Exploring Exponential Growth Exploring Exponential Growth and Decay Modelsand Decay Models

Acc. Coordinate Algebra / Geometry AAcc. Coordinate Algebra / Geometry A

Day 57Day 57

What am I going to learn?What am I going to learn?

Concept of an Concept of an exponential functionexponential functionModels for exponential Models for exponential growthgrowthModels for exponential Models for exponential decaydecayMeaning of an asymptoteMeaning of an asymptoteFinding the equation of Finding the equation of an exponential functionan exponential function

RecallRecall

Independent variableIndependent variable is another name for is another name for domain or input, which is typically but not domain or input, which is typically but not always represented using the variable, always represented using the variable, xx..

Dependent variableDependent variable is another name for is another name for range or output, which is typically but not range or output, which is typically but not always represented using the variable, always represented using the variable, yy..

What is an exponential function?What is an exponential function?

Obviously, it must have something to do Obviously, it must have something to do with an exponent!with an exponent!

An An exponential functionexponential function is a function is a function whose whose independent variableindependent variable is an is an exponent.exponent.

What does an exponential function What does an exponential function look like?look like?

BaseBase

ExponentExponentand and

Independent Independent VariableVariableJust some Just some

number number that’s not 0that’s not 0

Why not 0?Why not 0?

Dependent Dependent VariableVariable

Exponential FunctionExponential Function

A function in the form:A function in the form:

– BaseBase is Constant is Constant

– Exponent is the Independent VariableExponent is the Independent Variable

The Basis of BasesThe Basis of Bases

The The basebase of an exponential function carries of an exponential function carries much of the meaning of the function.much of the meaning of the function.

The base determines exponential growth The base determines exponential growth or decay.or decay.

The base is a positive number; however, it The base is a positive number; however, it cannot be 1. We will return later to the cannot be 1. We will return later to the reason behind this part of the definition .reason behind this part of the definition .

Exponential Growth and DecayExponential Growth and Decay

Exponential GrowthExponential Growth: Function in the form : Function in the form f(x) = abf(x) = abxx, with a>0 and b>1, with a>0 and b>1– Function increases as x increasesFunction increases as x increases

Exponential DecayExponential Decay: Function in the form : Function in the form f(x) = abf(x) = abxx, with a>0 and 0<b<1, with a>0 and 0<b<1– Function decreases as x increasesFunction decreases as x increases

Exponential GrowthExponential Growth

An exponential function models An exponential function models growthgrowth whenever its base > 1. (Why?)whenever its base > 1. (Why?)

If the base If the base bb is larger than 1, then is larger than 1, then bb is is referred to as the referred to as the growth factorgrowth factor..

What does Exponential Growth look like?What does Exponential Growth look like?

xx 22xx yy

-3-3 22-3-3

-2-2 22-2-2 ¼ ¼

-1-1 22-1-1 ½ ½

00 2200 11

11 2211 22

22 2222 44

33 2233 88

Consider Consider yy = 2 = 2xx

Table of Values:Table of Values:

Graph:Graph:1

8Cool Fact:Cool Fact:

All exponential All exponential growth growth

functions look functions look like this!like this!

Investigation: Tournament PlayInvestigation: Tournament Play

The NCAA holds an annual basketball The NCAA holds an annual basketball tournament every March.tournament every March.

The top 64 teams in Division I are invited The top 64 teams in Division I are invited to play each spring.to play each spring.

When a team loses, it is out of the When a team loses, it is out of the tournament.tournament.

Work with a partner close by to you and Work with a partner close by to you and answer the following questions.answer the following questions.

Investigation: Tournament PlayInvestigation: Tournament Play

Fill in the following Fill in the following chart and then graph chart and then graph the results on a piece the results on a piece of graph paper.of graph paper.

Then be prepared to Then be prepared to interpret what is interpret what is happening in the happening in the graph.graph.

After round After round xx Number of Number of teams in teams in

tournament tournament ((yy))

00 6464

11

22

33

44

55

66

Exponential DecayExponential Decay

An exponential function models An exponential function models decaydecay whenever its 0 < base < 1. (Why?)whenever its 0 < base < 1. (Why?)

If the base If the base bb is between 0 and 1, then is between 0 and 1, then bb is is referred to as the referred to as the decay factordecay factor..

What does Exponential Decay look What does Exponential Decay look like?like?

Consider Consider yy = (½) = (½)xx

Table of Values:Table of Values:

xx (½)(½)xx yy

-2-2 ½½-2-2 4 4

-1-1 ½½-1-1 2 2

00 ½½00 11

11 ½½11 ½ ½

22 ½½22 ¼ ¼

33 ½½33 1/81/8

Graph:Graph:

Cool Fact:Cool Fact: All All

exponential exponential decay decay

functions functions look like this!look like this!

Growth or DecayGrowth or Decay

( )x h

f x a b k

b>1b>1 0<b<10<b<1

Neither Growth Nor DecayNeither Growth Nor Decay

End BehaviorEnd Behavior

Notice the end behavior of the first graph-exponential Notice the end behavior of the first graph-exponential growth. Go back and look at your graph.growth. Go back and look at your graph.

As , _______ , which means

________________________________________

x f x

As , _______ , which means

_______________________________________

x f x

0as you move to the right, the graph goes up without bound.as you move to the right, the graph goes up without bound.

as you move to the left, the graph levels off-getting close to but as you move to the left, the graph levels off-getting close to but not touching the not touching the xx-axis -axis (y(y = 0). = 0).

End BehaviorEnd Behavior

Notice the end behavior of the second graph-Notice the end behavior of the second graph-exponential decay. Go back and look at your exponential decay. Go back and look at your graph.graph.

As , _______ , which means

________________________________________

x f x

As , _______ , which means

________________________________________

x f x

0as you move to the right, the graph levels off-getting close to but as you move to the right, the graph levels off-getting close to but

not touching the not touching the xx-axis -axis (y(y = 0). = 0).

as you move to the left, the graph goes up without bound.as you move to the left, the graph goes up without bound.

AsymptotesAsymptotes

One side of each of the graphs appears to One side of each of the graphs appears to flatten out into a horizontal line. flatten out into a horizontal line.

An An asymptoteasymptote is a line that a graph is a line that a graph approachesapproaches but never touches or but never touches or intersects.intersects.

AsymptotesAsymptotes

Notice that the left Notice that the left side of the graph gets side of the graph gets really close to really close to yy = 0 = 0 as .as .We call the line We call the line yy = 0 = 0 an asymptote of the an asymptote of the graph. Think about graph. Think about why the curve will why the curve will never take on a value never take on a value of zero and will never of zero and will never be negative.be negative.

x

AsymptotesAsymptotes

Notice the right side of Notice the right side of the graph gets really the graph gets really close to close to y y = 0 as = 0 as

..

We call the line We call the line yy = 0 = 0

an asymptote of the an asymptote of the graph. Think about why graph. Think about why the graph will never take the graph will never take on a value of zero and on a value of zero and will never be negative.will never be negative.

x

Let’s take a second look at the base Let’s take a second look at the base of an exponential function.of an exponential function.(It can be helpful to think about the base as the object that is (It can be helpful to think about the base as the object that is being multiplied by itself repeatedly.)being multiplied by itself repeatedly.)

Why can’t the base be negative?Why can’t the base be negative?

Why can’t the base be zero?Why can’t the base be zero?

Why can’t the base be one?Why can’t the base be one?

ExamplesExamplesDetermine if the function represents exponential Determine if the function represents exponential

growth or decay.growth or decay.

1.1.

2. 2.

3. 3.

5(3)xy

15

4x

y

2(4) xy

Exponential GrowthExponential Growth

Exponential DecayExponential Decay

Exponential DecayExponential Decay

Example 4 Writing an Exponential FunctionExample 4 Writing an Exponential FunctionWrite an exponential function for a graph that includes (2, Write an exponential function for a graph that includes (2, 2) and (3, 4). (Do each step on your own. We’ll show the 2) and (3, 4). (Do each step on your own. We’ll show the solution step by step.)solution step by step.)

(1) xy ab Use the general form.Use the general form.

2(2) 2 ab Substitute using (2, 2).Substitute using (2, 2).

2

2(3) a

b Solve for Solve for aa..

3(4) 4 ab Substitute using (3, 4).Substitute using (3, 4).

32

2(5) 4 b

b Substitute in for Substitute in for aa..

Example 4 Writing an Exponential FunctionExample 4 Writing an Exponential Function

Write an exponential function for a graph that Write an exponential function for a graph that includes (2, 2) and (3, 4).includes (2, 2) and (3, 4).

(6) 4 2 2b b Simplify.Simplify.

2

2 1(7)

2 2a Backsubstitute to get Backsubstitute to get aa..

1(8) (2)

2xy Plug in Plug in aa and and bb into the general into the general

formula to get equation.formula to get equation.