name: unit 2 • linear and exponential relationships lesson 6: comparing...

Unit 2 • Linear and exponentiaL reLationshipsLesson 6: Comparing Functions

naMe:

Assessment

CCSS IP Math I Teacher Resource U2-292

© Walch Education

Pre-AssessmentCircle the letter of the best answer.

1. Which of the following statements is true about the functions f(x) and g(x), shown in the table and graph below?

x f(x)–2 10 72 134 19

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10 g(x)

a. The function f(x) has a greater y-intercept than the function g(x).

b. The function g(x) has a greater rate of change than the function f(x).

c. The function g(x) has a greater y-intercept than the function f(x).

d. The rates of change for both f(x) and g(x) are equal.

continued


naMe:

Assessment

CCSS IP Math I Teacher Resource © Walch EducationU2-293

2. Which of the following statements is true about the functions f(x) and g(x)?

( )2

36= −f x x

x g (x)–4 100 74 48 1

a. The rate of change of f(x) is less than the rate of change of g(x).

b. The rate of change of f(x) is greater than the rate of change of g(x).

c. The y-intercept of f(x) is equal to the rate of change of g(x).

d. The y-intercept of f(x) is greater than the rate of change of g(x).

3. Which of the following statements is true about the functions f(x) and g(x) over the interval [0, 12]?

( ) 200 10.05

12

12

= +

f xx

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300 g(x)

a. The rates of change for the functions f(x) and g(x) are equal over the interval [0, 12].

b. The rate of change for the function f(x) is greater than the rate of change for the function g(x) over the interval [0, 12].

c. The rate of change for the function f(x) is less than the rate of change for the function g(x).

d. The rate of change for the functions cannot be determined. continued


naMe:

Assessment


© Walch Education

4. Which of the following statements is true about the functions f(x) and g(x)?

x f(x)–1 18.950 181 17.12 16.25

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30

2

4

6

8

10

12

14

16

18

20

g(x)

a. The y-intercept of the function f(x) is greater than the y-intercept for the function g(x).

b. The y-intercept of the function f(x) is less than the y-intercept for the function g(x).

c. The y-intercepts of the functions f(x) and g(x) are equal.

d. The y-intercepts of the functions cannot be determined.

continued


naMe:

Assessment


5. Which of the following statements is true about the functions f(x) and g(x) shown below?

0 4 8 12 16 20 24 28 32 36 40 44 48

20

40

60

80

100

120

140

160

180

200

220

240

260

280

300

320 g(x)

f(x)

a. The rate of change of the function f(x) is always greater than the rate of change of the function g(x).

b. The rate of change of the function g(x) will eventually be greater than the rate of change of the function f(x).

c. The rate of change of the function f(x) is never greater than the rate of change of the function g(x).

d. The rate of change of an exponential function cannot be determined.


Lesson 6: Comparing FunctionsUnit 2 • Linear and exponentiaL reLationships

© Walch Education

InstructionCommon Core State Standards

F–IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

F–LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. ★

Essential Questions

1. What different interpretations can be made from different representations of functions?

2. Why is comparing functions important?

3. How can you use characteristics of functions to compare functions?

WORDS TO KNOW

exponential function a function that has a variable in the exponent; the general form is f(x) = ab x, where a is the initial value, b is the rate of decay or growth, x is the time, and f(x) is the final output value

factor one of two or more numbers or expressions that when multiplied produce a given product

growth factor the multiple by which a quantity increases or decreases over time

interval a continuous series of values

linear function a function that can be written in the form f(x) = mx + b, in which m is the slope, b is the y-intercept, and the graph is a straight line

rate of change a ratio that describes how much one quantity changes with respect to the change in another quantity; also known as the slope of a line

slope the measure of the rate of change of one variable with respect to another

variable; sloperise

run2 1

2 1

=−−

= =y y

x x

y

x

x-intercept the point at which the line intersects the x-axis; written as (x, 0)

y-intercept the point at which the line intersects the y-axis; written as (0, y)


Instruction


Recommended Resources• Algebra 4 All. “Exponential Functions.”

http://walch.com/rr/CAU3L5ExponentialFunctions

This site provides links to several applets that allow users to explore exponential functions.

• Interactivate. “Graphit.”

http://walch.com/rr/CAU3L5Graphit

This interactive applet allows users to compare functions using tables, graphs, and/or equations.

• Math Open Reference. “Linear Function Explorer.”

http://walch.com/rr/CAU3L5LinearFunctions

Users of this interactive applet can change the values of the variables a and b to observe the changes in a given graph of linear functions.


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© Walch Education

Lesson 2.6.1: Comparing Linear Functions

Warm-Up 2.6.1Cecilia took her first parachute jump lesson last weekend. Her instructor gave her the graph below that shows her change in altitude in meters during a 5-second interval. Use the graph to answer the questions that follow.

0 1 2 3 4 5

300

600

900

1200

1500

1800

2100

2400

2700

Time (seconds)

Alt

itud

e (m

eter

s)

1. Estimate Cecilia’s average rate of change in altitude in meters per second.

2. What are the domain and range for this function?


Instruction


Lesson 2.6.1: Comparing Linear FunctionsCommon Core State Standard


Warm-Up 2.6.1 DebriefCecilia took her first parachute jump lesson last weekend. Her instructor gave her the graph below that shows her change in altitude in meters during a 5-second interval. Use the graph to answer the questions that follow.

0 1 2 3 4 5

300

600

900

1200

1500

1800

2100

2400

2700

Time (seconds)

Alt

itud

e (m

eter

s)


Instruction


© Walch Education

1. Estimate Cecilia’s average rate of change in altitude in meters per second.

Select two points from the graph and use the slope formula to calculate the rate of change.

Let (x1, y

1) = (1, 2400) and (x

2, y

2) = (3, 1800).

2 1

2 1

−−

y y

x x

Slope formula

1800 2400

3 1

−−

Substitute (1, 2400) and (3, 1800) for (x1, y

1) and (x

2, y

2).

600

2

−

Simplify as needed.

–300

The average rate of change is –300 meters per second.

Cecilia’s altitude is decreasing an average of 300 meters each second.

2. What are the domain and range for this function?

The domain is all possible x-values.

In this problem, the domain is all real numbers greater than or equal to 0, or x ≥ 0.

The range is all possible y-values.

In this problem, the range is all real numbers greater than or equal to 0, but less than or equal to 2,700. This can be written symbolically as 0 ≤ y ≤ 2700 or as y ≥ 0 and y ≤ 2700.

Connection to the Lesson

• Students will be determining the rate of change as well as the y-intercept from graphs, tables, equations, and verbal descriptions.

• Students will take this type of problem a step further and compare the rate of change and y-intercept to other linear functions.


Instruction


Prerequisite Skills

This lesson requires the use of the following skills:

• determining the slope of linear functions

• determining the intercepts of linear functions

Introduction

Remember that linear functions are functions that can be written in the form f(x) = mx + b, where

m is the slope and b is the y-intercept. The slope of a linear function is also the rate of change and can

be calculated using the formula slope 2 1

2 1

=−−

y y

x x. The y-intercept is the point at which the function

crosses the y-axis and has the point (0, y). The x-intercept, if it exists, is the point where the graph

crosses the x-axis and has the point (x, 0). The slope and both intercepts can be determined from

tables, equations, and graphs. These features are used to compare linear functions to one another.

Key Concepts

• Linear functions can be represented in words or as equations, graphs, or tables.

• To compare linear functions, determine the rate of change and intercepts of each function.

• Review the following processes for identifying the rate of change and y-intercept of a linear function.

Identifying the Rate of Change and the y-intercept from Context

1. Read the problem statement carefully.

2. Look for the information given and make a list of the known quantities.

3. Determine which information tells you the rate of change, or the slope, m. Look for words such as each, every, per, or rate.

4. Determine which information tells you the y-intercept, or b. This could be an initial value or a starting value, a flat fee, and so forth.

Identifying the Rate of Change and the y-intercept from an Equation

1. Simplify linear functions to follow the form f(x) = mx + b.

2. Identify the rate of change, or the slope, m, as the coefficient of x.

3. Identify the y-intercept, or b, as the constant in the function.


Instruction


© Walch Education

Identifying the Rate of Change and the y-intercept from a Table

1. Choose two points from the table.

2. Assign one point to be (x1, y

1) and the other point to be (x

2, y

2).

3. Substitute the values into the slope formula, 2 1

2 1

−−

y y

x x.

4. Identify the y-intercept as the coordinate in the form (0, y). If this coordinate is not given, substitute the rate of change and one coordinate into the form f(x) = mx + b and solve for b.

Identifying the Rate of Change and the y-intercept from a Graph

1. Choose two points from the graph.



2, y

2).


2 1

−−

y y

x x.

4. Identify the y-intercept as the coordinate of the line that intersects with the y-axis.

• When presented with functions represented in different ways, it is helpful to rewrite the information using function notation.

• You can compare the functions once you have identified the rate of change and the y-intercept of each function.

• Linear functions are increasing if the rate of change is a positive value.

• Linear functions are decreasing if the rate of change is a negative value.

• The greater the rate of change, the steeper the line will appear on the graph.

• A rate of change of 0 appears as a horizontal line on a graph.

Common Errors/Misconceptions

• incorrectly determining the rate of change

• assuming that a positive slope will be steeper than a negative slope

• interchanging the x- and y-intercepts


Instruction


Guided Practice 2.6.1Example 1

The functions f(x) and g(x) are described below. Compare the properties of each.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

f(x)

x g(x)–2 –10–1 –80 –61 –4


Instruction


© Walch Education

1. Identify the rate of change for the first function, f(x).

Let (0, 8) be (x1, y

1) and (4, 0) be (x

2, y

2).

Substitute the values into the slope formula.

2 1

2 1

−−

y y

x x

Slope formula

0 8

4 0

−−


1) and (x

2, y

2).

8

4

−

Simplify as needed.

–2

The rate of change for this function is –2.

2. Identify the rate of change for the second function, g(x).

Choose two points from the table. Let (–2, –10) be (x1, y

1) and let

(–1, –8) be (x2, y

2).


2 1

2 1

−−

y y

x x

Slope formula

( 8) ( 10)

( 1) ( 2)

− − −− − −

Substitute (–2, –10) and (–1, –8) for (x1, y

1) and (x

2, y

2).

2

1 Simplify as needed.

2

The rate of change for this function is 2.

3. Identify the y-intercept of the first function, f(x).

The graph crosses the y-axis at (0, 8), so the y-intercept is 8.


Instruction


4. Identify the y-intercept of the second function, g(x).

From the table, we can determine that the function would intersect the y-axis where the x-value is 0. This happens at the point (0, –6). The y-intercept is –6.

5. Compare the properties of each function.

The rate of change for the first function is –2 and the rate of change for the second function is 2. The first function is decreasing and the second is increasing, but the rates of change are equal in value.

The y-intercept of the first function is 8, but the y-intercept of the second function is –6. The graph of the second function crosses the y-axis at a lower point.


Instruction


© Walch Education

Example 2

Your employer has offered two pay scales for you to choose from. The first option is to receive a base salary of $250 a week plus 15% of the price of any merchandise you sell. The second option is represented in the graph below. Compare the properties of the functions.

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

Total price of merchandise sold ($)

Wee

kly

sala

ry ($

)

1. Identify the rate of change for the first function.

Determine which information tells you the rate of change, or the slope, m.

You are told that your employer will pay you 15% of the price of the merchandise you sell.

This information is the rate of change for this function and can be written as 0.15.

2. Identify the y-intercept for the first function.

Your employer has offered a base salary of $250 per week.

250 is the y-intercept of the function.


Instruction


3. Identify the rate of change for the second function.

Let (0, 200) be (x1, y

1) and (500, 300) be (x

2, y

2).


2 1

2 1

−−

y y

x x

Slope formula

300 200

500 0

−−


1) and (x

2, y

2).

100


1

50.2=

The rate of change for this function is 0.2.

4. Identify the y-intercept as the coordinate of the line that intersects the y-axis.

The graph intersects the y-axis at (0, 200). The y-intercept is 200.


The rate of change for the second function is greater than the first function. You will get paid more for the amount of merchandise you sell.

The y-intercept of the first function is greater than the second. You will get a higher base pay with the first function.

In the first function, you would receive a higher base salary, but get paid less for the amount of merchandise you sell.

In the second function, you would receive a lower base salary, but get paid more for the merchandise you sell.


Instruction


© Walch Education

Example 3

Two airplanes are in flight. The function f(x) = 400x + 1200 represents the altitude, f(x), of one airplane after x minutes. The graph below represents the altitude of the second airplane.

0 1 2 3 4 5 6 7 8 9 10

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

6000

Time (minutes)

Alt

itud

e (m

)

1. Identify the rate of change for the first function.

The function is written in f(x) = mx + b form; therefore, the rate of change for the function is 400.


Instruction


2. Identify the y-intercept for the first function.

The y-intercept of the first function is 1,200, as stated in the equation.

3. Identify the rate of change for the second function.

Choose two points from the graph.

Let (0, 5750) be (x1, y

1) and (5, 4500) be (x

2, y

2).


2 1

2 1

−−

y y

x x

Slope formula

4500 5750

5 0

−−


1) and (x

2, y

2).

1250

5

−

Simplify as needed.

–250

The rate of change for this function is –250.

4. Identify the y-intercept of the second function as the coordinate of the line that intersects with the y-axis.

The graph intersects the y-axis at (0, 5750).


The rate of change for the first function is greater than the second function. The rate of change for the first function is also positive, whereas the rate of change for the second function is negative. The first airplane is ascending at a faster rate than the second airplane is descending.

The y-intercept of the second function is greater than the first. The second airplane is higher in the air than the first airplane at that moment.


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© Walch Education

Problem-Based Task 2.6.1: Supply and DemandIdeal Electronics is determining the price of the newest tablet to hit the market. In an effort to make the most money and sell the most tablets, Ideal Electronics wants to price the tablet appropriately to the product’s supply and demand. Supply is the number of tablets that are available and demand is the amount that buyers are willing to pay. The relationship between supply and demand often influences the price of products.

Supply is modeled by the linear function f(x) = 0.3x + 100, where f(x) represents the price per tablet in dollars and x represents the number of tablets.

Demand is modeled in the table below, where g(x) represents the price per tablet in dollars and x represents the number of tablets.

x g (x)100 490300 370500 250600 190

Compare the properties of both of the functions described. At what point does the supply of tablets exceed the demand? Explain your reasoning.


naMe:


Problem-Based Task 2.6.1: Supply and Demand

Coachinga. What is the rate of change of the supply function?

b. What is the y-intercept of the supply function?

c. What is the rate of change of the demand function?

d. What is the y-intercept of the demand function?

e. How does the rate of change of the supply function compare to the rate of change of the demand function?

f. How does the y-intercept of the supply function compare to the rate of change of the demand function?

g. When graphing both functions, what does the x-axis represent?

h. What does the y-axis represent?

i. At what point are the supply and demand functions equal?

j. At what point does the supply function exceed the demand function?


Instruction


© Walch Education

Problem-Based Task 2.6.1: Supply and Demand

Coaching Sample Responsesa. What is the rate of change of the supply function?

The supply function is f(x) = 0.3x + 100, and is written in f(x) = mx + b form.

The rate of change of the function is the coefficient of x, or 0.3.

b. What is the y-intercept of the supply function?

The y-intercept of the supply function is the constant, or 100.

c. What is the rate of change of the demand function?

To determine this, choose two points from the table.

Let (100, 490) be (x1, y

1) and (600, 190) be (x

2, y

2).


2 1

2 1

−−

y y

x x

Slope formula

190 490

600 100

−−


1) and (x

2, y

2).

300

500

−

Simplify as needed.

3

50.6

−= −

The rate of change for this function is –0.6.

d. What is the y-intercept of the demand function?

The y-intercept is not listed in the table.

Calculate the y-intercept using one of the points from the table and the rate of change.

Let (100, 490) be (x1, y

1) and (0, y

2) be (x

2, y

2).


Instruction


2 1

2 1

=−−

my y

x x

Slope formula

0.6490

0 1002− =

−−

y

Substitute (100, 490), (0, y2), and –0.6 for (x

1, y

1), (x

2, y

2),

and m.

0.6490

1002− =

−−

y

Solve for m.

60 = y2 – 490

550 = y2

The y-intercept of the demand function is (0, 550).

e. How does the rate of change of the supply function compare to the rate of change of the demand function?

The rate of change of the supply function is a positive value of 0.3, whereas the demand function is a negative value of –0.6.

f. How does the y-intercept of the supply function compare to the y-intercept of the demand function?

The y-intercept of the supply function, (0, 100), is lower than the y-intercept of the demand function, (0, 550).

g. When graphing both functions, what does the x-axis represent?

The x-axis represents the number of tablets.

h. What does the y-axis represent?

The y-axis represents the price per tablet.


Instruction


© Walch Education

i. At what point are the supply and demand functions equal?

Graph each function using the information from parts a–h.

0 50 100 150 200 250 300 350 400 450 500 550 600 650 700 750 800

50

100

150

200

250

300

350

400

450

500

550

Number of tablets

Pric

e pe

r tab

let (

in d

olla

rs)

From the graph, we can see that the lines intersect at the point (500, 250).

At this point, the number of tablets demanded by the buyers is equal to the number of tablets the seller has.

When priced at $250, it can be expected that 500 tablets will be sold.

j. At what point does the supply function exceed the demand function?

After the point (500, 250), the supply function exceeds the demand function.

After that point, the number of tablets that are available is more than the number of tablets demanded by the buyers.

Recommended Closure Activity

Select one or more of the essential questions for a class discussion or as a journal entry prompt.


naMe:


Practice 2.6.1: Comparing Linear FunctionsCompare the properties of the linear functions.

1. Which function has a greater rate of change? Which function has the greater y-intercept? Explain how you know.

Function A

x f(x)–4 12–1 02 –123 –16

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)

2. Which function has a greater rate of change? Which function has the greater y-intercept?

Function A

x f(x)–8 10 24 2.58 3

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)

continued


naMe:


© Walch Education


Function A

( )1

43= +f x x

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)


Function A

f(x) = –5x

Function B

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

g(x)

continued


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Function A

The table below describes the profit in dollars that a restaurant makes for the beverages it sells.

Number of beverages sold (x)

Profit (f(x))

0 025 29.2550 58.5075 87.75

Function B

For each hamburger sold, the same restaurant makes a profit of $0.40.


Function A

A local newspaper began with a circulation of 1,300 readers in its first year. Since then, its circulation has increased by 150 readers per year.

Function B

The function g(x) = 225x + 950 represents the circulation of another newspaper where g(x) represents total subscriptions and x represents the number of years since its first year.


Function A

A rental store charges $40 to rent a steam cleaner, plus an additional $4 per hour.

Function B

The table below shows the total cost in dollars to rent a steam cleaner at a different rental store. g(x) represents the total cost after x hours.

Hours (x) Total cost (g(x))3 464 535 606 67

continued


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© Walch Education


Function A

The table shows the remaining balance in dollars, f(x), of the cost of car repairs after x months.

Months (x)Remaining

balance (f(x))0 15601 14302 13003 1170

Function B

The graph shows the remaining balance in dollars, g(x), of the cost of car repairs after x months.

0 1 2 3 4 5 6 7 8 9 10 11 12

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

Months

Rem

aini

ng b

alan

ce

9. Compare the properties of each function. What do the rate of change and y-intercept mean in terms of the scenarios?

Function A

The function f(x) = 7.5 – 0.25x represents the pounds of puppy food remaining, f(x), when the puppy is fed the same amount each day for x days.

Function B

The table represents the amount in pounds of puppy food remaining, g(x), when the puppy is fed the same amount each day for x days.

Days (x) Remaining food (g(x))4 95 8.756 8.57 8.25

continued


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10. Compare the properties of each function. What do the rate of change and y-intercept mean in terms of the scenarios?

Function A

Reggie bicycled 15 miles last week and plans to bicycle 20 miles each additional week.

Function B

The graph represents the total number of miles Zac plans to have bicycled by the end of each week.

0 1 2 3 4 5 6 7 8 9 10 11 12

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

Weeks

Mile

s


naMe:


© Walch Education

Lesson 2.6.2: Comparing Exponential Functions

Warm-Up 2.6.2A hard rubber ball will rebound to 75% of its height each time it bounces.

1. If the ball is dropped from a height of 200 centimeters, what will the height of each bounce be after 11 bounces? Create a table and a graph of the ball’s bounce rebound height over several bounces.

2. On which bounce will the rebound be less than 50 centimeters?


Instruction


Lesson 2.6.2: Comparing Exponential FunctionsCommon Core State Standard


Warm-Up 2.6.2 DebriefA hard rubber ball will rebound to 75% of its height each time it bounces.

1. If the ball is dropped from a height of 200 centimeters, what will the height of each bounce be after 11 bounces? Create a table and a graph of the ball’s bounce rebound height over several bounces.

Begin by creating a table with the number of bounces and the height of the ball after each ball.

At 0 bounces, the ball is at the initial height of 200 centimeters.

After each bounce, the height is 75% of the previous height.

To calculate the height of the ball after 1 bounce, find 75% of 200, or 200 • 0.75.

The height after the first bounce is 150 centimeters.

To calculate the height of the ball after 2 bounces, find 75% of the previous height, or 150 • 0.75.

The height after the second bounce is 112.5 centimeters.

Continue this way until the table is completed for 11 bounces.

Number of bounces Height of bounce (cm)0 2001 1502 112.53 84.3754 63.2815 47.466 35.67 26.78 209 15

10 1111 8


Instruction


© Walch Education

Use the table to plot the coordinates on a coordinate plane.

Label the x-axis “Number of bounces” and the y-axis “Height of bounce (cm).”

Plot each coordinate.

Notice that this scenario is best represented by the exponential function f(x) = 200(1 – 0.25) x.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10

20

30

40

50

60

70

80

90

100

110

120

130

140

150

160

170

180

190

200

Number of bounces

Hei

ght o

f bou

nce

(cm

)

2. On which bounce will the rebound be less than 50 centimeters?

We can see from both the table and the graph that the height of the fifth bounce will be less than 50 centimeters.


• As in the warm-up, students will create exponential functions from context.

• Students will use their knowledge of writing exponential functions to analyze properties of exponential functions in order to make comparisons.


Instruction


Prerequisite Skills


• determining the rate of change of exponential functions

• determining the intercepts of exponential functions

Introduction

Exponential functions are functions that can be written in the form f(x) = abx, where a is the initial

value, b is the rate of decay or growth, x is the time, and f(x) is the final output value. The growth

factor is the multiple by which a quantity increases or decreases over time. The rate of change of an

exponential function can be calculated using the formula slope = 2 1

2 1

−−

y y

x x, over a specified interval. An

interval is a continuous series of values. The y-intercept is the point at which the function crosses the

y-axis and has the point (0, y). Both the rate of change and y-intercept can be determined from tables,

equations, and graphs. Exponential functions can also be compared to one another using these features.

Key Concepts

• Exponential functions can be represented in words or as equations, graphs, or tables.

• To compare exponential functions, determine the rate of change and the intercepts of each function.

• Review the following processes for identifying the rate of change and the y-intercept of an exponential function.

Identifying the Rate of Change and the y-intercept from Context

1. Determine the interval to be observed.

2. Create a table of values by choosing appropriate x-values, substituting them, and solving for f(x).

3. Choose two points from the table.



2, y

2).


2 1

−−

y y

x x .

6. The result is the rate of change for the interval between the two points chosen.

7. Determine which information tells you the y-intercept, or b. This could be an initial value or a starting value, a flat fee, and so forth.


Instruction


© Walch Education

Identifying the Rate of Change and the y-intercept from Exponential Equations


2. Determine (x1, y

1) by identifying the starting x-value of the interval and substituting it

into the function.

3. Solve for f(x).

4. Determine (x2, y

2) by identifying the ending x-value of the interval and substituting it

into the function.

5. Solve for f(x).

6. Substitute (x1, y

1) and (x

2, y

2) into the slope formula,

2 1

2 1

−−

y y

x x, to calculate the rate

of change.

7. Determine the y-intercept by substituting 0 for x and solving for f(x).

Identifying the Rate of Change and the y-intercept from a Table




2, y

2).


2 1

−−

y y

x x .

4. The result is the rate of change for the interval between the two points chosen.

5. Identify the y-intercept as the coordinate in the form (0, y).

Identifying the Rate of Change and the y-intercept from a Graph


2. Identify (x1, y

1) as the starting point of the interval.

3. Identify (x2, y

2) as the ending point of the interval.

4. Substitute (x1, y

1) and (x

2, y

2) into the slope formula,

2 1

2 1

−−

y y

x x, to calculate the rate

of change.

5. Identify the y-intercept as the coordinate in the form (0, y).

• You can compare the functions once you have identified the rate of change and the y-intercept of each function.

• Exponential functions are increasing if the rate of change is a positive value.


Instruction


• Exponential functions are decreasing if the rate of change is a negative value.

• The greater the rate of change, the steeper the line connecting the points of the interval will appear on the graph.




• comparing functions over different intervals

• incorrectly applying the order of operations

• using the exponential growth model instead of exponential decay and vice versa


Instruction


© Walch Education


Compare the properties of each of the following two functions over the interval [0, 16].

Function A

0 2 4 6 8 10 12 14 16 18

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

1500

1600

1700

1800

1900

f(x)

Function B

x g(x)0 8504 976.558 1121.94

12 1288.9816 1480.88

1. Compare the y-intercepts of each function.

Identify the y-intercept of the graphed function, f(x).

The graphed function appears to cross the y-axis at the point (0, 850).

According to the table, g(x) has a y-intercept of (0, 850).

Both functions have a y-intercept of (0, 850).


Instruction


2. Compare the rate of change for each function over the interval [0, 16].

Calculate the rate of change over the interval [0, 16] for f(x).

Let (x1, y

1) = (0, 850).

Determine (x2, y

2) from the graph.

The value of y when x is 16 is approximately 1,600.

Let (x2, y

2) = (16, 1600).

Calculate the rate of change using the slope formula.

2 1

2 1

−−

y y

x x

Slope formula

1600 850

16 0

−−

Substitute (0, 850) and (16, 1600) for (x

1, y

1) and (x

2, y

2).

750


46.875

The rate of change for f(x) is approximately 47.

Calculate the rate of change over the interval [0, 16] for g(x).

Let (x1, y

1) = (0, 850).

Determine (x2, y

2) from the table.

The value of y when x is 16 is 1,480.88.

Let (x2, y

2) = (16, 1480.88).

(continued)


Instruction


© Walch Education


2 1

2 1

−−

y y

x x

Slope formula

1480.88 850

16 0

−−

Substitute (0, 850) and (16, 1480.88) for (x

1, y

1) and (x

2, y

2).

630.88


39.43

The rate of change for g(x) is 39.43.

The rate of change for the graphed function, f(x), is greater over the interval [0, 16] than the rate of change for the function in the table, g(x).

3. Summarize your findings.

The y-intercepts of both functions are the same; however, the graphed function, f(x), has a greater rate of change over the interval [0, 16].


Instruction


Example 2

A Petri dish contains 8 bacteria that double every 15 minutes. Compare the properties of the function that represents this situation to another population of bacteria, graphed below, that starts with 8 organisms over the interval [150, 210].

0 30 60 90 120 150 180 210

200

400

600

800

1000

1200

1400

1600

1800

2000

Minutes

Num

ber o

f bac

teri

a


According to the scenario, the starting number of bacteria for both functions is 8; therefore, the y-intercept is (0, 8).


Instruction


© Walch Education


Calculate the rate of change over the interval [150, 210] for the graphed function.

Determine (x1, y

1) from the graph.

The value of y when x is 150 is approximately 275.

Let (x1, y

1) = (150, 275).

Determine (x2, y

2) from the graph.

The value of y when x is 210 is approximately 1,000.

Let (x2, y

2) = (210, 1000).


2 1

2 1

−−

y y

x x

Slope formula

1000 275

210 150

−−


1, y

1) and (x

2, y

2).

725


≈ 12

The rate of change for the graphed function is approximately 12 bacteria per minute.

To determine the rate of change for the function in the scenario, first write a function rule to represent the situation.

( ) 8(2)15=f xx

Determine the value for y when x is 150 using the function.

( ) 8(2)15=f xx

Original function

( ) 8(2)150

15=f x Substitute 150 for x.

f(150) = 8(2)10 Simplify as needed.

(continued)


Instruction


f(150) = 8(1024)

f(150) = 8192

(x1, y

1) = (150, 8192)

Determine the value for y when x is 210 using the function.

( ) 8(2)15=f xx

Original function

( ) 8(2)210

15=f x Substitute 210 for x.


f(210) = 8(16,384)

f(210) = 131,072

(x2, y

2) = (210, 131,072)


2 1

2 1

−−

y y

x x

Slope formula

131,072 8192

210 150

−−

Substitute (150, 8192) and (210, 131,072) for (x

1, y

1) and (x

2, y

2).

122,880


2048

The rate of change for the function in the table is 2,048 bacteria per minute.

The rate of change for the graphed function is less steep over the interval [150, 210] than the rate of change for the scenario function.


The y-intercepts of both functions are the same; however, the graphed function is less steep over the interval [150, 210]. The bacteria in the graphed function are doubling at a slower rate than the first function described.


Instruction


© Walch Education

Example 3

A pendulum swings to 90% of its previous height. Pendulum A starts at a height of 50 centimeters. Its height at each swing is modeled by the function f(x) = 50(0.90)x. The height after every fifth swing of Pendulum B is recorded in the following table. Compare the properties of each function over the interval [5, 15].

x f(x)0 1005 59.05

10 34.8715 20.5920 12.16


Identify the y-intercept of Pendulum A.

The problem states that the pendulum starts at a height of 50 centimeters.

The y-intercept of the function is (0, 50).

Identify the y-intercept of Pendulum B.

The value of f(x) is 100 when x is 0.

The y-intercept of the function is (0, 100).


Instruction



Calculate the rate of change over the interval [5, 15] for Pendulum A.

Determine (x1, y

1) from the function.

f(x) = 50(0.90)x Original function

f(5) = 50(0.90)5 Substitute 5 for x.

f(5) = 29.52 Simplify as needed.

Let (x1, y

1) = (5, 29.52).

Determine (x2, y

2) from the function.

f(x) = 50(0.90)x Original function

f(15) = 50(0.90)15 Substitute 15 for x.

f(15) ≈ 10.29 Simplify as needed.

The value of y when x is 15 is approximately 10.29.

Let (x2, y

2) = (15, 10.29).


2 1

2 1

−−

y y

x x

Slope formula

10.29 29.52

15 5

−−

Substitute (5, 29.52) and (15, 10.29) for (x

1, y

1) and (x

2, y

2).

19.23

10–1.923

−= Simplify as needed.

The rate of change for Pendulum A’s function is approximately –1.923 centimeters per swing.

(continued)


Instruction


© Walch Education

Calculate the rate of change over the interval [5, 15] for Pendulum B.

Let (x1, y

1) = (5, 59.05).

Let (x2, y

2) = (15, 20.59).


2 1

2 1

−−

y y

x x

Slope formula

20.59 59.05

15 5

−−

Substitute (5, 59.05) and (15, 20.59) for (x

1, y

1) and (x

2, y

2).

38.46

103.846

−= −

Simplify as needed.

The rate of change for Pendulum B’s function is approximately –3.846 centimeters per swing.

The rate of change for Pendulum B is greater over the interval [5, 15] than the rate of change for Pendulum A.


The y-intercept of Pendulum A is less than the y-intercept of Pendulum B. This means that Pendulum B begins higher than Pendulum A. The rate of change for Pendulum A is less than the rate of change for Pendulum B. This means that Pendulum B is losing height faster than Pendulum A.


naMe:


Problem-Based Task 2.6.2: Analyzing Kidney Function Renal scans are common medical procedures used to measure how well a person’s kidneys are working. The patient is injected with a small amount of radioactive material. Then, a series of computerized scans are taken. The type of radioactive material used for the test depends on the specific kidney function being observed. Two types of radioactive materials used are technetium-99m and indium-113m.

The half-life, or decay rate, of indium-113m is 1.7 hours. The half-life of technetium-99m is

modeled by the function ( ) 5001

2

6

=

f x

x

, where f(x) represents the amount of material remaining

and x represents hours. There are 500 milligrams of each material. How does the half-life of

indium-113m compare with the function representing the half-life of technetium-99m? Which

material is decaying at a faster rate? Use a graph to help explain your reasoning.


naMe:


© Walch Education

Problem-Based Task 2.6.2: Analyzing Kidney Function

Coachinga. What is the starting amount of indium-113m?

b. At what rate does indium-113m decay?

c. What function represents the half-life of indium-113m?

d. What function represents the half-life of technetium-99m?

e. What is the initial amount of technetium-99m?

f. At what rate does technetium-99m decay?

g. Graph both functions on one coordinate plane.

h. Which function is decaying at a faster rate? Use the functions and your graph to explain your reasoning.


Instruction


Problem-Based Task 2.6.2: Analyzing Kidney Function

Coaching Sample Responsesa. What is the starting amount of indium-113m?

The starting amount of indium-113m is 500 milligrams.

b. At what rate does indium-113m decay?

Indium-113m has a half-life of 1.7 hours.

c. What function represents the half-life of indium-113m?

An exponential function in the form ( )=g x abx

t represents this situation, where a represents

the initial amount, b represents the rate of decay, x represents the number of hours, and t

represents the time period.

The function ( ) 5001

2

1.7

=

g x

x

represents the half-life of indium-113m.

d. What function represents the half-life of technetium-99m?

As stated in the problem, the function that represents the half-life of technetium-99m is

( ) 5001

2

6

=

f x

x

.

e. What is the initial amount of technetium-99m?

The initial amount of technetium-99m is 500 milligrams, as found in the equation.

f. At what rate does technetium-99m decay?

Technetium-99m has a half-life of 6 hours.

This value is found in the denominator of the exponent in the function given.


Instruction


© Walch Education

g. Graph both functions on one coordinate plane.

0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 42 44 46

25

50

75

100

125

150

175

200

225

250

275

300

325

350

375

400

425

450

475

500

Time in hours

Am

ount

of m

ater

ial i

nmg

Technetium-99m

Indium-113m

eria

l in

mg

h. Which function is decaying at a faster rate? Use the functions and your graph to explain your reasoning.

It can be observed from the graph that indium-113m decays at a faster rate than technetium-99m. The initial amount of each material is the same, 500 milligrams. As time passes, the amount of indium-113m decreases more quickly than technetium-99m. For example, after 2 hours, there are approximately 220 milligrams of indium-113m compared with 400 milligrams of technetium-99m. After 6 hours, there are approximately 40 milligrams of indium-113m compared with 100 milligrams of technetium-99m. It would take longer for technetium-99m to decay than for indium-113m to decay.




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Practice 2.6.2: Comparing Exponential FunctionsCompare the properties of the exponential functions.

1. Which function has a greater rate of change over the interval [2, 8]? Which function has the greater y-intercept? Explain how you know.

Function A

x f(x)0 14002 1546.924 1709.256 1888.628 2086.82

Function B

0 1 2 3 4 5 6 7 8 9 10

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

g(x)

2. Which function has a greater rate of change over the interval [0, 5]? Which function has the greater y-intercept?

Function A

( )1

2=

f xx

Function B

g(x) = 2x

continued


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© Walch Education

3. Compare the properties of each function over the interval [2, 8].

Function A

( ) 400 10.06

12

12

= +

f xx

Function B

x g(x)0 350.002 398.454 453.616 516.408 587.88

4. Compare the properties of each function over the interval [0, 5].

Function A

f(x) = 3(2) xFunction B

0 1 2 3 4 5 6 7 8 9 10

200

400

600

800

1000

1200

1400

1600

1800

2000

2200

2400

2600

2800

g(x)

continued


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5. Compare the properties of each exponential function over the interval [0, 10].

Function A

A fully inflated beach ball is losing 7.5% of its air every day. The beach ball originally contained 800 cubic inches of air.

Function B

0 5 10 15 20 25 30 35 40 45

50

100

150

200

250

300

350

400

450

500

550

600

650

700

750

800

Days

Am

ount

of a

ir (i

n3)

t of a

ir (i

n3 )


Function A

Jasmine received a job offer with a starting salary of $32,000 and a 1.5% increase every year.

Function B

A second job offer for Jasmine can be described by the function f(x) = 30,000(1 + 0.02) x.

continued


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© Walch Education


Function A

The enrollment of Eastern High School, f(x), after x years is modeled by the function f(x) = 1700(1 + 0.025) x.

Function B

The following table shows the enrollment of a rival high school, g(x), after x years.

x g(x)0 19001 18722 18433 18164 1789


Function A

The following table shows the value in dollars of a rare stamp, f(x), x years from the date purchased.

x f(x)0 521 54.082 56.243 58.494 60.83

Function B

The graph below models the value in dollars of a second rare stamp, g(x), after x years.

0 1 2 3 4 5 6 7 8 9 10

3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

51

54

57

60

63

66

Years

Valu

e of

sta

mp

($)

continued


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Function A

The value of a car in dollars, f(x), depreciates after each year, x. The following table shows the value of a car for each of the first 4 years after it was purchased.

x f(x)0 22,4501 19,3072 16,604.023 14,279.464 12,280.33

Function B

The value of a second car is modeled by the equation g(x) = 19,375(1 – 0.16) x, where g(x) represents the value of the car x years after the date it was purchased.


Function A

An investment of $1,000 earns interest at a rate of 3.75%, compounded monthly.

Function B

The value of a second investment is modeled in the following graph.

0 1 2 3 4 5 6 7 8 9 10

100

200

300

400

500

600

700

800

900

1000

1100

1200

1300

1400

Years

Inve

stm

ent i

n do

llars

($)


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© Walch Education

Lesson 2.6.3: Comparing Linear to Exponential Functions

Warm-Up 2.6.3Food-poisoning bacteria have to multiply to high numbers in food before there are enough bacteria to make someone sick. When conditions such as type of food, moisture level, temperature, and amount of time passed are ideal, the bacteria can double every 20 minutes.

1. Write a function to represent this scenario.

2. What is the rate of change for the first hour?

3. How many bacteria will there be after 6 hours if the initial number of bacteria is 1?


Instruction


Lesson 2.6.3: Comparing Linear to Exponential FunctionsCommon Core State Standard

F–LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. ★

Warm-Up 2.6.3 DebriefFood-poisoning bacteria have to multiply to high numbers in food before there are enough bacteria to make someone sick. When conditions such as type of food, moisture level, temperature, and amount of time passed are ideal, the bacteria can double every 20 minutes.

1. Write a function to represent this scenario.

This scenario is represented by an exponential function.

• The initial number of bacteria is 1.

• The bacteria doubles every 20 minutes, so the growth factor is 2.

• The time period is 20 minutes.

The function that represents this scenario is ( ) 1 2 20( )=f xx

.

2. What is the rate of change for the first hour?

Identify the interval.

The x-value for the start of the interval is 0.

The x-value for the end of the interval is 60, because there are 60 minutes in 1 hour.

When x is 0, f(x) = 1. This is stated in the problem.

To calculate the number of bacteria after 60 minutes, substitute 60 for x and evaluate the expression.

( ) 1 2 20( )=f xx

Original function

(60) 1 260

20( )=f Substitute 60 for x.


f(60) = 1(8)

f(60) = 8


Instruction


© Walch Education

Calculate the rate of change.

Let (x1, y

1) = (0, 1) and (x

2, y

2) = (60, 8).

2 1

2 1

−−

y y

x x

Slope formula

8 1

60 0

−−


1, y

1) and (x

2, y

2).

7


≈ 0.12

The rate of change for the first hour is approximately 0.12 bacteria per minute.

3. How many bacteria will there be after 6 hours if the initial number of bacteria is 1?

Determine how many minutes are in 6 hours.

6 hours •60minutes

1 hour

6 hours •60minutes

1 hour

6 • 60 minutes

360 minutes

Substitute 360 minutes into the function ( ) 1 2 20( )=f xx

.

( ) 1 2 20( )=f xx

Original function

(360) 1 2360

20( )=f Substitute 360 for x.


f(360) = 1(262,144)

f(360) = 262,144

After 6 hours, there will be 262,144 bacteria.


Instruction



• As in the warm-up, students will create exponential functions from context.

• Students will be determining the rate of change as well as the y-intercepts from graphs, tables, equations, and verbal descriptions.

• Students will use their knowledge of writing exponential functions to analyze these properties in order to compare exponential functions to linear functions.


Instruction


© Walch Education

Prerequisite Skills


• determining the rate of change of a function

• graphing functions

• identifying linear and exponential functions

IntroductionIn previous lessons, linear functions were compared to linear functions and exponential functions to exponential. In this lesson, the properties of linear functions will be compared to properties of exponential functions.

Key Concepts

• Linear functions are written in the form f(x) = mx + b.

• A factor is one of two or more numbers or expressions that when multiplied produce a given product.

• The variable of a linear function is a factor of the function.

• As the value of x increases, the value of f(x) will increase at a constant rate.

• The rate of change of linear functions remains constant.

• Exponential functions are written in the form g(x) = ab x.

• The variable of an exponential function is part of the exponent.

• As the value of x increases, the value of g(x) will increase by a multiple of b.

• As discussed previously, the rate of change of an exponential function varies depending on the interval observed.

• Graphs of exponential functions of the form g(x) = ab x, where b is greater than 1, will increase faster than graphs of linear functions of the form f(x) = mx + b.

• A quantity that increases exponentially will always eventually exceed a quantity that increases linearly.


Instruction





• interchanging the x- and y-intercepts

• assuming the rate of change of a function is linear by only referencing one interval


Instruction


© Walch Education


Which function increases faster, f(x) = 4x – 5 or g(x) = 4 x – 5? Justify your answer with a graph.

1. Make a general observation.

f(x) = 4x – 5 is a linear function of the form f(x) = mx + b.

The variable x is multiplied by the coefficient 4.

g(x) = 4 x – 5 is an exponential function of the form g(x) = ab x.

The variable x is the exponent.

2. Create a table of values.

Substitute values for x into each function.

f(x) = 4x – 5x f(x)

–2 –13–1 –90 –51 –12 3

g (x) = 4x – 5x g (x)

–2 –4.9375–1 –4.750 –41 –12 11


Instruction


3. Graph both functions on the same coordinate plane.

Use the tables of values created in order to plot both functions.

-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

10

f(x) = 4x – 5

g(x) = 4x – 5

4. Compare the rate of change of each function.

The graph of f(x) = 4x – 5 appears to be steeper than the graph of g(x) = 4 x – 5 until the point (1, –2). At this point, the graphs of both functions appear to be equal. Once x is greater than 1, the graph of g(x) = 4 x – 5 becomes steeper. From there, g(x) = 4x – 5 increases faster than f(x) = 4x – 5.


Instruction


© Walch Education

Example 2

At approximately what point does the value of f(x) exceed the value of g(x) if ( ) 2 4 20( )=f xx

and g(x) = 0.5x? Justify your answer with a graph.


( ) 2 4 20( )=f xx

is an exponential function of the form g(x) = ab x.


g(x) = 0.5x is a linear function of the form f(x) = mx + b.

The variable x is multiplied by the coefficient 0.5.

2. Create a table of values.


( ) 2 4 20f xx

( )=x f(x)0 22 2.304 2.646 3.03

g (x) = 0.5x

x g (x)0 02 14 26 3


Instruction




0 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32

2

4

6

8

10

12

14

16

18

( ) 2 4 20f xx

( )=g(x) = 0.5x

4. Identify the approximate point where f(x) is greater than g(x).

It can be seen from the graph that both functions are equal where x is approximately equal to 28. When x is greater than 28, f(x) is greater than g(x).


Instruction


© Walch Education

Example 3

Lena has been offered a job with two salary options. The first option is modeled by the function f(x) = 500x + 31,000, where f(x) is her salary in dollars after x years. The second option is represented by the function g(x) = 29,000(1.04) x, where g(x) is her salary in dollars after x years. If Lena is hoping to keep this position for at least 5 years, which salary option should she choose? Support your answer with a graph.


f(x) = 500x + 31,000 is a linear function of the form f(x) = mx + b.

The variable x is multiplied by the coefficient 500 and added to the constant 31,000.

g(x) = 29,000(1.04) x is an exponential function of the form g(x) = ab x.


Create a table of values.


f(x) = 500x + 31,000 x f(x)0 31,0002 32,0004 33,0006 34,000

g (x) = 29,000(1.04) x

x g (x)0 29,0002 31,366.404 33,925.906 36,694.25


Instruction




0 1 2 3 4 5 6

5

10

15

20

25

30

35

40

45

Years

g(x) = 29,000(1.04)x

f(x) = 500x + 31,000

Year

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alar

y (in

thou

sand

s of

dol

lars

)

3. Identify the approximate point where g(x) is greater than f(x).

It can be seen from the graph that after 3 years, g(x) is greater than f(x). If Lena is hoping to keep this position for at least 5 years, it is in her best interest to choose the salary option modeled by g(x) = 29,000(1.04) x.


naMe:


© Walch Education

Problem-Based Task 2.6.3: Future FinancesCole has just graduated from high school and is going to attend college while working on the family farm. The degree program he has chosen takes an average of 24 months to complete, but it’s possible that he could finish earlier or that he might need more time. His parents have offered to pay him a single payment for all his work on the farm once he graduates. The payment will be based on one of two options. The first option is to start at $0 and earn $3,495.25 a month for the remainder of his time on the farm. This means that if Cole were somehow able to graduate at the end of the first month, he would earn just $3,495.25; but for every month he works on the farm while he is in school, the amount of what he’ll eventually be paid increases by another $3,495.25. When he graduates, he will collect the amount earned for all the months he worked. The second option is to earn $0.01 the first month, and then double the previous month’s pay every month for the remainder of his time on the farm. With this pay plan, he can collect his payout for the month in which he graduates, but he cannot collect the money accumulated up to that point. Which option is the better choice for Cole?


naMe:


Problem-Based Task 2.6.3: Future Finances

Coachinga. How can you determine whether receiving an increase of $3,495.25 a month for the remainder

of Cole’s time on the farm is linear or exponential?

b. Write a function to model receiving $3,495.25 a month for the length of Cole’s time on the farm.

c. How can you determine whether receiving $0.01 for the first month and then receiving double the previous month’s pay is linear or exponential?

d. Write a function to model receiving $0.01 for the first month and then receiving double the previous month’s pay each month for the length of Cole’s time on the farm.

e. Graph both functions on the same coordinate plane.

f. When is it a better option to choose $0.01 for the first month and then double the previous month’s pay for every month after?

g. Is there a point on the graph where choosing either option results in the same payment?

h. If Cole finishes school in 21 months and collects his payment then, which is the better option?

i. If Cole finishes school in 27 months and collects his payment then, which is the better option?

j. Which is the better option for Cole?


Instruction


© Walch Education

Problem-Based Task 2.6.3: Future Finances

Coaching Sample Responsesa. How can you determine whether receiving an increase of $3,495.25 a month for the remainder

of Cole’s time on the farm is linear or exponential?

Receiving an increase of $3,495.25 a month for the remainder of his time on the farm is a constant pay rate.

If the rate of change remains the same, the function is linear.

b. Write a function to model receiving an increase of $3,495.25 a month for the length of Cole’s time on the farm.

Linear functions are of the form f(x) = mx + b.

• The rate of change for this function is 3,495.25.

• The y-intercept is 0.

The function that models receiving an increase of $3,495.25 a month is f(x) = 3495.25x.

c. How can you determine whether receiving $0.01 for the first month and then receiving double the previous month’s pay is linear or exponential?

Analyze the rates of change over the first few intervals.

Cole’s pay doubles each month.

The first month, his pay is $0.01.

The second month, his pay is $0.02.

The third month, his pay is $0.04.

The rate of change is not the same for each interval of the function, so the function can’t be linear.

This is an example of an exponential growth function.

d. Write a function to model receiving $0.01 for the first month and then receiving double the previous month’s pay each month for the length of Cole’s time on the farm.

Exponential growth functions are of the form g(x) = ab x.

• The initial amount is 0.01.

• The rate of growth is 2 because it is doubling.

The function that models receiving $0.01 for the first month is g(x) = (0.01)(2) x – 1.


Instruction


e. Graph both functions on the same coordinate plane.

01 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 306,00012,00018,00024,00030,00036,00042,00048,00054,00060,00066,00072,00078,00084,00090,00096,000102,000108,000114,000120,000126,000132,000138,000144,000150,000

f(x) = 3495.25x g(x) = 0.01(2)x–1

Tota

l pay

in d

olla

rs ($

)

Months of employment

f. When is it a better option to choose $0.01 for the first month and then double the previous month’s pay for every month after?

According to the graph, after about 24 months of employment, the value of the option of receiving $0.01 for the first month and then double the previous month’s salary after that becomes greater than the value of the option of being paid $3,495.25 each month. If Cole plans to be in school for more than two years, choosing to double his pay each month becomes the better option.

g. Is there a point on the graph where choosing either option results in the same payment?

At 24 months, or 2 years, the payments are the same regardless of the option chosen. Up until this point, the better option is to receive a payment of $3,495.25 a month. After this point, the better option is to receive $0.01 for the first month and then earn double the previous month’s pay each month afterward.


Instruction


© Walch Education

h. If Cole finishes school in 21 months and collects his payment then, which is the better option?

The better option before 24 months have passed is the linear function, f(x) = 3495.25x. Earnings for this option far exceed the second exponential option at this point. After 21 months, Cole would be better off choosing the first option, the one that increases his pay at the rate of $3,495.25 a month.

i. If Cole finishes school in 27 months and collects his payment then, which is the better option?

At this point, the exponential model far exceeds the linear model, so Cole should choose the second option, g(x) = 0.01(2)x – 1.

j. Which is the better option for Cole?

If Cole plans to take the average amount of time (2 years) to finish his schooling, either option will yield the same earnings. However, if he plans to finish early, he should choose the linear option that increases his earnings by $3,495.25 a month. If he takes longer than 2 years to finish his schooling, he should choose the exponential model which starts at $0.01 and bases his pay on doubling the previous month’s earnings.




naMe:


Practice 2.6.3: Comparing Linear to Exponential FunctionsUse what you know about linear and exponential functions to complete problems 1–6.

1. Which increases faster, f(x) = 5x or g(x) = 5x ? Justify your answer using a graph.

2. Which increases faster, f(x) = 2x – 4 or g(x) = 2x – 4? Justify your answer using a table of values.

3. Which decreases faster, f(x) = 200(0.97) x or g(x) = 200 – 3.6x? Justify your answer using a graph.

4. Which decreases faster, f(x) = 1000(0.85) x or g(x) = 1000 – 0.15x? Justify your answer using a table of values.

5. At what point does the value of f(x) exceed the value of g(x) if ( ) 2 8 4( )=f xx

and g(x) = 8x + 4? Justify your answer with a graph.

6. At what point does the value of f(x) exceed the value of g(x) if ( ) 500 1.035 12( )=f xx

and g(x) = 0.4x + 60? Justify your answer with a graph.

Use the following information to answer questions 7–10.

You are looking to invest $450. One savings option follows the function

f(x) = 450 + 450(0.055) x, where f(x) is the amount of money in savings after x years.

The second option is represented by the function ( ) 450 10.04

12

12

= +

g xx

, where g(x)

is the amount of money after x years.

7. Which increases faster, f(x) = 450 + 450(0.055) x or ( ) 450 10.04

12

12

= +

g xx

? Use a graph to explain your answer.

8. At what point does the value of g(x) exceed the value of f(x)?

9. If you were looking to invest your money for less than 10 years, which option would you choose? Explain your reasoning.

10. If you were looking to invest your money for more than 20 years, which option would you choose? Explain your reasoning.

name: unit 2 • linear and exponential relationships lesson 6: comparing...

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