Quantities and relationships

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Quantities and relationships

Skiers seek soft, freshly fallen snow because it gives a smooth “floating” ride. Of course, the ride up the mountain isn’t nearly as much fun—especially if the ski lifts are on the fritz!

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A Picture is Worth a thousand Words Understanding Quantities and Their Relationships . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

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A Sort of Sorts Analyzing and Sorting Graphs . . . . . . . . . . . . . . . . . . . . . 17

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© 2012 Carnegie Learning

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there Are Many Ways to represent Functions Recognizing Algebraic and Graphical Representations of Functions . . . . . . . . . . . . . . . . . . . . . . 35

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Function Families for 200, Alex… Recognizing Functions by Characteristics . . . . . . . . . . . . 53

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Chapter 1 Overview

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1.1

Understanding Quantities and Their Relationships

Technology

Highlights

Talk the Talk

Pacing

Peer Analysis

CCSS

Models

Lesson

Worked Examples

This chapter introduces students to the concept of functions. Lessons provide opportunities for students to explore functions, including linear, exponential, quadratic, linear absolute value functions, and linear piecewise functions through problem situations, graphs, and equations. Students will classify each function family using graphs, equations, and graphing calculators. Each function family is then defined and students will create graphic organizers that represent the graphical behavior and examples of each.

This lesson provides opportunities for students to explore quantities and their relationships with each other through eight different problem situations. N.Q.2 F.LE.1.b

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Questions ask students to identify the independent and dependent quantity for each and match a numberless graph to each scenario. Questions then focus students to compare and contrast the different graphs.

X

This lesson provides twenty-two different graphs (18 functions and 4 non-functions) for students to analyze and compare. Analyzing and Sorting Graphs

F.IF.1 F.IF.5

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Questions ask students to sort the graphs into different groups based on their own rationale, and then students will identify the grouping rationale of others. Finally, they distinguish between graphs of functions and non-functions.

X

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  Chapter 1  Quantities and Relationships

Technology

Talk the Talk

Highlights

Peer Analysis

Pacing

Worked Examples

CCSS

Models

Lesson

1

This lesson provides opportunities for students to explore graphical behavior and the form of the equation for different functions.

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F.IF.5 F.IF.9 A.REI.10 F.IF.1 F.IF.2 F.IF.7.a

Recognizing Functions by Characteristics

F.IF.1 F.IF.4 F.IF.7.a F.IF.9 F.LE.1.b F.LE.2 A.CED.2

2

X

X

Questions then ask students to sort the graphs based on the form of the equations. This leads students to identifying one of five different functions: linear, exponential, quadratic, linear absolute value, and linear piecewise. Finally, students paste each graph with its corresponding equation into the appropriate graphic organizer and describe the graphical behavior of each function. This lesson revisits the eight scenarios presented in the first lesson of this chapter. 1

Questions ask students to complete a table by identifying the function family represented by the scenario and the attributes of the function with respect to graphical behavior.

X

X

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1.4

Recognizing Algebraic and Graphical Representa­tions of Functions

Student will sort the eighteen graphs of functions identified in the previous lesson according to specific graphical behaviors and use a graphing calculator to match an equation to each graph.

Chapter 1  Quantities and Relationships 

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Skills Practice Correlation for Chapter 1

1 Lesson

Problem Set

Objectives Vocabulary

1.1

Understanding Quantities and Their Relationships

1–6

Determine independent and dependent quantities

7 – 12

Identify graphs that model scenarios

13 – 18

Label graphs modeling scenarios with independent and dependent quantities Vocabulary

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Analyzing and Sorting Graphs

1–6

Identify similarities of pairs of graphs

7 – 12

Determine whether graphs are discrete or continuous

13 – 18

Use the Vertical Line Test to determine if graphs represent functions Vocabulary

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Recognizing Functions by Characteristics

Rewrite functions using function notation

7 – 12

Identify graphs that represent functions

13 – 18

Determine whether functions are increasing, decreasing, constant, or a combination of increasing and decreasing based on its graph

19 – 24

Determine whether functions have an absolute minimum, absolute maximum, or neither based on its graph

25 – 30

Determine whether graphs represent linear, quadratic, exponential, linear absolute value, linear piecewise, or constant functions

1 – 10

Identify the appropriate function family based on characteristics

11 – 16

Create equations and graphs for functions given a set of characteristics

17 – 22

Identify the function family represented by graphs

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1.4

Recognizing Algebraic and Graphical Representa­ tions of Functions

1–6

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  Chapter 1  Quantities and Relationships

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Chapter 1  Quantities and Relationships 

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  Chapter 1  Quantities and Relationships

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A Picture Is Worth a Thousand Words Understanding Quantities and Their Relationships Learning Goals

Key Terms • dependent quantity • independent quantity

In this lesson, you will:

• Understand quantities and their relationships with each other.

• Identify the independent and dependent quantities for a problem situation. • Match a graph with an appropriate problem situation. • Label the independent and dependent quantities on a graph. • Review and analyze graphs. • Describe similarities and differences among graphs.

Essential Ideas • There are two quantities that change in problem situations.

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• When one quantity depends on another, it is said to be the dependent quantity. • The quantity that the dependent quantity depends upon is called the independent quantity. • The independent quantity is used to label the x-axis. • The dependent quantity is used to label the y-axis. • Graphs can be used to model problem situations.

Mathematics Common Core Standards N-Q Quantities Reason quantitatively and use units to solve problems. 2.  Define appropriate quantities for the purpose of descriptive modeling. F-LE Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems 1.  Distinguish between situations that can be modeled with linear functions and with exponential functions. b.  Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

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Overview

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© 2012 Carnegie Learning

Several problem situations are presented for students to identify the independent and dependent quantities. They are then given graphs that model each scenario and will match each graph to the appropriate scenario and label each axis using the independent and dependent quantities. Several questions are posed which focus the students on various characteristics of each graph, their similarities and differences. Some graphical behaviors are compared and discussed, such as increasing, decreasing, curved, linear, smooth (continuous), and maximum and minimum values.

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  Chapter 1  Quantities and Relationships

Warm Up

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Use the graph shown to answer each question. Time (number of hours playing game)

y 9 8 7 6 5 4 3 2 1 0

1

2

3

4 5 6 7 Time (days)

8

9

x

Emma bought a new video game. The graph describes the number of hours Emma spent playing the game over a period of 7 days. 1. How would you label the x-axis? The x-axis would be labeled Time (days). 2. How would you label the y-axis? The y-axis would be labeled Time (number of hours playing game). 3. Explain your reasoning for choosing each label. The scenario stated Emma played the game over a period of 7 days. I chose the x-axis to represent number of days because each point lies on a different day.

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4. What does the highest point on the graph represent with respect to the scenario? The highest point describes the number of hours Emma played the video game on the 3rd day, four hours. 5. What does the lowest point on the graph represent with respect to the scenario? The lowest point describes the number of hours Emma played the video game on the 7rd day, zero hours.

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  Chapter 1  Quantities and Relationships

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5.1 1.1

A Picture is Worth A trip to the Moon a thousand Words

Using tables to represent Understanding Quantities equivalent ratios and their relationships leArning gOAlS

Key terM Key terMS • ratio

In this lesson, you will:

• Understand Write ratios as quantities part-to-part and their and part-torelationships with

• dependent quantity • independent quantity

wholeother. each relationships.

• Identify Represent theratios independent using models. and dependent quantities for problem situation. • aUse models to determine equivalent ratios. • Match a graph with an appropriate problem situation. • Label the independent and dependent quantities on a graph.

• Review and analyze graphs. • Describe similarities and differences among graphs.

A

person who weighs 100 pounds on Earth would weigh only about 40 pounds on the planet Mercury and about 91 pounds on Venus. In fact, there are only three planets in our solar system where a 100-pound person would weigh more than 100 pounds: Jupiter, Saturn, and Neptune. On Saturn, a 100-pound person would weigh about 106 pounds, on Neptune, about 113 pounds, and on Jupiter, about 236 pounds! On Pluto——which is no longer considered a planet—–a 100-pound person would ow interesting would a website be without pictures or illustrations? Does an weigh less than 7 pounds. inviting image on a magazine cover make you more likely to buy it? Pictures and images just forperson drawing your attention, bring lifewere to text But what if aaren’t 100-pound could stand on thethough. surface They of thealso Sun? If that and stories. possible, then that person would weigh over 2700 pounds! More than a ton! What

© 2012 Carnegie Learning

© 2012 Carnegie Learning

H

causes these differences in weight? There is an old proverb that states that a picture is worth a thousand words. There is a lot of truth in this saying—and images have been used by humans for a long time to communicate. Just think: would you rather post a story of your adventure on a social media site, or post one picture to tell your thousand-word story in a glance?

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Problem 1

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In the first activity, students will identify the independent and dependent quantity in five different statements and describe their identification process. Next, students identify the independent and dependent quantities in eight different scenarios.

PROBLEM 1

1

What’s the Dependency?

Have you ever planned for a party? You may have purchased ice, gone grocery shopping, selected music, made food, or even cleaned in preparation. Many times, these tasks depend on another task being done first. For instance, you wouldn’t make food before grocery shopping, now would you? Let’s consider the relationship between:

• the number of hours worked and the money earned. • your grade on a test and the number of hours you studied.

Grouping • Ask a student to read

• the number of people working on a particular job and the time it takes to complete a job.

the introduction before Question 1. Discuss as a class.

• the number of games played and the number of points scored. • the speed of a car and how far the driver pushes down on the gas pedal.

• Have students complete Questions 1 and 2 independently. Then share the responses as a class.

There are two quantities that are changing in each situation. When one quantity depends on another in a problem situation, it is said to be the dependent quantity. The quantity that the dependent quantity depends upon is called the independent quantity.

Guiding Questions for Share Phase, Questions 1 and 2

1. Circle the independent quantity and underline the dependent quantity in each statement.

Which quantity forces the other quantity to change? 2. Describe how you can determine which quantity is the independent quantity and which quantity is the dependent quantity in any problem situation. The independent quantity is the quantity that stands alone and is not changed by the other quantities.

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Chapter 1

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Quantities and Relationships

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The dependent quantity depends on the independent quantity. The independent quantity causes a change in the dependent quantity.

Grouping Have students complete Question 3 with a partner. Then share the responses as a class.

Guiding Questions for Share Phase, Question 3

Candice is a building manager for the Crowley Enterprise office building. One of her responsibilities is cleaning the office building’s 200-gallon aquarium. For cleaning, she must remove the fish from the aquarium and drain the water. The water drains at a constant rate of 10 gallons per minute.

• independent quantity:

y

scenario?

• How is water measured in this scenario?

Water (gallons)

water (gallons)

Time (minutes)

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You have had your eye on an upgraded smart phone. However, you currently do not have the money to purchase it. Your cousin will provide the funding, as long as you pay him interest. He tells you that you only need to pay $1 in interest initially, and then the interest will double each week after that. You consider his offer and wonder: is this really a good deal? y

time (weeks)

Graph B

• dependent quantity: interest (dollars) Interest (dollars)

the money was borrowed change or determine the amount of interest paid, or does the amount of interest paid change or determine the amount of time the money was borrowed?

x

Smart Phone, but Is It a Smart Deal?

• independent quantity:

Smart Phone, but Is It a Smart Deal? • Does the amount of time

Graph H

• dependent quantity:

change or determine the number of gallons of water emptied, or does the gallons of water emptied change or determine the amount of time?

• How is time measured in this

1

Something’s Fishy

time (minutes)

Something’s Fishy • Does the amount of time

1

3. Read each scenario and then determine the independent and dependent quantities. Be sure to include the appropriate units of measure for each quantity.

Time (weeks)

x

• How is time measured in this scenario?

• How is interest measured in

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this scenario?

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was on the ski lift change or determine the distance the lift traveled, or does the distance the lift traveled change or determine the amount of time he was on the ski lift?

• How is time measured in this

Can’t Wait to Hit the Slopes! Andrew loves skiing—he just hates the ski lift ride back up to the top of the hill. For some reason the ski lift has been acting up today. His last trip started fine. The ski lift traveled up the mountain at a steady rate of about 83 feet per minute. Then all of a sudden it stopped and Andrew sat there waiting for 10 minutes! Finally, the ski lift began to ascend up the mountain to the top.

• independent quantity:

distance (feet)

• How is distance measured in this scenario?

the rope change or determine the length of each piece of rope, or does the length of each piece of rope change or determine the number of cuts in the rope?

• How is the length of the pieces of rope measured in this scenario?

Graph G

• dependent quantity:

scenario?

It’s Magic • Does the number of cuts in

y

time (minutes)

Distance (feet)

1

Time (minutes)

x

It’s Magic The Amazing Aloysius is practicing one of his tricks. As part of this trick, he cuts a rope into many pieces and then magically puts the pieces of rope back together. He begins the trick with a 20-foot rope and then cuts it in half. He then takes one of the halves and cuts that piece in half. He repeats this process until he is left with a piece so small he can no longer cut it. He wants to know how many total cuts he can make and the length of each remaining piece of rope after the total number of cuts.

• independent quantity: number of cuts

• dependent quantity: length of each piece of rope (feet)

y

Graph D

Length of Each Piece of Rope (feet)

Can’t Wait to Hit the Slopes! • Does the amount of time he 1

x

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Number of Cuts

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  Chapter 1  Quantities and Relationships

more times, what impact will it have on the maximum height of the baton?

• How is time measured in this scenario?

• How is the height of the baton measured in this scenario?

1

Baton Twirling

• independent quantity:

y

• dependent quantity: height of baton (feet)

purchase more songs, what impact will it have on the cost? scenario?

Graph F

time (seconds)

Music Club • If Jermaine wants to

• How is cost measured in this

1

Jill is a drum major for the Altadena High School marching band. She has been practicing for the band’s halftime performance. For the finale, Jill tosses her baton in the air so that it reaches a maximum height of 22 feet. This gives her 2 seconds to twirl around twice and catch the baton when it comes back down.

Height of Baton (feet)

Baton Twirling • If Jill wants to twirl around

Time (seconds)

x

Music Club Jermaine loves music. He can lip sync almost any song at a moment’s notice. He joined Songs When I Want Them, an online music store. By becoming a member, Jermaine can purchase just about any song he wants. Jermaine pays $1 per song.

• independent quantity:

y

number of songs

Graph A

• dependent quantity: Cost (dollars)

cost (dollars)

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Number of Songs

x

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distance she has to walk to school, or can she change the time it takes to walk to school?

1

• How is time measured in this scenario?

• How is distance measured in this scenario?

A Trip to School On Monday morning, Myra began her 1.3-mile walk to school. After a few minutes of walking, she walked right into a spider’s web—and Myra hates spiders! She began running until she ran into her friend Tanisha. She stopped and told Tanisha of her adventurous morning and the icky spider’s web! Then they walked the rest of the way to school.

• independent quantity:

y

Graph E

time (minutes)

• dependent quantity: distance traveled (miles)

Jelly Bean Challenge • Does Mr. Wright determine the number of jelly beans guessed, or the number of jelly beans they are off by?

Distance Traveled (miles)

A Trip to School • Can Myra change the

Time (minutes)

x

Jelly Bean Challenge Mr. Wright judges the annual Jelly Bean Challenge at the summer fair. Every year, he encourages the citizens in his town to guess the number of jelly beans in a jar. He keeps a record of everyone’s guesses and the number of jelly beans that each person’s guess was off by. Graph C y • independent quantity:

• dependent quantity: number of jelly beans the guess is off by

Number of Jelly Beans the Guess Is off By

number of jelly beans guessed

x

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Number of Jelly Beans Guessed

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  Chapter 1  Quantities and Relationships

Problem 2 Students are given eight different numberless graphs and will match each graph with the appropriate scenario from Problem 1. They then label each axis on every graph using the independent and dependent quantities, including the units of measurement.

Grouping • Ask a student to read

1 PROBLEM 2

Matching graphs and Scenarios

1

While a person can describe the monthly cost to operate a business, or talk about a marathon pace a runner ran to break a world record, graphs on a coordinate plane enable people to see the data. Graphs relay information about data in a visual way. If you read almost any newspaper, especially in the business section, you will probably encounter graphs. Points on a coordinate plane that are or are not connected with a line or smooth curve model, or represent, a relationship in a problem situation. In some problem situations, all the points on the coordinate plane will make sense. In other problem situations, not all the points will make sense. So, when you model a relationship on a coordinate plane, it is up to you to consider the situation and interpret the meaning of the data values shown. 1. Cut out each graph on the following pages. Then, analyze each graph, match it to a scenario, and tape it next to the scenario it matches. For each graph, label the x- and y-axes with the appropriate quantity and unit of measure. Then, write the title of the problem situation on each graph.

the introduction before Question 1. Discuss as a class.

What strategies will you use to match each graph with one of the eight scenarios?

• Have students complete Question 1 with a partner. Then share the responses as a class.

Guiding Questions for Share Phase, Question 1 • Why did you decide to use this graph to describe this scenario?

• What words in the scenario helped you to decide this was the appropriate graph?

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• Could more than one graph model this scenario? Why or why not?

• Did you need to use any graph twice?

• Is there any scenario that cannot be modeled using one of the graphs?

• How did you decide the label for the x-axis of the graph?

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• Is the independent quantity located on the x-axis or the y-axis? Does it make a difference? Explain.

• Is the dependent quantity located on the x-axis or the y-axis? Does it make a difference? Explain.

• How did you decide the label for the y-axis of the graph?

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  Chapter 1  Quantities and Relationships

1 Graph A Music Club

Interest (dollars) Number of Songs

x Time (weeks)

Graph C Jelly Bean Challenge

Number of Jelly Beans Guessed

x

x

Graph D It’s Magic

Length of Each Piece of Rope (feet)

y

Number of Jelly Beans the Guess Is off By

y

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Graph B Smart Phone, but Is It a Smart Deal y

Cost (dollars)

y

Number of Cuts

x

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  Chapter 1  Quantities and Relationships

1 Graph E A Trip to School

y

Time (minutes)

x

Time (seconds)

Graph G Can’t Wait to Hit the Slopes!

x

Graph H Something’s Fishy

Water (gallons)

y

Distance (feet)

y

Time (minutes)

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Graph F Baton Twirling

Height of Baton (feet)

Distance Traveled (miles)

y

x

Time (minutes)

x

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Problem 3 Students will examine each graph and answer questions related to similarities, differences, the placement of labels on each axis, and discrete or continuous data patterns.

1 PROBLEM 3

Oh, Say, Can you See (in the graphs)!

1

Now that you have matched a graph with the appropriate problem situation, let’s go back and examine all the graphs. 1. What similarities do you notice in the graphs? Answers will vary.

• The independent quantity is graphed on the x-axis while the dependent quantity is graphed on the y-axis.

Look closely when analyzing the graphs. What do you see?

• All the graphs are continuous.

Grouping • First, have students complete Questions 1 through 4 with a partner. Then share the responses as a class.

2. What differences do you notice in the graphs? Answers will vary.

• Some graphs contain straight lines, while some contain

• Next, have students

curves.

• Some graphs seem to move up as they go from left to

complete Question 5 with a partner. Then share the responses as a class.

right, some move down from left to right.

• Some graphs are made of pieces that go up, go down, or stay constant from left to right.

• Finally, have students complete Question 6 independently. Then share the responses as a class.

3. How did you label the independent and dependent quantities in each graph? I labeled the independent quantity on the x-axis and the dependent quantity on the y-axis in each graph.

Guiding Questions for Share Phase, Questions 1 through 4 • Is the independent quantity

4. Analyze each graph from left to right. Describe any graphical characteristics you notice.

always located on the same axis?

Answers will vary.

• • • • •

• Is the dependent quantity

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always located on the same axis?

• Which graphs contained

Some graphs only increase. Some graphs only decrease. Some graphs both increase and decrease. Some graphs have a minimum or maximum value. Some graphs increase or decrease at a constant rate.

straight lines?

• Which graphs contained curved lines?

• How would you describe the behavior of the graph from left to right?

• Which graphs could be described as increasing? Why?

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• Which graphs could be describe as decreasing? Why? • Are any graphs both increasing and decreasing? • Is it possible for a graph to be both increasing and decreasing at the same time?

• Can the curves on the graph be described as smooth curves? Why or why not? • Which graphs have a maximum value?

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Guiding Questions for Share Phase, Question 5 1 • Based on the scenarios, why

1

5. Compare the graphs for each scenario given and describe any similarities and differences you notice. a. Smart Phone, but Is It a Smart Deal? and Music Club

do both the Smart Phone, but Is It a Smart Deal? and Music Club graphs increase?

Answers will vary.

Think about all the different graphical characteristics you just identified.

Both graphs increase from left to right. The graph of the Smart Phone, but is It a Smart Deal? situation is a smooth curve, but the graph of the Music Club situation is a straight line.

• Based on the scenarios, why is the Smart Phone, but Is It a Smart Deal? a smooth curve, but the Music Club graph is a straight line?

b. Something’s Fishy and It’s Magic Answers will vary. Both graphs decrease from left to right. The graph of the Something’s Fishy situation is a straight line, but the graph of the It’s Magic situation is a smooth curve.

• Based on the scenarios, why do both the Something’s Fishy and It’s Magic graphs decrease?

c. Baton Twirling and Jelly Bean Challenge Answers will vary. The graphs have either a minimum or a maximum value. Both graphs increase and decrease.

• Based on the scenarios, why

The graph of the Baton Twirling situation is a smooth curve, but the graph of the Jelly Bean Challenge situation is made up of two straight lines.

is the Something’s Fishy graph a straight line, but the It’s Magic graph is a smooth curve?

6. Consider the scenario A Trip to School. a. Write a scenario and sketch a graph to describe a possible trip on a different day. Answers will vary. Scenario

• Based on the scenarios, why

Graph

do both the Baton Twirling and Jelly Bean Challenge graphs increase then decrease?

• Based on the scenarios, why is the Baton Twirling graph a smooth curve, but the Jelly Bean Challenge graph a straight line?

b. Compare your scenario and sketch with your classmates’ scenarios and sketches. What similarities do you notice? What differences do you notice?

as increasing or decreasing?

• Is your graph curved or linear in nature?

• Does your graph contain any horizontal line segments? If so, what does this represent in the scenario?

16 

Some graphs contain straight lines; some graphs contain different segments with varying degrees of steepness; and some graphs contain smooth curves. Be prepared to share your solutions and methods.

• How many different pieces are on your graph? • Does your graph contain any parallel line segments? What does this imply with respect to the scenario?

• If your graph contained a line segment having a negative slope, what would this imply with respect to the scenario?

• What point on your graph represents Myra’s home? • What point on your graph indicates that Myra arrived at school?

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

All the graphs increase from left to right.

Guiding Questions for Share Phase, Question 6 • Can your graph be described

Check for Students’ Understanding

1

Two graphs are shown. • One graph describes Molly’s height in inches over a period of years. • One graph describes Molly’s weight in pounds over a period of years. Graph 1    Graph 2 y

9

9

8

8

7

7

Height (inches)

Weight (pounds)

y

6 5 4 3

6 5 4 3

2

2

1

1

0

1

2

3

4 5 6 7 Time (years)

8

9

x

0

1

  

2

3

4 5 6 7 Time (years)

8

9

x

1. Match each graph with the appropriate scenario and explain your reasoning. Graph 1 describes Molly’s weight over a period of years because weight can increase and decrease. Graph 2 describes Molly’s height over a period of years because height eventually reaches a maximum and then remains the same. 2. Identify the independent and dependent quantities in Graph 1.

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The independent quantity in Graph 1 is time in terms of years. The dependent quantity in Graph 1 is the weight in terms of pounds. 3. Identify the independent and dependent quantities in Graph 2. The independent quantity in Graph 2 is time in terms of years. The dependent quantity in Graph 2 is the height in terms of inches 4. Label each axis with the appropriate quantity and unit.

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  Chapter 1  Quantities and Relationships

1.2

A Sort of Sorts Analyzing and Sorting Graphs

Learning Goals In this lesson, you will:

• Review and analyze graphs. • Determine similarities and differences among various graphs.

• Sort graphs by their similarities and rationalize

the differences between the groups of graphs. • Use the Vertical Line Test to determine if the graph of a relation is a function.

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Essential Ideas • A relation is the mapping between a set of

inputs and a set of outputs. • A function is a relation between a given set of elements, called the domain and the range, for which each element in the domain there exists exactly one element in the range. • The domain of a function is the set of all input values. • The range of a function is the set of all output values. • The Vertical Line Test is used to determine if the graph of a relation is a function.

Key Terms • relation • domain • range • function • Vertical Line Test • discrete graph • continuous graph

Mathematics Common Core Standards F-IF Interpreting Functions Understand the concept of a function and use function notation 1.  Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Interpret functions that arise in applications in terms of the context 5.  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

17A

Overview

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© 2012 Carnegie Learning

Students begin this lesson by cutting out twenty-two different graphs. They will sort the graphs into different groups based on their own rationale, compare their groupings with their classmates, and discuss the reasoning behind their choices. Next, four different groups of graphs are given and students analyze the groupings and explain possible rationales behind the choices made. The terms relation, function, domain, and range are defined. The Vertical Line Test is introduced and used to determine if various graphs are or are not functions.

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  Chapter 1  Quantities and Relationships

Warm Up

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1. Use the points on each graph to complete the corresponding table of values. Graph A y

x

y

1

2

5

2

1

4

2

3

3

0

3

4

4

21

4

5

x

y

6

1

2

5

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6

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7

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7 6

3 2 1 0

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21

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7

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9

x      

Graph B y 7

4 3 2 1 0 © 2012 Carnegie Learning

21

1

2

3

4

5

6

7

8

9

x      

2. Which graph has a different x-value for each y-value? Graph B has a different x-value for each y-value. 3. Which graph has the same x-values for different y-values? Graph A has the same x-values for different y-values. 4. If the equation x 5 2 were drawn on both graphs, how many points on each graph would intersect this line? The line would intersect two points on the graph in Question 1 and the line would intersect one point on the graph in Question 2.

1.2  Analyzing and Sorting Graphs 

  17C

© 2012 Carnegie Learning

1

17D 

  Chapter 1  Quantities and Relationships

1

1.2

A Sort of Sorts Analyzing and Sorting graphs

leArning gOAlS In this lesson, you will:

• Review and analyze graphs. • Determine similarities and differences among various graphs.

• Sort graphs by their similarities and rationalize the differences between the groups of graphs.

• Use the Vertical Line Test to determine if the graph of a relation is a function.

Key terMS • relation • domain • range • function • Vertical Line Test • discrete graph • continuous graph

A

re you getting the urge to start driving? Chances are that you’ll be studying for your driving test before you know it. But how much will driving cost you? For all states in the U.S., auto insurance is a must before any driving can take place. For most teens and their families, this more than likely means an increase in auto insurance costs. So how do insurance companies determine how much you will pay? The fact of the matter is that auto insurance companies sort drivers into different groups to determine their costs. For example, they sort drivers by gender, age, marital status, and driving experience. The type of car is also a factor. A sports vehicle or a luxury car is generally more expensive to insure than an economical car or a family sedan. Even the color of a car can affect the cost to insure it!

© 2012 Carnegie Learning

Do you think it is good business practice to group drivers to determine auto insurance costs? Or do you feel that each individual should be reviewed solely on the merit of the driver based on driving record? Do you think auto insurance companies factor in where a driver lives when computing insurance costs?

1.2  Analyzing and Sorting Graphs 

  17

Problem 1

1

Students cut out 22 graphs and will sort the graphs into different groupings of their choice. They then compare their groupings with their classmates’ groupings and explain their reasoning. Students are asked to create a list of all the different graphical behaviors used to sort the graphs.

1

PROBLEM 1

let’s Sort Some graphs

Mathematics is the science of patterns and relationships. Looking for patterns and sorting objects into different groups can provide valuable insights. In this lesson, you will analyze many different graphs and sort them into various groups. 1. Cut out the twenty-two graphs on the following pages. Then analyze and sort the graphs into different groups. You may group the graphs in any way you feel is appropriate. However, you must sort the graphs into more than one group! In the space provided, record the following information for each of your groups.

• Name each group of graphs. • List the letters of the graphs in each group.

Grouping • Ask a student to read the introduction before Question 1. Discuss as a class.

• Have students complete Questions 1 and 2 in groups. Then share the responses as a class.

• Provide a rationale why you created each group. Answers will vary.

Each coordinate plane is 10 units by 10 units.

Guiding Questions for Share Phase, Questions 1 and 2 • How many groups do you have?

• How did you decide what graph goes in each group?

• Do any of your groups contain a single graph? members disagree about any particular grouping?

• Were there any graphs that didn’t fit into a grouping?

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  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

• Did you and your group

1 A



22x 1 10, 2` # x , 3 f(x) 5 4, 3#x,7 22x 1 18 7 # x # 1`

1 f(x) 5 2 x2 1 2x 2

Domain: all real numbers

Domain: all real numbers

C

D



__

1x 1 4, __

2` # x , 2 2 f(x) 5 23x 1 11, 2 # x , 3 1 1x 1 __ __ 3#x#` 2 2

© 2012 Carnegie Learning

1

B

f(x) 5 |x|

Domain: all real numbers

Domain: all real numbers

E

F

x2 1 y2 5 16

f(x) 5 23x2 1 4, where x is an integer

Domain: 24 # x # 4

Domain: all integers

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

19

  19

© 2012 Carnegie Learning

1

20 

  Chapter 1  Quantities and Relationships

1 G

( __12 ) 2 5 x

f(x) 5 x

f(x) 5

Domain: all real numbers

Domain: all real numbers

J

I

f(x) 5 2| x |

x 5 y( y 2 3)( y 1 3)

Domain: all real numbers

Domain: 210 # x # 10

K

L

f(x) 5 2x,

© 2012 Carnegie Learning

1

H

__

where x is an integer

2 f(x) 5 2 x 1 5 3

Domain: all integers

Domain: all real numbers

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

21

  21

© 2012 Carnegie Learning

1

22 

  Chapter 1  Quantities and Relationships

1

__

f(x) 5 x2

f(x) 5 6 √x

Domain: all real numbers

Domain: x $ 0

O

P

f(x) 5 2x 1 3,

f(x) 5

where x is an integer Domain: all integers

Q

© 2012 Carnegie Learning

1

N

M

( __12 )

x

Domain: all real numbers

R

f(x) 5 22|x 1 2| 1 4

x52

Domain: all real numbers

Domain: x 5 2

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

23

  23

© 2012 Carnegie Learning

1

24 

  Chapter 1  Quantities and Relationships

1 S

T

22, 2` , x , 0 1 x 2 2, 0 # x , ` f(x) 5 __ 2 Domain: all real numbers

5

U

1

f(x) 5 x2 1 8x 1 12 Domain: all real numbers

V

© 2012 Carnegie Learning

f(x) 5 2, where x is an integer

f(x) 5 |x 2 3| 2 2

Domain: all integers

Domain: all real numbers

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

25

  25

© 2012 Carnegie Learning

1

26 

  Chapter 1  Quantities and Relationships

1

2. Compare your groupings with your classmates’ groupings. Create a list of the different graphical behaviors you noticed.

1

Answers may vary. Possible graphical behaviors:

© 2012 Carnegie Learning

• • • • • • • • • • • • •

always increasing from left to right always decreasing from left to right the graph both increases and decreases straight lines smooth curves discrete data values the graph has a maximum value the graph has a minimum value the graph is a function the graph is not a function the graph goes through the origin the graph forms a U shape the graph forms a V shape

Are any of the graphical behaviors shared among your groups? Or, are they unique to each group?

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

27

  27

Problem 2

1

Four different scenarios that show groups of graphs are given and students will explain the rationale behind the groupings, and errors in the reasoning behind a grouping. Rationales for grouping include graphs being discrete, having vertical symmetry, located in a single quadrant, and not being a function.

1

PROBLEM 2

i like the Way you think

1. Matthew grouped these graphs together.

F

K

O

U

Grouping Have students complete Questions 1 through 4 with a partner. Then share the responses as a class.

Guiding Questions for Share Phase, Question 1 • How are Matthew’s graphs different than other graphs you may have seen?

• Which of Matthew’s graphs

Why do you think Matthew put these graphs in the same group? Answers will vary. These graphs are made up of dots (discrete data).

are linear in appearance?

• Which of Matthew’s graphs are not linear in appearance?

• What kind of scenarios can

© 2012 Carnegie Learning

you think of for each of Matthew’s graphs?

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  Chapter 1  Quantities and Relationships

Guiding Questions for Share Phase, Question 2 • Where can you place the

1

2.

1

Ashley

line of symmetry on each of Ashley’s graphs?

I grouped these graphs together because they all show vertical symmetry. If I draw a vertical line through the middle of the graph, the image is the same on both sides.

• Do any of Ashley’s graphs have horizontal symmetry?

• Is the y-axis always the line of vertical symmetry?

M

F

D

T

V

a. Show why Ashley’s reasoning is correct. See graphs. Notice that for Graph V, the student may need to extend the graph to see the symmetry. b. If possible, identify other graphs that show vertical symmetry.

© 2012 Carnegie Learning

B, E, I, Q, and U

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

29

  29

Guiding Questions for Share Phase, Question 3 • Are Duane’s graphs only

1

1

what they appear to be or is only part of each graph visible?

3.

Duane

I grouped these graphs together because each graph only goes through two quadrants.

• What part of each of Duane’s graphs is not visible?

• Why do you think parts of

D

M

P

T

each graph are not visible?

• Do the lines and curves on Duane’s graph continue to infinity?

a. Explain why Duane’s reasoning is not correct.

b. If possible, identify other graphs that only go through two quadrants. G, I, J, K, N, R, and U

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  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

Even though it is not visible, Graph T continues into the first quadrant. Therefore, the graph goes through three quadrants. Each of the other graphs D, M, and P satisfy Duane’s reasoning.

Guiding Questions for Share Phase, Question 4 • How are Josephine’s graphs different from other graphs you have seen?

1

4. Judy grouped these four graphs together, but did not provide any rationale.

E

J

N

R

1

• For each x-value on one of Josephine’s graphs, how many y-values are there?

a. What do you notice about the graphs? Answers will vary. In each graph, for at least one value of x, there is more than one value of y.

b. What rationale could Judy have provided? Answers will vary.

© 2012 Carnegie Learning

The graphs are not functions.

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

31

  31

Problem 3

1

The terms relation, domain, range, and function are defined. The Vertical Line Test is introduced as a visual method used to determine whether a relation represented as a graph is a function. Discrete and continuous graphs are also defined. The Vertical Line Test is then used to sort the graphs from Problem 1 into two groups.

1

PROBLEM 3

Grouping • Have a student read the definitions. Discuss and complete Question 1 as a class.

• Have students complete Questions 2 and 3 with a partner. Then share the responses as a class.

Guiding Questions for Discuss Phase, Question 1 • If a graph passes the Vertical

Function Junction

A relation is the mapping between a set of input values called the domain and a set of output values called the range. A function is a relation between a given set of elements, such that for each element in the domain there exists exactly one element in the range.

So all functions are relations, but only some relations are functions. I guess it all depends on the domain and range.

The Vertical Line Test is a visual method used to determine whether a relation represented as a graph is a function. To apply the Vertical Line Test, consider all of the vertical lines that could be drawn on the graph of a relation. If any of the vertical lines intersect the graph of the relation at more than one point, then the relation is not a function.

A discrete graph is a graph of isolated points. A continuous graph is a graph of points that are connected by a line or smooth curve on the graph. Continuous graphs have no breaks. The Vertical Line Test applies for both discrete and continuous graphs. 1. Analyze the four graphs Judy grouped together. Do you think that the graphs she grouped are functions? Explain how you determined your conclusion. No. The graphs Judy grouped together are not functions. I used the Vertical Line Test for each graph and saw that the vertical line intersected the graph at more than one point in each graph. 2. Use the Vertical Line Test to sort the graphs in Problem 1 into two groups: functions and non-functions. Record your results by writing the letter of each graph in the appropriate column in the table shown. Functions

Non-Functions

A, B, C, D, F, G, H, I, K, L, M, O, P, Q, S, T, U, V

E, J, N, R

Line Test, what does that mean?

• If a graph fails the Vertical • Do Josephine’s graphs pass or fail the Vertical Line Test?

• For each x-value on one of Josephine’s graphs, how many y-values are there?

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© 2012 Carnegie Learning

Line Test, what does that mean?

Guiding Questions for Share Phase, Questions 2 and 3 • How many graphs did you sort into the non-function group?

• Did all of the graphs fit into one of the two groups? Can a graph be neither?

1

• the set of all real numbers, or • the set of integers.

sort into the function group?

• How many graphs did you

1

3. Each graph in this set of functions has a domain that is either:



For each graph, remember that the x-axis and the y-axis display values from 210 to 10 with an interval of 2 units.



Label each function graph with the appropriate domain. The graphs with the domain as the set of all real numbers are: A, B, C, D, G, H, I, K, L, M, P, Q, S, T, U, and V. The graphs with the domain as the set of integers are: F, O, R, and U.

• What do graphs of nonfunctions look like?

• What do graphs of functions look like?

• Are all curved graphs considered graphs of non-functions?

4. Clip all your graphs together and keep them for the next lesson.

Hang on to your graphs. You will need them for the next lesson.

• Can you think of a graph that is curved and is a function? What does it look like?

• Are all linear graphs considered graphs of functions?

• Can you think of a graph

© 2012 Carnegie Learning

that is linear and is a non-function? What does it look like?

1.2 Analyzing and Sorting Graphs

1.2  Analyzing and Sorting Graphs 

33

  33

Talk the Talk

1

Students will sketch a graph that is a function and one that is not a function to demonstrate their understanding.

1

talk the talk 1. Sketch a graph of a function. Explain how you know that it is a function. Answers will vary. y

Be original! Please don’t use any graphs from this lesson.

Grouping Have students complete Questions 1 and 2 independently. Then share the responses as a class.

Guiding Questions for Share Phase, Questions 1 and 2 • Does your graph pass or fail the Vertical Line Test?

• What is the significance of your graph if it passes the Vertical Line Test?

x

For each value of x, there is exactly one value of y. 2. Sketch a graph that is not a function. Explain how you know that it is not a function. Answers will vary. y

• What is the significance of your graph if it fails the Vertical Line Test?

• How many y-values does

x

each x-value have on your graph?

• Is your graph discrete or continuous?

Be prepared to share your solutions and methods.

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  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

For at least one value of x, there are two values of y.

Check for Students’ Understanding

1

1. Sketch a relation that is an example of a continuous function. y 9 8 7 6 5 4 3 2 1 0

1

2

3

4

5

6

7

8

9

x

2. Sketch a relation that is an example of a discrete function. y 9 8 7 6 5 4 3 2 © 2012 Carnegie Learning

1 0

1

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5

6

7

8

9

x

1.2  Analyzing and Sorting Graphs 

  34A

3. Sketch a relation that is an example of a continuous non-function.

1

y 9 8 7 6 5 4 3 2 1 0

1

2

3

4

5

6

7

8

9

x

4. Sketch a relation that is an example of a discrete non-function. y 9 8 7 6 5 4 3 2 1 1

2

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9

x © 2012 Carnegie Learning

0

34B 

  Chapter 1  Quantities and Relationships

There Are Many Ways to Represent Functions

1.3

Recognizing Algebraic and Graphical Representations of Functions Learning Goals In this lesson, you will:

• Write equations using function notation. • Recognize multiple representations

of functions. • Determine and recognize characteristics of functions. • Determine and recognize characteristics of function families.

© 2012 Carnegie Learning

Essential Ideas • The function notation f(x) indicates that x is

the independent variable. • When using a graphing calculator, equations must be written in function notation. • A function is said to be increasing when both the independent and the dependent variables are increasing. • A function is said to be decreasing when and the dependent variable decreases as the independent variable increases. • A function is said to be constant when the dependent variable of the function does not change or remains constant over the entire domain. • A family of functions is a group of functions that share certain attributes. • The family of linear functions includes functions of the form f(x) 5 m x 1 b, where m and b are real numbers and m is not equal to 0. • The family of exponential functions includes functions of the form f(x) 5 a ? bx, where a

Key Terms • function notation • increasing function • decreasing function • constant function • function family • linear functions • exponential functions • absolute minimum • absolute maximum • quadratic functions • linear absolute value functions • linear piecewise functions

and b are real numbers and b is greater than 0 but not equal to 1. • The family of quadratic functions includes functions of the form f(x) 5 ax2 1 bx 1 c, where a, b, and c are real numbers, and a is not equal to 0. • The family of linear absolute value functions includes functions of the form f(x) 5 a|x 1 b| 1 c, where a, b, and c are real numbers, and a is not equal to 0. • Linear piecewise functions include functions that have equation changes for different parts, or pieces, of the domain. • A function has an absolute minimum if there is a point whose y-coordinate is less than the y-coordinates of every other point on the graph. • A function has an absolute maximum if there is a point whose y-coordinate is greater than the y-coordinates of every other point on the graph.

35A

Mathematics Common Core Standards

1

F-IF Interpreting Functions Interpret functions that arise in applications in terms of the context 5.  Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Analyze functions using different representations 9.  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). A-REI Reasoning with Equations and Inequalities Represent and solve equations and inequalities graphically 10.  Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). F-IF Interpreting Functions Understand the concept of a function and use function notation 1.  Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 2.  Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Analyze functions using different representations 7.  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a.  Graph linear and quadratic functions and show intercepts, maxima, and minima.

Function notation is introduced. Steps are provided to guide students through entering and viewing the graph of a linear function on a graphing calculator. The terms increasing function, decreasing function, and constant function are defined. Students will sort the graphs from the previous lesson into groups using these terms and match each graph with its appropriate equation written in function notation. The terms function family, linear function, and exponential function are then defined. Next, the terms absolute minimum and absolute maximum are defined. Students will continue to sort the remaining graphs into groups using these terms and match each graph with its appropriate equation written in function notation. The terms quadratic function and linear absolute value function are then defined. Finally, linear piecewise functions are defined and students match the remaining graphs to their appropriate functions. In the final activity, students will complete a graphic organizer for each function family that describes the graphical behavior and displays the various graphical examples.

35B 

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

Overview

Warm Up

1

1. Graph the equation y 5 x Graph A y 8 6 4 2 0

28 26 24 22

2

4

6

8

4

6

8

x

22 24 26 28

2. Graph the equation y 5 2x Graph B y 8 6 4 2 0

28 26 24 22

2

x

22 24 26

© 2012 Carnegie Learning

28

3. Describe the impact the negative sign has on the graph of the equation y 5 x. The negative sign reverses the direction of the graph, or flips the graph upside down.

1.3  Recognizing Algebraic and Graphical Representations of Functions 

  35C

4. Graph the equation y 5 x2

1

Graph A y 8 6 4 2 0

28 26 24 22

2

4

6

8

4

6

8

x

22 24 26 28

5. Graph the equation y 5 2x2 Graph B y 8 6 4 2 0

28 26 24 22

2

x

22 24 26

6. Describe the impact the negative sign has on the graph of the equation y 5 x2. The negative sign reverses the direction of the graph, or flips the graph upside down.

35D 

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

28

there Are Many Ways to represent Functions

1

1.3

recognizing Algebraic and graphical representations of Functions leArning gOAlS In this lesson, you will:

• Write equations using function notation. • Recognize multiple representations of functions.

• Determine and recognize characteristics of functions.

• Determine and recognize characteristics of function families.

Key terMS • function notation • increasing function • decreasing function • constant function • function family • linear functions • exponential functions • absolute minimum • absolute maximum • quadratic functions • linear absolute value functions • linear piecewise functions

J

ust about everything you see, hear, or own has a name. It’s not just people who have names—streets have names, cars have names, even trees have names. So where do these names come from and why were they chosen? There are many naming conventions we use in our society. The purpose of these naming conventions is to provide useful information about the object being named. For example, just saying “I live on a street” does not provide much information. However, saying “I live on East Main Street” makes it much more clear where you live.

© 2012 Carnegie Learning

Think about other objects and their names. Why do you think they were named the way they were? What information is provided by their names? Would another name suit the object better?

1.3  Recognizing Algebraic and Graphical Representations of Functions 

  35

Problem 1

1

Function notation is defined. A scenario is described and then expressed in function notation. Step by step procedures are given to help students enter a function into a graphing calculator and adjust the window to view a complete graph.

1

PROBLEM 1

Functions can be represented in a number of ways. An equation representing a function can be written using function notation. Function notation is a way of representing functions algebraically. This form allows you to more efficiently identify the independent and dependent quantities. The function f(x) is read as “f of x” and indicates that x is the independent variable.

Consider orders for a custom T-shirt shop. U.S. Shirts charges $8 per shirt plus a one-time charge of $15 to set a T-shirt design. The equation y 5 8x 1 15 can be written to model this situation. The independent variable x represents the number of shirts ordered, and the dependent variable y represents the total cost of the order, in dollars.

Ask a student to read the information. Discuss as a class.

You know this is a function because for each number of shirts ordered (independent value) there is exactly one total cost (dependent value) associated with it.

Guiding Questions for Discuss Phase, Problem 1 • Can all equations be written • What is an example of an

Because this situation is a function, you can write y 5 8x 1 15 in function notation. f( x) 5 8x 1 15 The cost, defined by f, is a function of x, the number of shirts ordered.

So can I use any two letters when representing a function?

A common way to name a function is f(x). However, you can choose any variable to name a function. You could write the T-shirt cost function as C(s) 5 8s 1 15, where the cost, defined as C, is a function of s, the number of shirts ordered.

© 2012 Carnegie Learning

equation that cannot be written in function form?

Remember, you can only write functions in function notation. So sorry, non-functions! You’ll still need to be written as equations.

Let’s look at the relationship between an equation and function notation.

Grouping

in function notation? Why not?

A new Way to Write Something Familiar

36 

  Chapter 1  Quantities and Relationships

Guiding Questions for Discuss Phase, Problem 1 • Do you think a non-function can be graphed on the graphing calculator? Explain.

• What will happen if the Ymin is not changed to 220? What impact will this have on the graph of the function?

• What will happen if the Yscl is not changed to 2? What impact will this have on the graph of the function?

• When viewing a graph on the graphing calculator, how will you know the Xscl or the Yscl needs to be adjusted?

• When viewing a graph on the graphing calculator, how will you know the Xmin or the Xmax needs to be adjusted?

1

Let’s graph the function f(x) 5 8x 1 15 on a calculator by following the steps shown.

You can use a graphing calculator to graph a function. Step 1: Press Y=. Your cursor should be blinking on the line \Y1=. Enter the equation. To enter a variable like x, press the key with

• What will happen if the Ymax is not changed to 20? What impact will this have on the graph of the function?

1

You can input equations written in function notation into your graphing calculator. Your graphing calculator will list different functions as Y1, Y2, Y3, etc.

X, T, Ø, n once.

The way you set the window will vary each time depending on the equation you are graphing.

Step 2: Press WINDOW to set the bounds and intervals you want displayed. Step 3: Press GRAPH to view the graph.

The Xmin represents the least point on the x-axis that will be seen on the screen. The Xmax represents the greatest point that will be seen on the x-axis. Lastly, the Xscl represents the intervals. Similar names are used for the y-axis (Ymin, Ymax, and Yscl). A convention to communicate the viewing WINDOW on a graphing calculator is shown. Xmin: 210 Xmax: 10 Ymin: 220 Ymax: 20

} }

[210, 10]

[220, 20]

}

[210, 10] 3 [220, 20]

• When viewing a graph on the graphing calculator, how will you know the Ymin or the Ymax needs to be adjusted?

• What does it mean to view a complete graph? © 2012 Carnegie Learning

• When viewing a graph on the graphing calculator, how will you know the graph is not a complete graph?

• Why is it important to view a

1.3

Recognizing Algebraic and Graphical Representations of Functions

37

complete graph?

1.3  Recognizing Algebraic and Graphical Representations of Functions 

  37

Problem 2

1

Students will distinguish between increasing functions, decreasing functions, constant functions, and combinations of increasing, decreasing, or constant functions using a sorting activity and the graphs from the previous lesson. Focusing only on the seven graphs of increasing functions, decreasing functions, and constant functions, they match each graph with the appropriate equation written in function notation, and then sort these graphs again into two groups based on the equation of each function. The terms family of functions, linear functions, and exponential functions are described and students identify which group is best represented using these terms.

1

PROBLEM 2

Up, Down, or neither?

In the previous lesson, you determined which of the given graphs represented functions. Gather all of the graphs from the previous lesson that you identified as functions. A function is described as increasing when the dependent variable increases as the independent variable increases. If a function increases across the entire domain, then the function is called an increasing function. A function is described as decreasing when the dependent variable decreases as the independent variable increases. If a function decreases across the entire domain, then the function is called a decreasing function.

Record the function letter in the appropriate column of the table shown.

If the dependent variable of a function does not change or remains constant over the entire domain, then the function is called a constant function. 1. Analyze each graph from left to right. Sort all the graphs into one of the four groups:

• increasing function, • decreasing function, • constant function, • a combination of increasing, decreasing, or constant. Increasing Function

Decreasing Function

Constant Function

Combination of Increasing, Decreasing, or Constant

G, K

H, L, O, P

U

A, B, C, D, F, I, M, Q, S, T, V

Grouping • Ask a student to read the definitions. Discuss as a class.

• Have students complete

Guiding Questions for Share Phase, Questions 1 through 3 • How would you describe the graph of an increasing function? • Are some graphs increasing at a faster rate than others? How can you tell? • How would you describe the graph of a decreasing function? • Are some graphs decreasing at a faster rate than others? How can you tell? • How would you describe the graph of a constant function?

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  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

Questions 1 through 3 with a partner. Then share the responses as a class.

• Did you have trouble entering any of the functions in the graphing calculator? If so, which ones?

• When entering the functions into the graphing calculator, which functions required the use of parenthesis?

• When entering the functions into the graphing calculator, how do you know when you need to use parenthesis?

• What criteria did you use to sort the graphs into two groups?

1

2. Each function shown represents one of the graphs in the increasing function, decreasing function, or constant function categories. Enter each function into a graphing calculator to determine the shape of its graph. Then match the function to its corresponding graph by writing the function directly on the graph that it represents.

1

• f(x) 5 x

Be sure to correctly interpret the domain of each function. Also, remember to use parentheses when entering fractions.

Graph G

( )

x • f(x) 5 __12 2 5

Graph H

• f(x) 5 2x, where x is an integer Graph K 2

• f(x) 5 2__3 x 1 5 Graph L

• f(x) 5 2x 1 3, where x is an integer Graph O

( )

x • f(x) 5 __12

Graph P

• f(x) 5 2, where x is an integer Graph U 3. Consider the seven graphs and functions that are increasing functions, decreasing functions, or constant functions. a. Sort the graphs into two groups based on the equations representing the functions and record the function letter in the table. Group 1

Group 2

G, L, O, U Linear/Constant

H, K, P Exponential

b. What is the same about all the functions in each group? Answers may vary. All the functions in Group 1 form straight lines.

© 2012 Carnegie Learning

All the functions in Group 2 form smooth curves. All the functions in Group 2 involve exponents.

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Recognizing Algebraic and Graphical Representations of Functions

1.3  Recognizing Algebraic and Graphical Representations of Functions 

39

  39

Grouping

1

Ask a student to read the definitions before Question 4. Discuss and complete Questions 4 and 5 as a class.

Congratulations! You have just sorted the graphs into their own function families. A function family is a group of functions that share certain characteristics.

1

The family of linear functions includes functions of the form f(x) 5 mx 1 b, where m and b are real numbers.

Place these two groups of graphs off to the side. You will need them again in Problem 4. Wait for it. . .

The family of exponential functions includes functions of the form f(x) 5 a . bx, where a and b are real numbers, and b is greater than 0 but is not equal to 1. 4. Go back to your table in Question 3 and identify which group represents linear and constant functions and which group represents exponential functions. See table. 5. If f(x) 5 mx 1 b, represents a linear function, describe the m and b values that produce a constant function. If m 5 0 and b is any real number, then the result will be a constant function.

Students will sort the graphs from the combination category in Problem 2 into three groups having the characteristics of absolute minimum, absolute maximum, and having no absolute minimum or absolute maximum. Focusing only on the eight graphs containing absolute minimums, absolute maximums, or having no absolute minimums or absolute maximums, they match each graph with the appropriate equation written in function notation, and then sort these graphs again into two groups based on containing absolute minimums and absolute maximums. The terms quadratic functions and linear absolute value functions are defined and students identify which graphs are best represented using these terms.

40 

PROBLEM 3

least, greatest, or neither?

A function has an absolute minimum if there is a point that has a y-coordinate that is less than the y-coordinates of every other point on the graph. A function has an absolute maximum if there is a point that has a y-coordinate that is greater than the y-coordinates of every other point on the graph. 1. Sort the graphs from the Combination category in Problem 2 into three groups:

• those that have an absolute minimum value, • those that have an absolute maximum value, and • those that have no absolute minimum or maximum value. Then record the function letter in the appropriate column of the table shown.

Think about the graphical behavior of the function over its entire domain.

Absolute Minimum

Absolute Maximum

No Absolute Minimum or Absolute Maximum

D, M, T, V

B, F, I, Q

A, C, S

Grouping • Ask a student to read the definitions. Discuss as a class. • Have students complete Questions 1 through 3 with a partner. Then share the responses as a class.

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

Problem 3

Guiding Questions for Share Phase, Questions 1 through 3 • What impact does the negative sign in front of the lead terms coefficient have on the graph of the function?

• Where can the absolute value function be found on the graphing calculator?

• Did you have trouble entering any of the functions in the graphing calculator? If so, which ones?

• When entering the functions into the graphing calculator, which functions required the use of parenthesis?

• What criteria did you use to sort the graphs into two groups?

• How would you describe the graph of a quadratic function?

• How would you describe the graph of an absolute value function?

1

2. Each function shown represents one of the graphs with an absolute maximum or an absolute minimum value. Enter each function into your graphing calculator to determine the shape of its graph. Then match the function to its corresponding graph by writing the function directly on the graph that it represents.

• f(x) 5 x2 1 8x 1 12

1

When entering an absolute value function into your calculator, Press 2nd Catalog, abs(. Then type the absolute value portion of the equation and press ). For example, the second function would look like this: \Y1=abs(x - 3) - 2

Graph T

• f(x) 5 |x 2 3| 2 2 Graph V

• f(x) 5 x2 Graph M

• f(x) 5 |x | Graph D

• f(x) 5 2|x | Graph I

• f(x) 5 23x2 1 4, where x is integer Graph F 1

• f(x) 5 2__2 x2 1 2x Graph B

• f(x) 5 22|x 1 2| 1 4 Graph Q 3. Consider the graphs of functions that have an absolute minimum or an absolute maximum. (Do not consider Graphs A and C yet.) a. Sort the graphs into two groups based on the equations representing the functions and record the function letter in the table. Group 1

Group 2

B, F, M, T Quadratic

D, I, Q, V Linear Absolute Value

• How would you describe the graph of a linear piecewise function?

• Do you think all piecewise © 2012 Carnegie Learning

functions are linear in nature? Explain.

b. What is the same about all the functions in each group? Answers may vary. All the functions in Group 1 are made of smooth curves. All the functions in Group 2 are made of two straight lines. All the functions in Group 2 involve absolute value.

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Recognizing Algebraic and Graphical Representations of Functions

1.3  Recognizing Algebraic and Graphical Representations of Functions 

41

  41

Grouping

1

Ask a student to read the information before Question 4 aloud. Discuss and complete Question 4 as a class.

Congratulations! You have just sorted functions into two more function families.

1

The family of quadratic functions includes functions of the form f(x) 5 ax2 1 bx 1 c, where a, b, and c are real numbers, and a is not equal to 0. The family of linear absolute value functions includes functions of the form f(x) 5 a|x 1 b| 1 c, where a, b, and c are real numbers, and a is not equal to 0. 4. Go back to your table in Question 3 and identify which group represents quadratic functions and which group represents linear absolute value functions. See table.

Problem 4

PROBLEM 4

The remaining three graphs are defined as linear piecewise functions. Graphing calculator instructions are provided to demonstrate how to enter linear piecewise functions. Students will match the functions to the corresponding graphs.

Piecing things together

Analyze each of the functions shown. These functions represent the last three graphs of functions from the no absolute minimum and no absolute maximum category.

5

22x 1 10, 2` # x , 3 3#x,7 22x 1 18, 7 # x # 1`

• f(x) 5 4, Graph A

5

22,

• f(x) 5 __1x 2 2, 2

2` , x , 0 0#x,`

Graph S

Grouping Ask a student to read the information aloud. Discuss as a class.

5

1x 1 4, __ 2

• f(x) 5 23x 1 11, 1x 1 __ 1 __ 2

2

If your graphing calculator dœs not have an infinity symbol, you can enter the biggest number your calculator can compute using scientific notation. On mine, this is 9 × 1099. I enter this by pressing 9 2nd EE 99, which is shown on my calculator as 9E99.

2` # x , 2 2#x,3 3#x#`

Graph C These functions are part of the family of linear piecewise functions. Linear piecewise functions include functions that have equation changes for different parts, or pieces, of the domain.

So then for negative infinity, would I use -9 times 10 to the 99th power? How do I enter that?

© 2012 Carnegie Learning

Because these graphs each contain compound inequalities, there are additional steps required to use a graphing calculator to graph each function.

42 

  Chapter 1  Quantities and Relationships

1

Let’s graph the piecewise function:

5

22x 1 10, f(x) 5 4, f(x) 5 22x 1 18,

1

2` # x , 3 3#x,7 7 # x # 1`

You can use a graphing calculator to graph piecewise functions.

The first section is (–2x + 10).

Step 1: Press Y=. Enter the first section of the function within parentheses. Then press the division button.

The first part of the inequality is (-` # x).

Step 2: Press the ( key twice and enter the first part of the compound inequality within parentheses. Step 3: Enter the second part of the compound inequality within parentheses and then

Use the 2ND TEST to locate the inequality symbols. The TEST button is on the same key as the MATH button.

type two closing parentheses. Press GRAPH here to see the first section of the piecewise function. Step 4: Enter the remaining sections of the piecewise functions as Y2 and Y3.

The second part of the compound inequality is (x < 3). The entire line should look like: \Y1 = (-2x + 10)/((-` # x) (x < 3))

Grouping

© 2012 Carnegie Learning

Have students complete Questions 1 and 2 with a partner. Then share the responses as a class.

Guiding Questions for Share Phase, Questions 1 and 2 • How would you describe the

By completing the first piecewise function, you can now choose the graph that matches your graphing calculator screen. 1. Enter the remaining functions into your graphing calculator to determine the shapes of their graphs.

You will need these graphs again in Problem 5. Wait for it. . .

2. Match each function to its corresponding graph by writing the function directly on the graph that it represents.

1.3

Recognizing Algebraic and Graphical Representations of Functions

43

graph of a linear piecewise function?

• Do you think all piecewise functions are linear in nature? Explain.

1.3  Recognizing Algebraic and Graphical Representations of Functions 

  43

1 1

Congratulations! You have just sorted the remaining functions into one more function family. The family of linear piecewise functions includes functions that have equation changes for different parts, or pieces, of the domain.

© 2012 Carnegie Learning

You will need these graphs again in Problem 5. Wait for it. . .

44 

  Chapter 1  Quantities and Relationships

Problem 5 Students will paste their equations and linear, exponential, quadratic, linear absolute value, and linear piecewise graphs into appropriate graphic organizers and describe the graphical behavior of each function.

1 PROBLEM 5

We Are Family!

1

You have now sorted each of the graphs and equations representing functions into one of five function families: linear, exponential, quadratic, linear absolute value, and linear piecewise. 1. Glue your sorted graphs and functions to the appropriate function family Graphic Organizer on the pages that follow. Write a description of the graphical behavior for each function family.

Hang on to your graphic organizers. They will be a great resource moving forward!

Grouping Have students complete Question 1 with a partner. Then share the responses as a class.

Guiding Questions for Share Phase, Question 1 • Which families of functions contain curves?

• Which families of functions contain straight lines?

• Does a linear function contain an absolute minimum or absolute maximum? Explain.

• Does an exponential function

© 2012 Carnegie Learning

contain an absolute minimum or absolute maximum? Explain.

You’ve done a lot of work up to this point! You’ve been introduced to linear, exponential, quadratic, linear absolute value, and linear piecewise functions. Don’t worry—you don’t need to know everything there is to know about all of the function families right now. As you progress through this course, you will learn more about each function family. Be prepared to share your solutions and methods.

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Recognizing Algebraic and Graphical Representations of Functions

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  45

© 2012 Carnegie Learning

1

46 

  Chapter 1  Quantities and Relationships

1 1

graphical Behavior

Definition The family of linear functions includes functions of the form f(x) 5 mx 1 b, where m and b are real numbers.

Increasing / Decreasing: Linear functions can be increasing or decreasing over the entire domain. Maximum / Minimum: Linear functions have no maximum or minimum point. Curve / Line: Linear functions are made up of straight lines.

linear Functions

G

L

__

f(x) 5 x

2 f(x) 5 2 x 1 5 3

Domain: all real numbers

Domain: all real numbers U

© 2012 Carnegie Learning

O

f(x) 5 2x 1 3,

f(x) 5 2,

where x is an integer

where x is an integer

Domain: all integers

Domain: all integers

examples

1.3

Recognizing Algebraic and Graphical Representations of Functions

1.3  Recognizing Algebraic and Graphical Representations of Functions 

47

  47

1 1

graphical Behavior

Definition The family of exponential functions includes functions of the form f(x) 5 a ∙ b x, where a and b are real numbers, and b is greater than 0 but not equal to 1.

Increasing / Decreasing: Exponential functions can be increasing or decreasing over the entire domain. Maximum / Minimum: Exponential functions do not have a maximum or minimum. Curve / Line: Exponential functions are made of smooth curves.

exponential Functions

H

K

f(x) 5 2x,

1 25 f(x) 5 ( __ ) x

where x is an integer

2 Domain: all real numbers

Domain: all integers

P

( __ )

examples

48 

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

x f(x) 5 1 2 Domain: all real numbers

1 1

graphical Behavior

Definition The family of quadratic functions includes functions of the form, f(x) 5 ax2 1 bx 1 c where a, b, and c are real numbers, and a is not equal to 0.

Increasing / Decreasing: Quadratic functions can increase then decrease or decrease then increase over the domain. Maximum / Minimum: Quadratic functions can have either a maximum or a minimum depending on the shade of the parabola. Curve / Line: Quadratic functions are made of smooth curves.

Quadratic Functions

F

B

__

f(x) 5 23x2 1 4, where x is an integer

1 f(x) 5 2 x2 1 2x 2 Domain: all real numbers

Domain: all integers T

© 2012 Carnegie Learning

M

f(x) 5 x2

f(x) 5 x2 1 8x 1 12

Domain: all real numbers

Domain: all real numbers

examples 1.3

Recognizing Algebraic and Graphical Representations of Functions

1.3  Recognizing Algebraic and Graphical Representations of Functions 

49

  49

1 1

graphical Behavior

Definition The family of linear absolute value functions includes functions of the form f(x) 5 a|x 1 b| 1 c, where a, b, and c are real numbers, and a is not equal to 0.

Increasing / Decreasing: Linear absolute value function can increase then decrease, or decrease then increase over the domain. Maximum / Minimum: Linear absolute value functions can have either an absolute maximum or an absolute minimum. Curve / Line: Linear absolute value functions are made of straight lines.

linear Absolute Value Functions

D

I

f(x) 5 |x|

f(x) 5 2| x |

Domain: all real numbers

Domain: all real numbers

f(x) 5 22|x 1 2| 1 4

f(x) 5 |x 2 3| 2 2

Domain: all real numbers

Domain: all real numbers

examples

50 

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

V

Q

1 1

graphical Behavior

Definition The family of linear piecewise functions includes functions that have equation changes for different parts, or pieces, of the domain.

Increasing / Decreasing: Linear piecewise functions can have pieces that are increasing, decreasing, or constant. Maximum / Minimum: Linear piecewise functions may or may not have a maximum or a minimum. Curve / Line: Linear piecewise functions are made up of pieces that are straight lines and line segments.

linear Piecewise Functions

C

A

5

1x 1 4, __

2` # x , 2 2 f(x) 5 23x 1 11, 2 # x , 3 1 1x 1 __ __ 3#x#` 2 2 Domain: all real numbers

5

22x 1 10, 2` # x , 3 f(x) 5 4, 3#x,7 22x 1 18 7 # x # 1` Domain: all real numbers S

22, 2` , x , 0 f(x) 5 __ 1x 2 2, 0 # x , ` 2 Domain: all real numbers

© 2012 Carnegie Learning

5

examples

1.3

Recognizing Algebraic and Graphical Representations of Functions

1.3  Recognizing Algebraic and Graphical Representations of Functions 

51

  51

© 2012 Carnegie Learning

1

52 

  Chapter 1  Quantities and Relationships

Check for Students’ Understanding

1

Molly thinks the two graphs shown belong to the linear absolute value function family. Do you agree or disagree? Explain your reasoning. I disagree with Molly. The graphs are not functions. They do not pass the Vertical Line Test. Graph A

Graph B

y

y

7

7

6

6

5

5

4

4

3

3

2

2

1

1

0 21

1

2

3

4

5

6

7

8

9

0

x

21

2

3

4

5

6

7

8

9

x

  

© 2012 Carnegie Learning



1

1.3  Recognizing Algebraic and Graphical Representations of Functions 

  52A

© 2012 Carnegie Learning

1

52B 

  Chapter 1  Quantities and Relationships

1.4

Function Families for 200, Alex… Recognizing Functions by Characteristics Learning Goals In this lesson, you will:

• Recognize similar characteristics among function families. • Recognize different characteristics among function families. • Determine function types given certain characteristics.

Essential Ideas • Graphs described as smooth curves are

© 2012 Carnegie Learning

associated with an exponential function or a quadratic function. • Graphs described as straight lines are associated with a linear function or a linear absolute value function. • Graphs described as containing an absolute maximum or an absolute minimum are associated with a quadratic function or a linear absolute value function.

Mathematics Common Core Standards F-IF Interpreting Functions Understand the concept of a function and use function notation 1.  Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its

domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). Interpret functions that arise in applications in terms of the context 4.  For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Analyze functions using different representations 7.  Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. a.  Graph linear and quadratic functions and show intercepts, maxima, and minima.

53A

9.  Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

1

F-LE Linear, Quadratic, and Exponential Models Construct and compare linear, quadratic, and exponential models and solve problems 1.  Distinguish between situations that can be modeled with linear functions and with exponential functions. b.  Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. 2.  Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). A-CED Creating Equations Create equations that describe numbers or relationships 2.  Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Overview

© 2012 Carnegie Learning

Students are given characteristics of specific functions describing their graphical behaviors and they will name the possible function family/families to fit each description. Next, they are given characteristics of functions and will write an equation and sketch a graph to fit each description. Finally, using the scenarios from the first lesson of the chapter, they complete a table by listing the function family associated with each graph, and describing all of the attributes of the function with respect to increasing or decreasing, containing absolute minimums or absolute maximums, and a domain that is either discrete or continuous.

53B 

  Chapter 1  Quantities and Relationships

Warm Up

1

1. Sketch a graph that can be described as a linear function. Graph A y 8 6 4 2 0

28 26 24 22

2

4

6

8

x

22 24 26 28

2. Write an equation to fit this graph. y5x 3. Sketch a graph that can be described as a decreasing linear function. Graph B y 8 6 4 2 0

28 26 24 22

2

4

6

8

x

© 2012 Carnegie Learning

22 24 26 28

4. Write an equation to fit this graph. y 5 2x

1.4  Recognizing Functions by Characteristics 

  53C

5. Sketch a graph that can be described as a decreasing linear function with a slope of 3.

1

Graph C y 8 6 4 2 0

28 26 24 22

2

4

6

8

x

22 24 26 28

6. Write an equation to fit this graph. y 5 23x 7. Sketch a graph that can be described as a decreasing linear function with a slope of 3 and a y-intercept of 25. Graph D y 8 6 4 2 0

28 26 24 22

2

4

6

8

x

22

26 28

8. Write an equation to fit this graph. y 5 23x 2 5

53D 

  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

24

Function Families for 200, Alex…

1

1.4

recognizing Functions by Characteristics leArning gOAlS In this lesson, you will:

• Recognize similar characteristics among function families. • Recognize different characteristics among function families. • Determine function types given certain characteristics.

S

© 2012 Carnegie Learning

© 2012 Carnegie Learning

ince the debut of television in the early 1950s, Americans have had a love/hate relationship with the game show. One of the original game shows that aired was Name that Tune. The game was played when two contestants were given a clue about a song. Then, one opponent would “bid” that the song could be named in a certain number of notes played. The other opponent could either beat the number of notes “bid” from the opponent, or they could tell their opponent to “name that tune!” Do you like game shows? If so, what are your favorite game shows?

53

1.4  Recognizing Functions by Characteristics 

  53

Problem 1

1

Several functions are described and students are asked to identify the function family/ families best associated with each description.

1

PROBLEM 1

Grouping Have students complete Questions 1 and 2 with a partner. Then share the responses as a class.

Guiding Questions for Share Phase, Questions 1 and 2 • What does the graph of a smooth curve look like?

• What is does a graph that is

name that Function!

1. Use the characteristic(s) provided to choose the appropriate function family or families from the word box shown.

linear function family

exponential function family

quadratic function family

linear absolute value function family

a. The graph of this function family: • is a smooth curve. exponential function or quadratic function b. The graph of this function family: • is made up of one or more straight lines. linear absolute value function or linear function c. The graph of this function family: • increases or decreases over the entire domain. linear function or exponential function d. The graph of this function family: • has a maximum or a minimum. quadratic function or linear absolute value function

not a smooth curve look like?

• What does a function that increases over the entire domain look like?

• What does a function that decreases over the entire domain look like? • What does a function that has a maximum look like?

• What does a function that

2. A second characteristic has been added to the graphical description of each function. Name the possible function family or families given the graphical characteristics. a. The graph of this function family:

• has an absolute minimum or absolute maximum, and • is a smooth curve. quadratic function b. The graph of this function family:

• either increases or decreases over the entire domain, and • is made up of a straight line. linear function

• What is the difference between the appearance of a function that has a maximum and a function that has an absolute maximum?

• What does a function that has an absolute minimum look like?

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  Chapter 1  Quantities and Relationships

© 2012 Carnegie Learning

has an absolute maximum look like?

1

c. The graph of this function family:

1

• is a smooth curve, and • either increases or decreases over the entire domain. exponential function d. The graph of this function family:

• has either an absolute minimum or an absolute maximum • has symmetry, and • is made up of 2 straight lines linear absolute value function

© 2012 Carnegie Learning

Each function family has certain graphical behaviors with some behaviors common among different function families. Notice, the more specific characteristics that are given, the more specifically you can Name that Function!

1.4 Recognizing Functions by Characteristics

1.4  Recognizing Functions by Characteristics 

55

  55

Problem 2

1

Students are given several characteristics of a function and asked to write an equation and sketch a graph for each set of criteria. Students should work in pairs in such a way that both partners are able to simultaneously sketch their own graph and only after the graphs are sketched and the equations are written should they share their responses with each other to discuss similarities and differences.

1

PROBLEM 2

graph that Function!

1. Use the given characteristics to create an equation and sketch a graph. Use the equations given in the box as a guide. Then share your graph with your partner. Discuss similarities and differences between your graphs. When creating your equation, use a, b, and c values that are any real numbers between 23 and 3. Do not use any functions that were used previously in this chapter. Answers may vary.

Linear function f(x) 5 mx 1 b Exponential function f(x) 5 a . bx Quadratic function f(x) 5 ax2 1 bx 1 c Linear Absolute Value Function f(x) 5 a|x 1 b| 1 c

Questions 1 and 2 with a partner. Then share the responses as a class.

a. Create an equation and sketch a graph that:

• is a function, • is exponential,

Don’t forget about the function family graphic organizers you created if you need some help.

• is continuous, and • is decreasing.

Guiding Questions for Share Phase, Questions 1 and 2 • If the equation is described

Equation:

( __ )

f(x) 5 1 4

y 8 6

as a function, what does that imply?

• If the equation is described an exponential function, what does that imply? • If the equation is described as a continuous function, what does that imply?

• If the equation is described as a decreasing function, what does that imply?

• Is there more than one correct equation and graph that fits this list of criteria?

• If the equation is described contains a minimum, what does that imply?

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  Chapter 1  Quantities and Relationships

x

4 2 0

28 26 24 22

2

4

6

8

x

22 24 26 28

© 2012 Carnegie Learning

Grouping • Have students complete

• If the equation is described is discrete, what does that mean to you?

• If the equation is described as a linear absolute value function, what does that imply?

b. Create an equation and sketch a graph that:

• has a minimum, • is discrete, and

Equation: f(x) 5 |x 2 1| 1 2 where x is an integer y

as linear, what does that imply? Is it a function?

6 5

• If the equation is described

4

as increasing, what does that imply?

3 2

• If the equation is described

1 0

24 23 22 21

• Is it possible to compose

• is increasing, and

Equation: f(x) 5 2x 2 1 where x is an integer y 8 6 4

• What is an example of

© 2012 Carnegie Learning

• How is your list of characteristics different than your partner’s list of characteristics?

x

• is a function.

on your list contradict each other?

correct sketch that matches all of the characteristics on your list?

4

• is discrete,

• Do any of the characteristics

• Is there more than one

3

c. Create an equation and sketch a graph that:

• is linear,

two characteristics that contradict each other?

2

22

• How many characteristics did you list for your function? a list of characteristics that do not describe a function? Explain.

1

21

• If the equation is described as quadratic what does that imply? Is it a function?

1

• is a linear absolute value function.

• If the equation is described

as continuous, what does that imply? Is it a function?

1

Is the domain the same or different for each function?

2 0

28 26 24 22

2

4

6

8

x

22 24 26 28

1.4 Recognizing Functions by Characteristics

1.4  Recognizing Functions by Characteristics 

57

  57

1

d. Create an equation and sketch a graph that:

1

• is continuous, • has a maximum, • is a function, and • is quadratic. Equation: f(x) 5 2x2 y 8 6 4 2 0

28 26 24 22

2

4

6

8

x

22 24 26 28

e. Create an equation and sketch a graph that:

• is not a function, • is continuous, and • is a straight line. Equation: x 5 3 y 8 6 4 2 0

28 26 24 22

2

4

6

8

x

22 24

58

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  Chapter 1  Quantities and Relationships

Quantities and Relationships

© 2012 Carnegie Learning

26 28

1

2. Create your own function. Describe certain characteristics of the function and see if your partner can sketch it. Then try to sketch your partner’s function based on characteristics provided.

1

y

0

Talk the Talk

Talk the Talk

introduction. Discuss and complete Question 1 as a class.

© 2012 Carnegie Learning

Students use the scenarios from the first lesson of this chapter to complete a table which lists the family of functions associated with each scenario, and the characteristics of the graphical behaviors of the graph of each scenario.

Grouping • Ask a student to read the

• Have students complete © 2012 Carnegie Learning

x

Throughout this chapter, you were introduced to five function families: linear, exponential, quadratic, linear absolute value, and linear piecewise. Let’s revisit the first lesson in this chapter: A Picture Is Worth a Thousand Words. Each of the scenarios in this lesson represents one of these function families. 1. Describe how each scenario represents a function. For every input value there is exactly one output value.

Recall that each of the graphs representing the scenarios was drawn with either a continuous line or a continuous smooth curve to model the problem situation.

2. Complete the table to describe each scenario. a. Identify the appropriate function family. b. Based on the problem situation, identify whether the graph of the function should be discrete or continuous. c. Create a sketch of the mathematical model. d. Identify the graphical behavior.

Question 2 with a partner. Then share the responses as a class. 1.4  Recognizing Functions by Characteristics 

Guiding Questions for Discuss Phase, Question 1 • How many output values are there for every input value?

Guiding Questions for Share Phase, Question 2 • How many of the scenarios are associated with a linear function? • How many of the scenarios are associated with a quadratic function? • How many of the scenarios are associated with an exponential function? • How many of the scenarios are associated with a linear piecewise function? • How many of the scenarios are associated with an absolute value function? • How many of the scenarios can be described as continuous?

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© 2012 Carnegie Learning

  Chapter 1  Quantities and Relationships

Linear Piecewise Function

Exponential Function

Can’t Wait to Hit the Slopes!

It’s Magic

© 2012 Carnegie Learning

Exponential Function

Smart Phone, but Is It a Smart Deal?

Discrete

Continuous

Discrete

y

y

y

Number of Cuts

Graph D

Time (minutes)

Graph G

Time (weeks)

Graph B

Time (minutes)

Graph H

x

x

x

x

can be described as decreasing? No minimum or maximum

No minimum or maximum

No minimum or maximum

No minimum or maximum

Absolute Minimum or Absolute Maximum

• How many of the scenarios

y

can be described as increasing?

Continuous

• How many of the scenarios

Linear Function

Scenario

Sketch of the Mathematical Model

contain a minimum?

Decreasing

Combination of increasing and constant

Increasing

Decreasing

Increasing, Decreasing, Constant, or Combination

Graphical Behavior

• How many of the scenarios

Something’s Fishy

Function Family

Domain of the Real-World Situation: Discrete or Continuous

contain a maximum?

Water (gallons)

60 

1

Interest (dollars)

• How many of the scenarios

Distance (feet)

can be described as discrete?

Length of Each Piece of Rope (feet)

1 • How many of the scenarios

  Chapter 1  Quantities and Relationships

27/03/14 3:17 PM

Linear Piecewise Function

Linear Absolute Value Function

Jelly Bean Challenge

Linear Function

Music Club

A Trip to School

Quadratic Function

Scenario Baton Twirling

Function Family

© 2012 Carnegie Learning

Discrete

Continuous

Discrete

Continuous

Domain of the Real-World Situation: Discrete or Continuous

y

y

y

y

Number of Jelly Beans Guessed

Graph C

Time (minutes)

Graph E

Number of Songs

Graph A

Time (seconds)

Graph F

x

x

x

x

Sketch of the Mathematical Model

Height of Baton (feet) Cost (dollars) Distance Traveled (miles)

8044_TIG_CH01_001-070.indd 61 Number of Jelly Beans the Guess Is off By

© 2012 Carnegie Learning

Absolute minimum

No minimum or maximum

No minimum or maximum

Absolute maximum

Absolute Minimum or Absolute Maximum

Decreasing and increasing

Combination of increasing and constant

Increasing

Increasing and decreasing

Increasing, Decreasing, Constant, or Combination

Graphical Behavior

1

1

Be prepared to share your solutions and methods.

1.4  Recognizing Functions by Characteristics    61

1.4  Recognizing Functions by Characteristics 

  61

27/03/14 3:17 PM

Check for Students’ Understanding

1

List possible characteristics for each of the following graphs. 1. Graph A y 7 6 5 4 3 2 1 0 21

1

2

3

4

5

6

7

8

9

x

• The graph is not a function. • The graph is continuous. • The graph is piecewise. • The domain of the graph is all x-values greater than or equal to 1 and less than or equal to 4.

© 2012 Carnegie Learning

• The range of the graph is all y-values greater than or equal to 21 and less than or equal to 5.

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  Chapter 1  Quantities and Relationships

2.

1

Graph B y 7 6 5 4 3 2 1 0 21

1

2

3

4

5

6

7

8

9

x

• The graph is not a function. • The graph is continuous. • The graph is piecewise. • The domain of the graph is all x-values greater than or equal to 4 and less than or equal to 7.

© 2012 Carnegie Learning

• The range of the graph is all y-values greater than or equal to 21 and less than or equal to 5.

1.4  Recognizing Functions by Characteristics 

  62A

© 2012 Carnegie Learning

1

62B 

  Chapter 1  Quantities and Relationships

Chapter 1 Summary

1

Key terMS • dependent quantity (1.1) • independent quantity (1.1) • relation (1.2) • domain (1.2) • range (1.2) • function (1.2) • Vertical Line Test (1.2) • discrete graph (1.2)

1.1

• continuous graph (1.2) • function notation (1.3) • increasing function (1.3) • decreasing function (1.3) • constant function (1.3) • function family (1.3) • linear functions (1.3) • exponential functions (1.3)

• absolute minimum (1.3) • absolute maximum (1.3) • quadratic functions (1.3) • linear absolute value functions (1.3)

• linear piecewise functions (1.3)

identifying the Dependent and independent Quantities for a Problem Situation Many problem situations include two quantities that change. When one quantity depends on another, it is said to be the dependent quantity. The quantity that the dependent quantity depends upon is called the independent quantity.

Example Caroline makes $8.50 an hour babysitting for her neighbors’ children after school and on the weekends.

© 2012 Carnegie Learning

The dependent quantity is the total amount of money Caroline earns based on the independent quantity. The independent quantity is the total number of hours she babysits.

63

1.1

1

labeling and Matching a graph to an Appropriate Problem Situation Graphs relay information about data in a visual way. Connecting points on a coordinate plane with a line or smooth curve is a way to model or represent relationships. The independent quantity is graphed on the horizontal or x-axis, while the dependent quantity is graphed on the vertical, or y-axis. Graphs can be straight lines or curves, and can increase or decrease from left to right. When matching with a problem situation, consider the situation and the quantities to interpret the meaning of the data values.

Example Pedro is hiking in a canyon. At the start of his hike, he was at 3500 feet. During the first 20 minutes of the hike, he descended 500 feet at a constant rate. Then he rested for half an hour before continuing the hike at the same rate. Time is the independent quantity and elevation is the dependent quantity.

Elevation (feet)

y

Time (minutes)

Analyzing and Comparing types of graphs Looking for patterns can help when sorting and comparing graphs. A discrete graph is a graph of isolated points. A continuous graph is a graph of points with no breaks in it. The points are connected by a straight line or smooth curve. Some graphs show vertical symmetry (if a vertical line were drawn through the middle of the graph the image is the same on both sides). Other possible patterns to look for include: only goes through two quadrants, always increasing from left to right, always decreasing from left to right, straight lines, smooth curves, the graph goes through the origin, the graph forms a U shape, the graph forms a V shape.

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© 2012 Carnegie Learning

1.2

x

Examples A discrete graph with vertical symmetry:

y

y

0

1.2

1

A continuous graph with a smooth curve increasing from left to right:

x

x

0

Using the Vertical line test When Determining Whether a relation is a Function A relation is the mapping between a set of input values called the domain and a set of output values called the range. A function is a relation between a given set of elements for which each element in the domain has exactly one element in the range. The Vertical Line Test is a visual method used to determine whether a relation represented as a graph is a function. To apply the Vertical Line Test, consider all of the vertical lines that could be drawn on the graph of a relation. If any of the vertical lines intersect the graph of the relation at more than one point, then the relation is not a function.

Examples

© 2012 Carnegie Learning

A line drawn vertically through the graph touches more than one point. The graph does not represent a function.

y

y

0

A line drawn vertically through the graph only touches one point. The graph represents a function.

x

0

x

Chapter 1 1  Summary Summary 

  65

1.3

1

Writing equations Using Function notation Functions can be represented in a number of ways. An equation representing a function can be written using function notation. Function notation is a way of representing functions algebraically. This form allows you to more efficiently identify the independent and dependent quantities. The function f(x) is read as “f of x” and indicates that x is the independent variable. Remember, you know an equation is a function because for each independent value there is exactly one dependent value associated with it.

Example Write this equation using function notation: y 5 2x 1 5 The dependent variable (y), defined by f, is a function of x, the independent variable. f(x) 5 2x 1 5

1.3

Determining Whether a graph represents a Function that is increasing, Decreasing, or Constant A function is described as increasing when both the independent and dependent variables are increasing. If a function increases across the entire domain, then the function is called an increasing function. A function is described as decreasing when the dependent variable decreases as the independent variable increases. If a function decreases across the entire domain, then the function is called a decreasing function. If the dependent variable of a function does not change or remains constant over the entire domain, then the function is called a constant function.

Example The function shown in the graph is a decreasing function because the dependent variable (y) decreases as the independent variable (x) increases.

0

66 

  Chapter 1  Quantities and Relationships

x

© 2012 Carnegie Learning

y

1.3

Determining Whether a graph represents a Function with an Absolute Maximum or Absolute Minimum

1

A function has an absolute minimum if there is a point that has a y-coordinate that is less than the y-coordinates of every other point on the graph. A function has an absolute maximum if there is a point that has a y-coordinate that is greater than the y-coordinates of every other point on the graph.

Example The function shown in the graph has an absolute maximum because the y-coordinate of the point (0, 5) is greater than the y-coordinates of every other point on the graph. y

0

1.3

x

Distinguishing Between Function Families A function family is a group of functions that share certain characteristics. The family of linear functions includes functions of the form f(x) 5 ax 1 b, where a and b are real numbers.

© 2012 Carnegie Learning

The family of exponential functions includes functions of the form f(x) 5 a · bx, where a and b are real numbers, and b is greater than 0, but not equal to 1. The family of quadratic functions includes functions of the form f(x) 5 ax2 1 bx 1 c, where a, b, and c are real numbers, and a is not equal to 0. The family of linear absolute value functions includes functions of the form f(x) 5 a|x 1 b| 1 c, where a, b, and c are real numbers, and a is not equal to 0. The family of linear piecewise functions includes functions that has an equation that changes for different parts, or pieces, of the domain.

Chapter 1 1  Summary Summary 

  67

Examples

1

The function is quadradic.

The function is linear. y

y

0

x

y

0

0

x

y

x

The function is exponential.

0

x

The function is linear absolute value.

0

68 

  Chapter 1  Quantities and Relationships

x

The function is linear piecewise.

© 2012 Carnegie Learning

y

1.4

identifying a Function given its Characteristics

1

Certain characteristics of a graph such as whether it increases or decreases over its domain, has an absolute minimum or maximum, is a smooth curve or not, or other characteristics, can help when determining if a function is linear, exponential, quadratic, or linear absolute value.

Example The graph of a function f(x) is a smooth curve and has an absolute minimum. Thus, the function is quadradtic.

1.4

graphing a Function given its Characteristics Use the given characteristics to create an equation and sketch a graph. Linear function f(x) 5 mx 1 b Exponential function f(x) 5 a · bx Quadratic function f(x) 5 ax2 1 bx 1 c Linear Absolute Value Function f(x) 5 a|x 1 b| 1 c

Example Create an equation and sketch a graph that has:

• an absolute maximum • and is a linear absolute value function. f(x) 5 2|x|

© 2012 Carnegie Learning

y

0

x

Chapter 1 1  Summary Summary 

  69

1.4

1

identifying a Function given its graph Certain characteristics of a graph such as whether it increases or decreases over its domain, has an absolute minimum or maximum, is a smooth curve or not, or other characteristics, can help when determining if a graph represents a linear, exponential, quadratic, or linear absolute value function.

Example The graph shown is a linear absolute value function. It is discrete. The graph decreases and then increases. It has an absolute minimum. y

x

© 2012 Carnegie Learning

0

70 

  Chapter 1  Quantities and Relationships

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Quantities and relationships

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