Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

3.1 Exponential and Logistic Modeling (Day 2) Use exponential and logistic functions to model real world data.

Exponential Population Model If a population is increasing at a constant percentage rate each year, then it can be modeled as: Where is the initial population, r is a decimal, and t is time in years.

Examples a) Initial population = 5 Increasing rate of 17% Then we can write the formula as

You try! b) Initial Val. = 24 Inc. rate of 12% c) Initial Val. = 14 Dec. rate of 3%

Finding the percentage rate. Given: What’s the percentage rate…

Find percentage rate.

Half-life Radioactive decay is represented with an exponential model with a base of ½, or a half life. This model is represented by is the initial amount of substance is the half-life of the substance

Example Express amount remaining as a function of time: The half life of a radioactive substance is 30 days, if we have 10 grams of this radioactive substance initially, how long would it take to have 1 gram left.

Logistic Growth Modeling Where a, c, and k are positive and c is the limit to growth.

Population Growth The population of New York state can be modeled by where P is the population in millions and t is the number of years since Based on the model, a) What was the population of New York in 1850? b) What will it be in the year 2010? c) What is New York’s maximum sustainable population (limit of growth)?

Turn to Page 298 Do Problem 46.

Warm-up (10 min. – NO TALKING) 1) Determine the exponential function with initial value of 27 and rate of increase is 12% 2) Determine the exponential function with initial value of 450 and rate of decrease is 32%. 3) What is the half-life (in years) given 4) How much substance will there be in 21 years? 5) What is the maximum sustainable population given the model

Warm-up (10 min. – NO TALKING) 1) y = 27(1.12) t 2) y = 45(0.68) t 3) 7 4) 25 5) 3050 Give a score out of 5 and turn in!!

Homework Answers 41-44, 49,50, 55,56 41) y-intercept: (0,4) Hor. Asym: y = 0, y = 12 42) y-intercept: (0,3) Hor. Asym: y = 0, y = 18 43) y-intercept: (0,4) Hor. Asym: y = 0, y = 16 44) y-intercept: (0,3) Hor. Asym: y = 0, y = 9 49) D: (- ,  ); R(0,5); Continuous; Always increasing; symmetric about (0.69,2.5); Bounded above by 5 & below by 0; No local extrema; Asymptote: y =0 & y = 5; end behavior approaches 5 when x  & 0 when x  - . 50) D: (- ,  ); R(0,6); Continuous; Always increasing; symmetric about (0.69,3); Bounded above by 6 & below by 0; No local extrema; Asymptote: y =0 & y = 5; end behavior approaches 6 when x  & 0 when x  - . 55) In ) (a) 1,794,558 (b) 19,161,673 (c) 19,875,00

Exponential & Logistic Modeling Review determining an exponential function given two points on its graph Use regression to determine exponential and logistic models Use exponential & logistic models in a real-life context

Finding an exponential function. Determine an exponential function f(x) given (0,5) and (3, 320) are on the graph.

Finding an exponential function. Determine an exponential function f(x) given (0,5) and (3, 320) are on the graph. Solution:y = 3(4) x FirstSecond 5 = a(b) 0 y = 5(b) x 5 = a(1) 320 = 5(b) 3 5 = a 64 = (b) 3 4 = bHow? What number cubed is 64? Got it!!?

Exponential Model Determine an equation for the exponential function whose values are given below. Use this function to determine the missing value. xg(X) ?

Exponential Model Determine an equation for the exponential function whose values are given below. Use this function to determine the missing value. g(x) = -5.8(0.8) x xg(X)

Logistic Model Use logistic regression [Stat  Calc  (B)] to predict the maximum sustainable populations for the two countries. Will Mexico’s population surpass that of the United States? Justify your answer. YearUSMexico

Let’s do #45 on pg. 298

homework Page odd, 19, 29-33odd, 40,44,45