Quiz 3-1

1.1. This data can be modeled using anThis data can be modeled using an exponentialexponential equation equation xabxf )(

Find ‘a’ Find ‘a’ and ‘b’and ‘b’

2.2. Where does cross the y-axis ?Where does cross the y-axis ?xxg )5.0(3)( 3. Is g(x) an exponential 3. Is g(x) an exponential growthgrowth or or decaydecay function? function?

4. Convert to exponential notation:4. Convert to exponential notation: 5243log x

5. Convert to logarithmic notation:5. Convert to logarithmic notation: 35x

3.2

Exponential and Logistic Modeling

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 whyExponential functions model unrestricted growth (money) and decay (radioactive material); Logistic functions model restricted growth,

(spread of disease, populations and rumors)

Factoring

1. f(x) = 3 + 3x1. f(x) = 3 + 3x

Your turnYour turn: Factor the following: Factor the following

2. g(y) = 5 + 5y2. g(y) = 5 + 5y

A population is changing at a constant percentage rate r, where r is the percent rate (in decimal form).

Constant Percentage Rate

Time (years)Time (years) PopulationPopulation

0 “initial population” 0)0( PP

1 rPPPP *)1( 001

rPPPP *)2( 112

)1)(1(02 rrPP 20 )1( rP

2

Your turnYour turn: 3. Factor P(1): 3. Factor P(1) )1(01 rPP

Your turnYour turn: 5. Write P(2) in terms of P(0) only.: 5. Write P(2) in terms of P(0) only.

Your turnYour turn: 4. Factor P(2): 4. Factor P(2) )1(12 rPP

Constant Percentage Rate

Time (years)Time (years) PopulationPopulation0 “initial population”0P1

303 )1( rPP

202 )1( rPP 2

)1(01 rPP

Your turnYour turn: 6. What do you think P(3) will be?: 6. What do you think P(3) will be?

3

4 404 )1( rPP

trPtP )1()( 0 t

If a population is changing at a constantIf a population is changing at a constant percentage rate ‘r’ each year, then:percentage rate ‘r’ each year, then:

is the population as a function of time. is the population as a function of time.

Exponential Population Model

trPtP )1()( 0

Finding Growth and Decay Rates

Is the following population model an Is the following population model an exponential growth or decayexponential growth or decay function? Find function? Find the constant percentage growth (decay) rate.the constant percentage growth (decay) rate.

ttP )0136.1(248,782)(

trPtP )1()( 0 ttP )0136.01(248,782)(

‘‘r’ > 0, therefore this is r’ > 0, therefore this is exponential growthexponential growth..

‘‘r’ = 0.0136 or 1.36%r’ = 0.0136 or 1.36%

Finding an Exponential Function

Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.

trPtP )1()( 0 100 P ‘‘r’ = 0.05r’ = 0.05

ttP )05.1(10)( xxf )05.1(10)( or or

Your Turn:

The population of “Smallville” in the yearThe population of “Smallville” in the year 1890 was 6250. Assume the population1890 was 6250. Assume the population increased at a rate of 2.75% per year.increased at a rate of 2.75% per year.

• What is the population in 1915 ? What is the population in 1915 ?

Modeling Bacteria Growth

Suppose a culture of 100 bacteria cells are put into a Suppose a culture of 100 bacteria cells are put into a petri dish and the culture petri dish and the culture doublesdoubles every hour. every hour.

Predict whenPredict when the number of bacteria will be 350,000. the number of bacteria will be 350,000.

trPtP )1()( 0 P(0) = 100 P(0) = 100 P(t) = 350000 P(t) = 350000

t)2(100350000

P(1) = 2*P(0)P(1) = 2*P(0)

tPtP )2()( 0

Doubles withDoubles withEvery time intervalEvery time interval

3500105.3101

105.32 3

2

5

t 35002 t

Solving an Exponential Equation

35002 t

Your calculator doesn’t have base 2 (it mightYour calculator doesn’t have base 2 (it might in some of the catalog of functions)in some of the catalog of functions)

3500log2log 22 x

3500log2x

Change of Base FormulaChange of Base Formula: : c

aa

b

bc log

loglog

77.112log

3500logx t = 11 hours, 46 minutest = 11 hours, 46 minutes

Your Turn:

8. When did the population reach 50,000 ?8. When did the population reach 50,000 ?

The population of “Smallville” in the yearThe population of “Smallville” in the year 1890 was 6250. Assume the population1890 was 6250. Assume the population increased at a rate of 2.75% per year.increased at a rate of 2.75% per year.

Exponential RegressionStat p/b Stat p/b gives lists gives lists

Enter the data: Enter the data: Let L1 be years since initial valueLet L1 be years since initial value

Let L2 be populationLet L2 be population

Stat p/b Stat p/b calc p/b calc p/bscroll down to exponential regressionscroll down to exponential regression

““ExpReg” displayed:ExpReg” displayed:enter the lists: “L1,L2”enter the lists: “L1,L2”

xabxf )( The calculator will display the The calculator will display the values for ‘a’ and ‘b’.values for ‘a’ and ‘b’.

Modeling U.S. Population Using Exponential Regression

Use the 1900-2000 data and exponential regression to predict the U.S. population for 2003. (Don’t enter the 2003

value).

Let P(t) = population, Let P(t) = population, “ “t” years after 1900.t” years after 1900.

Enter the data into yourEnter the data into your calculator and use calculator and use exponential regressionexponential regression to determine the model (equation).to determine the model (equation).

Modeling U.S. Population Using Exponential Regression

9. What is your equation?9. What is your equation?

10.10. What is your predicted populationWhat is your predicted population in 2003 ?in 2003 ?

Your turn:Your turn:

11.11. Why isn’t your predicted valueWhy isn’t your predicted value the same as the actual value ofthe same as the actual value of 290.8 million?290.8 million?

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 growthWe must use Logistic function if the growth is limited !!!is limited !!!

Modeling a Rumor

Roy High School has about 1500 students.Roy High School has about 1500 students. 5 students start a rumor, which spreads 5 students start a rumor, which spreads logistically so thatlogistically so that

Models the number of students who haveModels the number of students who have heard the rumor by the end of ‘t’ days, whereheard the rumor by the end of ‘t’ days, where ‘ ‘t’ = 0 is the day the rumor began to spread.t’ = 0 is the day the rumor began to spread.

)291(

1500)(

9.0 tetS

How many students have heard the rumorHow many students have heard the rumor by the end of day ‘0’ ?by the end of day ‘0’ ?

How long does it take for 1000 students toHow long does it take for 1000 students to have heard the rumor ?have heard the rumor ?

Rumors at RHS)291(

1500)(

9.0 tetS

How many students have heard the rumorHow many students have heard the rumor by the end of day ‘0’ ?by the end of day ‘0’ ?

50)291(

1500

)291(

1500)0(

)0(9.0

e

S

How long does it take for 1000 students toHow long does it take for 1000 students to have heard the rumor ?have heard the rumor ?

)291(

15001000

)(9.0 te

Your turn:Your turn:

12. “t” = ? (days)12. “t” = ? (days)

HOMEWORK