $81,000. From the year 1971
to the year 2007, the average selling price rose
3.7 percent per year.
1. Write a formula for the average selling price P of a house in
Cobb County t years after 1971.
t
P = 81,000(1.037 )
2. What was the average selling price for a house in Cobb
County in 2007?
$299,585.13
3.7% is the growth rate, 1.037 is the growth factor.
A function is exponential if it shows constant percentage growth or decay.
1. For an exponential function with (yearly, monthly, etc.) percentage
growth rate r (as a decimal), the growth factor is a = 1 + r
2. For an exponential function with (yearly, monthly, etc.) percentage
decay rate r (as a decimal), the decay factor is a = 1 - r
In our investment example, we added .08 to 1 to obtain the growth factor
W = 5,000(1.08t) where W = worth of the investment and t = number of years invested
8% is the percentage growth rate (or just growth rate).
1 + .08 = 1.08 is the yearly growth factor (or just growth factor).
In our second fruit fly example, we subtracted .25 from 1 to obtain the decay factor
N = 1,000(.75t) where N = number of fruit flies remaining and t = number of days
25% is the percentage decay rate (or just decay rate).
1 – .25 = .75 is the daily decay factor (or just decay factor).
From 1900 to 1960, the population of Orlando, Florida was growing at a
rate of 3% per year. Orlando’s population in 1900 was 13,850.
1. Write a formula to express population P after t years (from 1900 to 1960)
as an exponential function. P = 13850(1.03t)
2. What is the yearly growth factor?
1.03
3. Estimate the population of Orlando in 1925?
Based on the model, the population of Orlando in 1925 was 28,999.
From 1900 to 1960, the population of Orlando, Florida was growing at a
rate of 3% per year. Orlando’s population in 1900 was 13,850.
1. Write a formula to express population P after t years (from 1900 to 1960)
as an exponential function. P = 13850(1.03t)
2. What is the yearly growth factor?
1.03
3. Estimate the population of Orlando in 1925?
Based on the model, the population of Orlando in 1925 was 28,999.
4. In 1950 Podunk, N.J. and Orlando had the same population.
However, from 1950 on, Podunk began losing people at the rate
of 4% a year. Write an exponential function for the population
of Podunk t years after 1950.
P = 60717(.96t)
Section 4.1
Homework:
Read Section 4.1 (through the top of page 315)
Page 322 # S-1 through S-12
Page 323 # 1, 2, 3, 4, 13
Graphs on the
calculator only
Only parts a and b.