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**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:

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**Exponential and Logistic Modeling**

3.2 Exponential and Logistic Modeling

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**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)

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**Factoring Your turn: Factor the following 1. f(x) = 3 + 3x**

2. g(y) = 5 + 5y

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**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.

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**Constant Percentage Rate**

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

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**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.

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**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%

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**Finding an Exponential Function**

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

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**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 ?

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**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)

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**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

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**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 ?

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**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’.

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**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).

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**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?

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**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 !!!

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**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 ?

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**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)

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HOMEWORK

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