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

2. Exponential and Logistic Modeling
3.2
Exponential and Logistic Modeling

3. What you’ll learn about
Constant Percentage Rate and Exponential Functions
Exponential Growth and Decay Models
Using Regression to Model Population
Other Logistic Models
… and why
Exponential functions model unrestricted growth (money)
and decay (radioactive material);
Logistic functions model restricted growth,
(spread of disease, populations and rumors)

4. Factoring Your turn: Factor the following 1. f(x) = 3 + 3x
2. g(y) = 5 + 5y

5. Constant Percentage Rate
A population is changing at a constant percentage rate r, where r is the percent rate (in decimal form).
Time (years)
Population
“initial population” 1
Your turn: Factor P(1) 2
Your turn: Factor P(2)
Your turn: Write P(2) in terms of P(0) only.

6. Constant Percentage Rate
Time (years)
Population
“initial population” 1 2
Your turn: What do you think P(3) will be? 3 4 t

7. Exponential Population Model
If a population is changing at a constant
percentage rate ‘r’ each year, then:
is the population as a function of time.

8. 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%

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

10. Your Turn: 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 ?

11. 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.
P(0) = 100
P(t) =
Doubles with
Every time interval
P(1) = 2*P(0)

12. Solving an Exponential Equation
Your calculator doesn’t have base 2 (it might
in some of the catalog of functions)
Change of Base Formula:
t = 11 hours, 46 minutes

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

14. 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’.

15. 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).

16. Modeling U.S. Population Using Exponential Regression
Your turn:
9. What is your equation?
What is your predicted population
in 2003 ?
Why isn’t your predicted value
the same as the actual value of
290.8 million?

17. Maximum Sustainable Population
Exponential growth is unrestricted, but population growth often is not. For many populations, the growth begins exponentially, but eventually slows and approaches a limit to growth called the maximum sustainable population.
We must use Logistic function if the growth
is limited !!!

18. Modeling a Rumor Roy High School has about 1500 students.
5 students start a rumor, which spreads
logistically so that
Models the number of students who have
heard the rumor by the end of ‘t’ days, where
‘t’ = 0 is the day the rumor began to spread.
How many students have heard the rumor
by the end of day ‘0’ ?
How long does it take for 1000 students to
have heard the rumor ?

19. Rumors at RHS How many students have heard the rumor
by the end of day ‘0’ ?
How long does it take for 1000 students to
have heard the rumor ?
Your turn:
12. “t” = ? (days)

20. HOMEWORK