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Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3) 4) 5) 6) t 9765 . 0 5 . 9 1.1053 3.05 t x x e 26 x e 6 17

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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.

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

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

Page 2: 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.

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

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

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

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

ttP )17.1(5)(

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

You try!b) Initial Val. = 24

Inc. rate of 12%

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

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

Finding the percentage rate.Given:

What’s the percentage rate…

ttP 213.15.17)(

Page 7: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

Find percentage rate.

t9765.05.9

1.10533.05 t

Page 8: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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

Page 9: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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

Page 10: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

Logistic Growth Modeling

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

kxeacxf

1

)(

Page 11: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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)(

Page 12: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

Turn to Page 298Do Problem 46.

Page 13: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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)(

Page 14: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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

Page 15: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

Finding an exponential function. Determine an exponential function

f(x) given (0,5) and (3, 320) are on the graph.

Page 16: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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!!?

Page 17: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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 ?

Page 18: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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

Page 19: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

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

Page 20: Warm-up Identify if the function is Exp. Growth or Decay 1) 2) Which are exponential expressions? 3)4) 5)6)

Let’s do #45 on pg. 298