Warm-up Identify if the function is Exp. Growth or

Decay1)

2)Which are exponential expressions?3) 4)

5) 6)

t9765.05.9

1.10533.05 t

x

x

e

26x

e

6

17

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.

trPtP 1)( 0

0P

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

ttP )17.1(5)(

You try!b) Initial Val. = 24

Inc. rate of 12%

c) Initial Val. = 14Dec. rate of 3%

Finding the percentage rate.Given:

What’s the percentage rate…

ttP 213.15.17)(

Find percentage rate.

t9765.05.9

1.10533.05 t

Half-lifeRadioactive 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 substanceis the half-life of the substance

halftt

Stf

21)( 0

0Shalft

ExampleExpress 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.

30

21101

t

Logistic Growth Modeling

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

kxeacxf

1

)(

Population GrowthThe population of New York state can be

modeled by

where P is the population in millions and t is the number of years since 1800. 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)?

tetP 035005.0993.571

875.19)(

Turn to Page 298Do 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) given4) How much substance will there be in 21

years?5) What is the maximum sustainable population

given the model

7

21200)(

t

tf

tetP 07189.0912.161

3050)(

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

First Second5 = a(b)0

y = 5(b)x

5 = a(1) 320 = 5(b)3

5 = a 64 = (b)3

4 = b How? 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.

x g(X)-2 -9.0625-1 -7.250 -5.81 -4.642 -3.7125 ?

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

x g(X)-2 -9.0625-1 -7.250 -5.81 -4.642 -3.7125 -

1.900544

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.

Year US Mexico

1900 76.2 13.61950 92.2 25.81960 106.0 34.91970 123.2 48.21980 226.5 66.81990 248.7 88.12001 281.4 101.9

Let’s do #45 on pg. 298