1. Quiz 3-1a This data can be modeled using an exponential equation 
Find ‘a’ and ‘b’
Where does cross the y-axis ?
3. Is r(x) an exponential growth or decay function?
4. Convert r(x) to exponential base ‘e’ notation:

2. Applications of Exponential Functions and the Logistic Function
3.1B
Applications of Exponential Functions
and
the Logistic Function

3. Exponential Function Initial value Input variable Growth factor:
What does ‘b’ equal
In order for it to be “growth”?
What is the value of ‘y’ where
the graph crosses the y-axis?
What does ‘b’ equal
In order for it to be “decay”?

4. Your turn: 1. Where does it cross the y-axis?
Graph the functions:
1. Where does it cross the y-axis?
2. What is the “intial value of f(t) ?

5. Population Growth
If population grows at a constant percentage rate over a
year time frame, (the final population is the initial population
plus a percentage of the orginial population) then the
population at the end of the first year would be:
Percent rate of change
(in decimal form)
At the end of the second year the population would be:

6. Population Growth
Quadratic equation!

7. Population Growth
Quadratic equation!

8. Population Growth
Special cubic!

9. Population Growth Percent rate of change (in decimal form) time
Population (as a
function of time)
Initial
population
Growth
rate
Growth factor:
Initial value

10. Word problems There are 4 quantities in the equation:
1. Population “t” years/min/sec in the future
2. Initial population
3. Growth rate
4. time
The words in the problem will give you three of the four
quantities. You just have to “plug them in” to the equation
and solve for the unknown quantity.

11. Population Growth Percent rate of change (in decimal form) time
Population (at time
“t”) in the future
Initial
population
Growth
rate
The initial population of a colony of bacteria is The population increases by 50% every hour. What is the population after 5 hours?
Unknown value

12. Simple Interest (savings account)
time
Amount (as a
function of time)
Initial amount
(“principle”)
Interest
rate
A bank account pays 3.5% interest per year. If you initially invest $200, how much money will you have after 5 years?
Unknown value

13. Your turn: 3.
A bank account pays 14% interest per year. If you initially invest $2500, how much money will you have after 7 years?
The population of a small town was 1500 in The population increases by 3% every year. What is the population in 2009? 4.

14. Solve by graphing San Jose, CA Year Population 1990 782,248
,248
,193
Assuming exponential growth, when will
the population equal 1 million?
Let ‘t’ = years since 1990
We must find the growth factor ‘b’
‘b’ =
Unknown value

15. Example 1,000,000 ‘t’ = approximately 18 18 years AFTER 1990  2008
Later in the chapter we will learn how to solve for the
unknown exponent algebraically.

16. Your Turn: 5. When did the population reach 50,000 ?
The population of “Smallville” in the year 1890 was Assume the population increased at a rate of 2.75% per year.
5. When did the population reach 50,000 ?

17. Your turn: 6. Assuming exponential growth, when will Year Population
,709,873
,006,550
USA
Your turn:
6. Assuming exponential growth, when will
the population exceed 400 million?
We must find the growth factor ‘b’
‘b’ =
43 yrs after
t = 0 (1990)
2033

18. Your turn: 7. Assuming exponential growth, when will Year Population
million
million
USA
Your turn:
7. Assuming exponential growth, when will
the population exceed 400 million?
We must find the growth factor ‘b’
‘b’ =
140.2 yrs after
t = 0 (1900)
2040.2

19. Finding an Exponential Function
$500 was deposited into an account that pays “simple interest” (interest paid at the end of the year).
25 years later, the account contained $ What was the percentage rate of change?
Unknown value

20. Your Turn:
8. The population of “Smallville” in the year 1890 was Assume the population increased at a rate of 2.75% per year. What is the population in 1915 ?
9. The population of “Bigville” in the year 1900 was 25,200. In 1955 the population was 37,200. What was the percentage rate of change?
10. The population of “Ghost-town” in the year 1900 was In 1935 the population was What was the percentage rate of change?

21. Finding Growth and Decay Rates
Is the following population model an
exponential growth or decay function?
Find the constant percentage growth (decay) rate.
‘r’ > 0, therefore this is exponential growth.
‘r’ = or 1.36%

22. Your turn: 11. Is it growth or decay?
b = 1.5
b > 0
Growth!
12. Find the constant percentage growth (decay) rate.
‘r’ = or 50%  % rate of growth is 50%
‘r’ > 0, therefore this is exponential growth.

23. Finding an Exponential Function
Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.
‘r’ = 0.05 or

24. Modeling Bacteria Growth
Suppose a culture of 100 bacteria cells are put into a petri dish and the culture doubles every hour.
Predict when the number of bacteria will be 350,000.
What is the growth factor?
P(0) = 100
P(t) =

25. Modeling Bacteria Growth
Suppose a culture of 100 bacteria cells are put into a petri dish and the culture doubles every hour.
Predict when the number of bacteria will be 350,000.
t = 11 hours hrs
Where do the two
graphs cross?
t = 11 hours hrs
t = 11 hours + 46 min

26. Your turn: 13. A family of 10 rabbits doubles every 2 years. When
will the family have 225 members?
b = 2
t = 7.8 years
t = 7 years 6 months

27. Modeling U.S. Population Using Exponential Regression
Use the data and exponential regression to predict the U.S. population for (Don’t enter the 2003 value).
Let P(t) = population,
“t” years after 1900.
Enter the data into your
calculator and use
exponential regression
to determine the model (equation).

28. Exponential Regression
Stat p/b  gives lists
Enter the data:
Let L1 be years since initial value
Let L2 be population
Stat p/b  calc p/b
scroll down to exponential regression
“ExpReg” displayed:
enter the lists: “L1,L2”
The calculator will display the
values for ‘a’ and ‘b’.

29. Your turn: 14. What is your equation?
15. What is your predicted population in 2003 ?
16. Why isn’t your predicted value the same as the
actual value of million?

30. HOMEWORK