Table of Contents

Exponential Growth Introduction

Exponential Relationships in Equations, Tables and Graphs

Growth Rates and Growth Factors

Exponential Decay

Rules of Exponents

Click on the topic to go to that section

Exponential Growth vs. Linear Growth

Exponential GrowthIntroduction

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Making Confetti

The members of the Drama Club need to make confetti to throw after the final act in the play. They start by cutting a sheet of colored paper in half. Then, they stack the two pieces and cut them in half. They stack the resulting four pieces and cut them in half. They repeat this process, creating smaller and smaller pieces of paper.

Cut One Cut Two Cut Three

How many pieces of confetti are made after 1 cut?How many pieces of confetti are made after 2 cuts?How many pieces of confetti are made after 3 cuts?

How many pieces of confetti after 4 cuts?

How many pieces of confetti after 5 cuts?

After each cut, the members of the Drama Club count the pieces of confetti and record the results in a table.

The members of the Drama Club want to predict the number of pieces of confetti after any number of cuts.

Look at the pattern in the way the number of pieces of confetti changes with each cut. Use your observations to extend your table to show the number of pieces of confetti for up to 10 cuts.

Suppose the members of the Drama Club make 20 cuts. How many pieces of confetti would they have?

How many pieces of confetti would they have if they made 30 cuts?

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As opening night quickly approaches, the members of the Drama Club need to speed up the process of making confetti.

They decide to cut the sheet of colored paper into thirds instead of cutting it in half. Then, they stack the three pieces and cut them in thirds.

They repeat this process of cutting into thirds, creating smaller and smaller pieces of paper.

First Cut Second Cut Third Cut

How many pieces of confetti are made after 1 cut?

How many pieces of confetti are made after 2 cuts?

How many pieces of confetti are made after 3 cuts?

3

9

After each cut, the members of the Drama Club count the pieces of confetti and record the results in a table.

How many pieces of confetti are made after: 3 cuts? 4 cuts? 5 cuts?10 cuts?

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How is the process the same when the members cut the original sheet into halves and when they cut the first sheet into thirds?

How is the process different?

Is there a way to predict how many pieces of confetti they will have after any number of cuts?

These problems are an example of exponential growth.

53

5 is the Base

3 is the Exponent

Exponential Form

53 = 5 x 5 x 5 = 125

In the expression 53:

5 is the Base: The number being repeatedly

multiplied

3 is the Exponent: How many times to multiply the base

53 = 5 x 5 x 5 = 12553 is in Exponent Form

5 x 5 x 5 is in Expanded Form

125 is in Standard Form

Vocabulary

Common Powers

An exponent is also called a Power

An exponent of 2 is called a "square"72 is read "7 squared"

A power of 3 is called a "cube"53 is read "5 cubed"

Write each expression in exponential form.

a. 2 x 2 x 2

b. 6 x 6 x 6 x 6 x 6

c. 9 x 9 x 9 x 9 x 9 x 9 x 9 x 9

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Example

Write each expression in standard form.

a. 27

b. 35

c. 1.54

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Example

Explain how the meanings of 52, 25 and 5(2) differ.

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1 Evaluate

A 53

B 15

C 125

D 35

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2 Evaluate

A 35

B 243

C 15

D 53

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3 Evaluate

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4 Evaluate eight squared

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5 Evaluate three cubed

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6 Evaluate four raised to the seventh power

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Return to Table of Contents

Exponential Growth vs

Linear Growth

The Five Million Dollar Mission

Let's Imagine...

What would your dream job be?Any profession Working for any person/company

Write down your dream job at the top of your page

You will be working this job for 30 days straight so choose wisely!!!

Payroll Options

You must decide what payment option you would like before beginning your dream job:

Option 1: You receive $35,000 a day for the next thirty days

Option 2: You make $0.01 on the first day and then your salary will double every day for the next thirty days (You receive $0.01 on the first day of work, $0.02 on the second day of work, $0.04 on the third day of work, etc.)

Write down the number of the payroll option you prefer next to your dream job at the top of the page

7 What payment option did you choose?

A Option 1: $35,000 a day

B Option 2: $0.01 day 1 and doubles each day after

Getting Paid

Why did you choose Option 1?

Why did you choose Option 2?

Let's see who would get paid more by the end of the 30 days.

Option 1

30 days x $35,000 a day = $1,050,000

CONGRATULATIONS, you are a MILLIONAIRE!!!

This means for 7 days worth of work you earned $1.27.If you worked 40 hours in week 1, a typical number of hours for a work week, how much money have you made per hour?Do you want to keep this payroll option?

Option 2

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Week 2

Although this is more money than the previous week, this is still a small amount of money for working 7 days.

Anyone want to change to Option 1?!?

Let's Keep Going

What patterns/trends do you notice with this payment plan?

What do you predict will happen by day 30?

8 Now, what payment Option would you prefer?

A Option 1: $35,000 a day

B Option 2: $0.01 day 1 and doubled each day after

After 30 days, those that chose payment Option 1 will only have $1,050,000.

After 4 weeks we are up to $2,684,354.55 for payment Option 2 and we still have two more days to get paid!

Two More Days of Pay!!!

Although those workers that chose Option 2 got paid $0.01 on day 1 of work, they ended up making significantly more money than the workers that chose Option 1.

Why do you think this occurred?

Linear Growth

Payment Option 1 is an example of Linear Growth.

Linear Growth: Constant rate of change during a given interval

Rate of Change = $35,000

Given Interval = Every day

We can display linear growth in three forms:Table Equation

Graph

Equation

y = 35000x

Different Representations: Linear Growth

Day Pay for the Day

1 $35,000

2 $35,000

3 $35,000

4 $35,000

5 $35,000

6 $35,000

7 $35,000

Table

Graph

Linear Growth

Payment Option 1 pays the same amount of money each day.

How does each form of the payment option represent the linear growth?

y = 35000x

Day Pay for the Day

1 $35,000

2 $35,000

3 $35,000

4 $35,000

5 $35,000

6 $35,000

7 $35,000

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Exponential Growth

Payment Option 2 is an example of Exponential Growth.

Exponential Growth: Rate of change increases at a constantly

growing rate

Constantly Growing Rate = Doubling previous days pay

We can also display the exponential growth in three forms:Table Equation

Graph

Equation Day Pay for the Day

1 $0.01

2 $0.02

3 $0.04

4 $0.08

5 $0.16

6 $0.32

7 $0.64

Table

Graph

Different Representations: Exponential Growth

Exponential Growth

Payment Option 2 increases the Pay for the Day EACH day

How does each form of the payment option represent the exponential growth?

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Day Pay for the Day

1 $0.01

2 $0.02

3 $0.04

4 $0.08

5 $0.16

6 $0.32

7 $0.64

Linear vs Exponential

y = 35000x

Day Pay for the Day

1 $35,000

2 $35,000

3 $35,000

4 $35,000

5 $35,000

6 $35,000

7 $35,000

Day Pay for the Day

1 $0.01

2 $0.02

3 $0.04

4 $0.08

5 $0.16

6 $0.32

7 $0.64

9 What type of growth has an increasing rate of change?

A Exponential Growth

B Linear Growth

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10 Which equation(s) below represents linear growth? (Choose all that apply)

A

B y = 25x + 3

C

D y = 2x

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11 Which graph(s) below depicts linear growth? (Choose all that apply)

A B

C D

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12 Choose the table(s) below that represents exponential growth. (Choose all that apply)

A B

C D

x y

0 0

1 3

2 11

3 27

4 59

5 123

x y

0 1.00

1 1.50

2 2.25

3 3.38

4 5.06

5 7.59

x y

0 0.00

1 0.50

2 1.00

3 1.50

4 2.00

5 2.50

x y

0 25

1 125

2 225

3 325

4 425

5 525

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Exponential Relationships in Equations, Tables and

Graphs

Return to Table of Contents

Related Materials

Fish Exponential Growth.pdf

Exponential Table "Graph" Equation.pdf

There are handouts that can be used along with this section. They are located under the heading labs on the Exponential page of PMI Algebra.

Click for link to materials.

By now you have explored exponential growth and seen how it compares to linear growth.

We will take a closer look at tables, graphs and patterns found in exponential relationships.

Our exploration will lead us to recognize how the starting values and growth factors for exponential relationships are reflected in tables, graphs, and equations.

Exponential Equations can be written in the form:

y = a(bx)

y = a(bx)

y-interceptLocation where the graph of the equation will intersect the y-axis

Growth FactorThe quantity increasing at a growing rate

Equations

Examples of Exponential Equations

Identify the y-intercept and growth factor for each equation

y-intercept = 25

growth factor = 4

y-intercept = 3

growth factor = 17

y-intercept = 2

growth factor = 8

y-intercept = 6

growth factor = 3

Click

Click

Click

Click

Click

Click

Click

Click

Create the exponential equation from the provided information.

Examples

y-intercept = 9growth factor = 11

y-intercept = 32growth factor = 4

y-intercept = 7growth factor = 8

y-intercept = 8growth factor = 7

Click

Click

Click

Click

In nature, populations have the tendency to grow or decline exponentially.

We will explore several organisms where this type of growth is evident.

What do x, y, 5 and 2 represent in this equation?

The equation for the yearly growth of the rabbit population in a farmer's field is: y = 5 (2x)

x Number of years

y Number of bunnies

5 Initial amount of bunnies

2 Growth factor

Example

How many rabbits will be in the farmer's field after 3 years?

y = 5 (23) y = 5(8)y = 40 rabbits

The equation for the growth of the rabbit population in a farmer's field is: y = 5 (2x)

13 In the equation, what does the a represent?

A The exponent

B The growth factor

C The linear equation

D The y-intercept

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14 Identify the growth factor in the following equation:

A x

B y

C 9

D 56 an

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15 Create an exponential equation using the given information:

growth factor = 2 y-intercept = 7

A

B

C

D

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During Spring and Summer, some plants grow rapidly. One particular plant, a flower named the Black-eyed Susan grows exponentially. Black-eyed Susans (Rudbeckia hirta), the state flower of Maryland, can be recognized by their bright yellow flowers and dark center.

They can grow over three feet tall and have green leaves up to six inches long. This flower flourishes during the summer months of June through August and needs little care other than rich soil, plenty of sunlight and water.

In the bird garden at Monongahela Middle School, Mr. Evans planted several Black-eyed Susans one summer. The next summer he noticed that the flowers had reproduced significantly and were taking up a larger portion of the garden. Mr. Evans and his class wrote the following equation to represent the growth of the Black-eyed Susans over time:

n = 10(3t)

In this equation, n represents the number of flowers after t time in years.

With your group, consider the following questions and be prepared to share how you arrived at your response.

 n = 10(3t)

a. How many flowers did Mr. Evans and the class plant the first year?

b. What is the growth factor of the Black-eyed Susan flower in the garden?

c. How many flowers will be in the garden after 5 years?

d. In how many years will there be 270 flowers in the garden?

Solutions

 n = 10(3t)

a. How many flowers did Mr. Evans and the class plant the first year? n = 10(3t)n = 10(31)n = 10(3)n = 30 flowers

b. What is the growth factor of the Black-eyed Susan flower in the garden?Growth Factor = 3

 n = 10(3t)c. How many flowers will be in the garden after 5 years?  n = 10(3t)n = 10(35)n = 10(243)n = 2,430 flowers

d. In how many years will there be 270 flowers in the garden? n = 10(3t)270 = 10(3t)27 = (3t)t = 3 years

Solutions Continued

16 What is the value of y when x = 6 for the given relationship?

y = 2(3x)

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17 We are given the equation, . What is the value of c when x = 3?

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18 What is the value of m if n = 1,728 in the equation:

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19 A newly discovered microbe has a growth factor of 5 for every hour. If we have a petri dish with 4 of the microbes on it, what would the equation be to represent this scenario?

Let m = the number of microbes and t = time in hours

A

B

C

D

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20 A newly discovered microbe has a growth factor of 5 for every hour. If we have a petri dish with 4 of the microbes on it, how many would we expect to see after 9 hours have passed?

A 18

B 180

C 1,310,720

D 7,812,500

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Tables

y = initial amount (growth factor)x

Now we will explore growth factors in exponential tables.

We have already explored growth factor in exponential equations:

Year # of Frogs

2 4

3 12

4 36

5 108

6 324

Growth Factor in TablesIdentify the growth factor by choosing any two successive values in the dependent variable's column:

= 34(3) = 12

= 312(3) = 36

= 336(3) = 108

Since the previous value is always multiplied by 3 to obtain the next value, the growth factor is 3.

Year # ofLadybugs

1 10

2 37

3 137

4 506

5 1874

What is the growth factor in the given table?

= 3.73.7

3.73.7

The growth factor is 3.7

Creating Exponential Equations From a Table

Create an equation for the given table:

x y

0 6

1 18

2 54

3 162

4 486

5 1,458

Step 1: Identify the growth factor

Step 2: Identify the y-intercept

Step 3: Create the equation

Solution

x3

x3

x3

growth factor

x y

0 6

1 18

2 54

3 162

4 486

5 1,458

Step 1: Identify the growth factor

6 to 18 18 to 54 486 to 1,458 etc

Multiply by 3

growth factor = 3

Solution Continued

Step 2: Identify the y-intercept

The y-intercept occurs when the x value is 0y-intercept = 6

Step 3: Create the equation

Use the growth factor and the y-intercept to create the exponential equation

y-intercept

x y

0 6

1 18

2 54

3 162

4 486

5 1,458

Example

Create an equation for the given table

x y

3 686

4 4,802

5 33,614

6 235,298

7 1,647,086

Step 1: Identify the growth factor

Step 2: Identify the y-intercept

The y-intercept is not given in the table but we can evaluate it using the growth factor.

growth factor = 7

x = 2 x = 1x = 0

y = 98 y = 14 y = 2

Step 3: Create equation

Solution

x y

3 686

4 4,802

5 33,614

6 235,298

7 1,647,086

21 What is the growth factor of the given relationship?

x y

1 20

2 40

3 80

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22 What is the y-intercept of the given relationship?

x y

1 20

2 40

3 80

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23 Which equation matches the given table?

A y = 5(2x) B y = 2(5x)

C y = 10(2x) D y = 2x

x y

0 5

1 10

2 20

3 40

4 80

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Hint: Identify the y-intercept and growth factor firstClick

24 Create an equation for the provided table:

A y = 40(2x) B y = 2(40x)

C y = 5(2x) D y = 2(5x)

Hint: Identify the y-intercept and growth factor firstClick

x y

3 40

4 80

5 160

6 320

7 640

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25 Create an equation for the provided table:

A y = 12,288(4x) B y = 4(3x)

C y = 4x D y = 3(4x)

x y

6 12,288

7 49,152

8 196,608

9 786,432

10 3,145,728

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The snakehead fish is a predatory species whose natural habitat is found in the waters of Africa and Asia.

Recently, environmentalist and sport fisherman in the United States have become concerned when it was discovered that the snakehead fish has been introduced to American waters.

Exponential Graphs

These fish are known as invasive fish because they have no natural predators, they can survive on land for up to four days, and they can travel distances of up to one-quarter mile across wetlands.

In addition, the reproduction rate of the snakehead is extremely high, allowing them to completely take over bodies of water wiping out whole populations of other species of fish.

The snakehead will double its population yearly. Currently, the Fish and Game Commission estimates that there are 100 snakehead fish living in the Schuylkill River.

Complete the table below and create a graph to chart the yearly growth of the snakehead population:

Year Population

0 100

1

2

3

4

How does the y-intercept appear in the graph

How does the growth factor for the snakehead population appear in the graph?

Year Population

0 100

1 200

2 400

3 800

4 1600

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Step 1: Identify y-intercept

y-intercept = 100

Step 2: Identify any other point on the graph

(2, 400)

Example

Create an equation from the provided graph:

Example Continued

Step 3: Substitute the y-intercept and point into the exponential equation then solve for by = a(bx)400 = 100(b2)4 = b2b = 2Step 4: Rewrite equation using the y-intercept and growth factor

y = 100(2x)

Example

Step 1: Identify y-intercept

y-intercept = 5

Create an equation from the provided graph given that the y-intercept is 5:

Step 2: Identify any other point on the graph

(1, 15)

Step 3: Substitute the y-intercept and point into the exponential equation then solve for by = a(bx)15 = 5(b1)3 = b1b = 3

Step 4: Rewrite equation using the y-intercept and growth factor

y = 5(3x)

26 Which graph represents an exponential equation?

A

B

C

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27 Choose the graph that corresponds to the following equation:

A

B

C

D

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28 Choose the equation that corresponds to the graph below:

A

B

C

D

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29 Choose the equation that corresponds to the graph below:

A

B

C

D Not Enough Information is Provided

Hint: Choose points on the graph that are exactClick

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Growth Rates and

Growth Factors

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Growth Rate vs Growth Factor

Growth Rate was previously defined as: The quantity increasing at a growing rate.

Another way to define Growth Rate is: The percent increase.

For Example: The student population increased by 2%, therefore the Growth Rate is 2%

The Growth Factor is the sum of the percent increase and 100%.

For Example: The student population increased by 2%, therefore the Growth Factor is 0.02 + 1.00 = 1.02

***Growth Factor utilizes percents in decimal form***

Growth Rate Growth Factor

5% = 0.05 + 1.00 = 1.05

Growth Factor Growth Rate

1.05 = 1.05 - 1.00 = 0.05 = 5%

Example

Examples

What is the Growth Rate of 27%?

27%

What is the Growth Factor of 27%?

1.00 + 0.27 = 1.27

What is the Growth Rate if the Growth Factor is 1.084?

1.084 - 1.00 = 0.084 = 8.4%

Example

Taylor was researching interest rates at local banks in preparation for opening a savings account. He discovered that The Main Street Bank was offering 0.25% interest on new savings accounts.

What is the Growth Rate for this bank's savings accounts?0.25%

What is the Growth Factor for this bank's savings accounts?0.0025 + 1

1.0025

30 What is the growth factor for a growth rate of 15%?

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31 What is the growth rate for a growth factor of 1.5?

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32 What is the growth factor for a growth rate of 95%?

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33 What is the growth rate for a growth factor of 2.35?

____%

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34 What is the growth factor in the equation:

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Ashley invested $1,200 at a 4% yearly interest rate. How much money will she have at the end of four years?

Years 0 1 2 3 4

Money $1,200.00

$1,248.00

This problem can be solved using either Growth Rate or Growth Factor.

Application Example

Method 1: Growth Rate

Balance(Growth Rate) = Interest$1,200.00(0.04) = $48.00

Balance + Interest = Total at End of Year 1$1,200.00 + $48.00 = $1,248.00

Method 2: Growth Factor

Balance(Growth Factor) = Total at End of Year 1$1,200.00(1.04) = $1,248

Method Comparison

Years 0 1 2 3 4

Money

$1,200.00

$1,248.00

$1,297.92

$1,349.84

$1,403.83

Solution

Complete the rest of the table using the Growth Factor Method.

Application Example

The number of cell phone users in Centerville has been increasing over the last several years. The table below documents the number of cell

phone users by year.

Analyzing the table and rate of growth can enable Centerville to make predictions about future usage levels.

Approximately how many users can Centerville expect to have on their network in 2014 if they experience similar levels of growth?

Year 2008 2009 2010 2011 2012 2013

Number of Users

498 872 1,527 2,672 4,677 8,186

=Growth FactorValue

Previous Value

Recall:

Solution

Year 2008 2009 2010 2011 2012 2013

Number of Users

498 872 1,527 2,672 4,677 8,186

Solution Continued

Year 2008 2009 2010 2011 2012 2013

Number of Users

498 872 1,527 2,672 4,677 8,186

Growth Factor 1.75

Year 2014 = 8,186(1.75)

Year 2014 14,326 people

Years 0 1 2 3 4

Money

$1,700.00

$1,802.00

$1,910.12

$2,024.72

$2,146.20

Example

Nicholas invest $1,700.00 at a 6% yearly interest rate. How much money will he have at the end of four years?

How many methods can you find to complete the table below?

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35 Your parents open a savings account for your college education with an initial deposit of $2,000 at a growth rate of 7%. What would the growth factor on that account be?

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36 Suppose there is an initial deer population in the forest of 750,000 deer and the growth factor for the populatoin is 1.3 per year. How large would the deer population be in two year?

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37 What is the equation for the table below?

A

B

C

D

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x 0 1 2 3

y 30 57 108 206

38 Evaluate the missing number in the table below:

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x 0 1 2 3

y 20 32 81.92

39 According to CNN.com, Facebook increased their total number of employees by 11% during the first quarter of 2011. If the company started the year with 3,082 employees and planned to hire at the same rate for the second, third and fourth quarter, how many employees would they expect to have at the end of 2011?

http://money.cnn.com/2012/04/23/technology/facebook-q1/index.htm

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Exponential Decay

Return to Table of Contents

Related Materials

ELIM&MANATION.pdf

Decay Rate"Factor PDF.pdf

Exponential Decay PDF.pdf

These are handouts that can be used along with this section. They are located under the heading labs on the Exponential page

of PMI Algebra.

Click for link to materials.

If we are given a cupful of M&M's and eat half of them every hour, what happens to the number of M&M's each hour?

The number of M&M's decreases by half. Click

Decay

Does this pattern remain consistent?

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Decay Continued

Is the M&M problem linear relationship? Explain your reasoning.

No, you are constantly dividing by 2 or multiplying by .

If the relationship was linear, there would be constant subtraction/addition NOT multiplication/division.

This relationship represents .

Exponential Decay: An exponential relationship in which the quantity decreases at each stage.

Decay Rate is the percent decrease.

For Example: The car's value decreased by 2% each month, therefore the decay rate is 2%.

The Decay Factor is the difference of 100% and the percent decrease.

For Example: The car's value decreased by 2% each month, therefore the decay factor is 1.00 -

0.02 = 0.98

***Decay Factor utilizes percents in decimal form***

Decay Rate vs Decay Factor

Decay Rate Decay Factor

5% = 1.00 - 0.05 = 0.95

Decay Factor Decay Rate

0.95 = 1.00 - 0.95 = 0.05 = 5%

Example

If we started with 1,000 M&M's, how many would we have after 5 hours?

Hour M&M's

0 1,000

1 500

2 250

3 125

4

5 31

Example

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Equation

x = number of hoursy = number of M&M's

If your experiment started with 1,000 M&M's, what is the equation that relates hours to the number of M&M's?

Equations, Tables and Graphs

Hour M&M's

0 1,000

1 500

2 250

3 125

4

5 31

How is the exponential decay represented in each form?

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Example

Half-life is the amount of time it takes for half of a sample to decay.

The half-life of caffeine in a healthy adult can take as long as 6 hours. Victoria is a healthy adult and consumed a Red Bull Energy Drink at 3 pm on Friday, which contains 80 mg of caffeine.

A. How many hours would it take for her body to reduce the caffeine to 5 mg if Victoria does not consume any additional caffeine?

B. What time will it be when her caffeine level drops to 5 mg?

SolutionA. How many hours would it take for her body to reduce the caffeine to 5 mg if Victoria does not consume any additional caffeine?

B. What time will it be when her caffeine level drops to 5 mg?

Hour Caffeine Level

0 80 mg

1 40 mg

2 20 mg

3 10 mg

4 5 mg

Hour 0 1 2 3 4

Time 3:00 pm

4:00 pm

5:00 pm

6:00 pm

7:00 pm

40 If you start with 200 M&M's and you eat half of them every 10 minutes, how many M&M's will you have after 30 minutes?

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41 Given the same scenario as the previous question, how much time has elapsed if there are only 3 M&M's left?

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42 A scientist has just discovered a new substance and is studying it to determine its half-life. The table provides the data collected during the experiment. What is the missing value?

Interval 0 1 2 3 4

Substance Weight

750.0 375.0 93.75 46.88

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A person takes a dose of Tylenol for a headache. The medicine begins to break down in their blood system. With each hour, there is less medicine in the blood. The table and graph show the amount of medicine in the person's bloodstream.

800

600

400

200

0

0

1 2 3 4 5 6 7 8

Time (Hour)

Dose (mg)

0 800

1 200

2 50

3 12.5

4 3.125

5 0.78125

Medicine Activity

Time (Hour)

Dose (mg)

0 800

1 200

2 50

3 12.5

4 3.125

5 0.78125

How much medicine is initially in the person's body?

800 mg

How much medicine is left in the person's system after 1 hour?

200 mg

How much medicine is left in the person's system after 4 hours?

3.125 mg

Click

Click

Click

Decay Rate

What is the Decay Rate of the medicine?

800 - 200 = 600 mg of medicine removed

200 - 50 = 150 mg of medicine removed

Therefore the Decay Rate is 75%

Decay Rate Continued

Time (Hour)

Dose (mg)

0 800

1 200

2 50

3 12.5

4 3.125

5 0.78125

Time (Hour)

Dose (mg)

0 800

1 200

2 50

3 12.5

4 3.125

5 0.78125

Decay Factor

What is the decay factor for this medicine?

Create an equation that represents the decay for the medicine.

Decay Factor =

x y

0 1001 852 72.253 61.44 52.25 44.36 37.7

Example

The table below shows an exponential relationship:

What is the decay rate?

100 - 85 = 15 85 - 72.25 = 12.75

Decay Rate = 15%

What is the decay factor?

1.00 - 0.15 = 0.85

What is the equation for the table?x y

0 1001 852 72.253 61.44 52.25 44.36 37.7

Example Continued

43 Identify the decay factor.

Step  M&M's

0 128

1 64

2 32

3 16

4 8

5 4

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44 If the decay rate is 20%, what is the decay factor?

an

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45 If the decay factor is 0.73, what is the decay rate?

an

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46 Choose the equation that matches the given table:

A

B

C

D

Interval Weight

0 725.00

1 217.50

2 65.25

3 19.58

4 5.87

5 1.76

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47 Choose the graph that matches the provided equation:

A

B

C

D

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Rules of Exponents

Return to Table of Contents

Rules of Exponents

Exponential Table Questions.pdf

Exponential Table.pdf

Exponential Test Review.pdf

Materials

There are handouts that can be used along with this section. They are located under the heading labs on the Exponential page of PMI Algebra.

Click for link to materials.

The Exponential Table

x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x

1

2

3

4

5

6

7

8

x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x

1

2 4

3

4 16

5

6 64

7

8 Why is 22 x 24 = 26?

Question 1

HINT: Write the expressions in expanded form and see if you can explain why.

A. 23 x 22 = 25

(2 x 2 x 2) x (2 x 2) = (2 x 2 x 2 x 2 x 2)

B. 34 x 33 = 37  

(3 x 3 x 3 x 3) x (3 x 3 x 3) = (3 x 3 x 3 x 3 x 3 x 3 x 3)

C. am x an = am+n

Multiplying with Same Base

x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x

1

2 4 25 100

3

4

5

6

7

8 Why is 22 x 52 = 102?

HINT: Write the expressions in expanded form and see if you can explain why.

Question 2

A. 23 x 33 = 63

(2 x 2 x 2) x (3 x 3 x 3) = (2 x 3)(2 x 3)(2 x 3)

B. 54 x 64 = 304  

(5 x 5 x 5 x 5) x (6 x 6 x 6 x 6) = (5 x 6)(5 x 6)(5 x 6)(5 x 6)

C. am x bm = (a x b)m

Multiplying with Same Exponent

x 1x 2x 3x 4x 5x 6x 7x 8x 9x 10x

1

2 9

3

4

5

6

7

8 6561

Why is (32)4 = 38?

HINT: Write the expressions in expanded form and see if you can explain why.

Question 3

A. (22)3 = 26

(2 x 2)4 = (2 x 2)(2 x 2)(2 x 2)

B. (53)4 = 512

(5 x 5 x 5)4 = (5 x 5 x 5)(5 x 5 x 5)(5 x 5 x 5)(5 x 5 x 5)

C. (am)n = am(n)

Multiplying Exponents with One Base

Why is 56 ÷ 54 = 52?

Question 4x 1x 2x 3x 4x 5x 6x 7x 8x 9x

10x

1

2 25

3

4 625

5

6 15,625

7

8 HINT: Write the expressions in expanded form and see if you can explain why.

A. 35  32

B. 46   44 

C. 

=

=

33

42

am

an= am-n

Dividing with Same Base

Exponent Operations

32 x 34 = 36

52 x 32 = 152

am x an = am+n

am x bm = (ab)m

(43)2 = 46

Examples

3533 = 32

(am)n = amn

aman = am-n

Operations

48 Simplify: 43 x 45

415

42

48

47

A

B

C

D

an

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49

52

521

54

510

A

B

C

D

Simplify: 57 ÷ 53

an

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50 Simplify:

A

B

C

D

an

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51 Simplify:

A

B

C

D

an

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52 The expression (x2z3)(xy2z) is equivalent to

A x2y2z3

B x3y2z4

C x3y3z4

D x4y2z5

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53 If the number of molecules in 1 mole of a substance is 6.02 X 1023, then the number of molecules in 100 moles is

A 6.02 X 1021

B 6.02 X 1022

C 6.02 X 1024

D 6.02 X 1025

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54 The expression is equivalent to

A 2w5

B 2w8

C 20w5

D 20w8

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55 If x = - 4 and y = 3, what is the value of x - 3y2?

A -13

B -23

C -31

D -85 an

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56 When -9x5 is divided by -3x3, x ≠ 0, the quotient is

A –3x2

B 3x2

C –27x15

D 27x8

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Additional Exponent Rules

Exponent of Zero

Exponent of One

Fraction Exponents

Additional Exponent Rules

Negative Exponents

andand

Simplify:

Examples

Simplify:

Examples

Rewrite the expression below in three different forms:

4(4)(4)(4)(4)

1024

57 What is the value of 2-3?

A

B

C -6

D -8

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58 Which expression is equivalent to x-4?

A

B x4

C -4x

D 0

an

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59 Which expression is equivalent to x-1(y2)?

A xy2

B

C

D xy-2

an

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60 What is the value of j in the expression below?

an

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61 Solve for the value of x in the expression below:

an

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62 Choose the expression that is NOT equivalent to:

(Choose all that apply)

A

B

C

D

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63 Choose the expression that is equivalent to:

A

B

C

D

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