Presentation on theme: "Quiz 3-1 This data can be modeled using an Find ‘a’"— Presentation transcript:
1Quiz 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:
2Exponential and Logistic Modeling 3.2Exponential and Logistic Modeling
3What you’ll learn about Constant Percentage Rate and Exponential FunctionsExponential Growth and Decay ModelsUsing Regression to Model PopulationOther Logistic Models… and whyExponential functions model unrestricted growth (money)and decay (radioactive material);Logistic functions model restricted growth,(spread of disease, populations and rumors)
4Factoring Your turn: Factor the following 1. f(x) = 3 + 3x 2. g(y) = 5 + 5y
5Constant 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”1Your turn: Factor P(1)2Your turn: Factor P(2)Your turn: Write P(2) in terms of P(0) only.
6Constant Percentage Rate Time (years)Population“initial population”12Your turn: What do you think P(3) will be?34t
7Exponential Population Model If a population is changing at a constantpercentage rate ‘r’ each year, then:is the population as a function of time.
8Finding Growth and Decay Rates Is the following population model anexponential growth or decay function? Findthe constant percentage growth (decay) rate.‘r’ > 0, therefore this is exponential growth.‘r’ = or 1.36%
9Finding an Exponential Function Determine the exponential function with initial value = 10, increasing at a rate of 5% per year.‘r’ = 0.05or
10Your Turn: The population of “Smallville” in the year 1890 was Assume the populationincreased at a rate of 2.75% per year.What is the population in 1915 ?
11Modeling 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) = 100P(t) =Doubles withEvery time intervalP(1) = 2*P(0)
12Solving an Exponential Equation Your calculator doesn’t have base 2 (it mightin some of the catalog of functions)Change of Base Formula:t = 11 hours, 46 minutes
13Your Turn: The population of “Smallville” in the year 1890 was Assume the populationincreased at a rate of 2.75% per year.8. When did the population reach 50,000 ?
14Exponential Regression Stat p/b gives listsEnter the data:Let L1 be years since initial valueLet L2 be populationStat p/b calc p/bscroll down to exponential regression“ExpReg” displayed:enter the lists: “L1,L2”The calculator will display thevalues for ‘a’ and ‘b’.
15Modeling U.S. Population Using Exponential Regression Use the data and exponential regression to predict the U.S. population for (Don’t enter the 2003value).Let P(t) = population,“t” years after 1900.Enter the data into yourcalculator and useexponential regressionto determine the model (equation).
16Modeling U.S. Population Using Exponential Regression Your turn:9. What is your equation?What is your predicted populationin 2003 ?Why isn’t your predicted valuethe same as the actual value of290.8 million?
17Maximum 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 growthis limited !!!
18Modeling a Rumor Roy High School has about 1500 students. 5 students start a rumor, which spreadslogistically so thatModels the number of students who haveheard 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 rumorby the end of day ‘0’ ?How long does it take for 1000 students tohave heard the rumor ?
19Rumors at RHS How many students have heard the rumor by the end of day ‘0’ ?How long does it take for 1000 students tohave heard the rumor ?Your turn:12. “t” = ? (days)