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Quiz This data can be modeled using an exponential equation exponential equation Find ‘a’ and ‘b’ 2.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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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 1. f(x) = 3 + 3x Your turn: Factor the following 2. g(y) = 5 + 5y

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A population is changing at a constant percentage rate r, where r is the percent rate (in decimal form). Constant Percentage Rate Time (years) Population 0 “initial population” 1 2 Your turn: 3. Factor P(1) Your turn: 5. Write P(2) in terms of P(0) only. Your turn: 4. Factor P(2)

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Constant Percentage Rate Time (years) Population 0 “initial population” 1 2 Your turn: 6. What do you think P(3) will be? 3 4 t

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If 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

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Finding Growth and Decay Rates Is the following population model an exponential growth or decay function? Find exponential growth or decay function? Find the constant percentage growth (decay) rate. 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 1890 was 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 ?

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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) = P(1) = 2*P(0) Doubles with Every time interval

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

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

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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 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’. 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 “t” years after Enter the data into your calculator and use calculator and use exponential regression exponential regression to determine the model (equation). to determine the model (equation).

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Modeling U.S. Population Using Exponential Regression 9. What is your equation? 10. What is your predicted population in 2003 ? in 2003 ? Your turn: 11. Why isn’t your predicted value the same as the actual value of the same as the actual value of million? 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 !!! is limited !!!

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Modeling a Rumor Roy High School has about 1500 students. 5 students start a rumor, which spreads 5 students start a rumor, which spreads logistically so that logistically so that Models the number of students who have heard the rumor by the end of ‘t’ days, where heard 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. How 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 to have heard the rumor ? have heard the rumor ?

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Rumors at RHS How 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 to have heard the rumor ? have heard the rumor ? Your turn: 12. “t” = ? (days)

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HOMEWORK

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