Linear Functions Animal trackers are experts at identifying animals by their footprints. From tracks they can also sometimes tell what direction the animal was heading, the age of the animal, and even if it is male or female! Can you guess what animal made the tracks shown in the picture?.1 Patterns, Patterns, Patterns Developing Sequences of Numbers from Diagrams and Contexts... 3. Every Graph Tells a Story Describing Characteristics of Graphs...55.3 To Be or Not To Be a Function? Defining and Recognizing Functions...67. Scaling a Cliff Linear Functions... 85.5 U.S. Shirts Using Tables, Graphs, and Equations, Part 1... 91.6 Hot Shirts Using Tables, Graphs, and Equations, Part...99.7 What, Not Lines? Introduction to Non-Linear Functions... 107 1
Chapter Overview This chapter develops the understanding that a function is a rule that assigns to each input exactly one output. Realworld problems, tables, graphs, and equations are used to model linear function relationships. Non-linear functions are introduced to contrast with linear functions. Lesson CCSS Pacing Highlights Models Worked Examples Error Analysis Talk the Talk Technology.1 Developing Sequences of Numbers from Diagrams and Contexts 8.F.1 1 This lesson presents sequences to develop an understanding for growth patterns that either increase or decrease. Questions provide contexts to help students determine each sequence. X X X. Describing Characteristics of Graphs 8.F.1 8.F.5 1 This lesson provides a sorting activity of various graphical displays to develop student understanding of characteristics of graphs. Questions ask students to distinguish between discrete and continuous, linear and non-linear, and increasing and decreasing. X This lesson defines relations and develops the understanding of functions..3 Defining and Recognizing Functions 8.F.1 8.F. 8.F.3 8.F.5 1 Questions ask students to recognize if relations presented as mappings, sets of ordered pairs, tables, equations, and graphs are functions. A sorting activity utilizing graphs is used to solidify student understanding of functions. X X X X. Linear Functions 8.F.1 8.F. 8.F.3 8.F. 8.F.5 1 This lesson focuses on the development of linear functions using a real-world situation. Questions ask students to represent linear functions using input-output tables and graphs, and then to identify the independent and dependent variables. X 1A Chapter Linear Functions
Lesson CCSS Pacing Highlights Models Worked Examples Error Analysis Talk the Talk Technology.5 Using Tables, Graphs, and Equations, Part 1 8.F.1 8.F. 8.F.3 8.F. 8.F.5 1 This lesson continues the development of linear functions. Questions ask students to model a realworld situation using a sentence, a table, a graph, and an equation. A graphic organizer is provided to summarize the advantages and disadvantages of each representation. X X.6 Using Tables, Graphs, and Equations, Part 8.F.1 8.F. 8.F.3 8.F. 8.F.5 1 This lesson continues the development of linear functions; this time using a realworld situation with rational numbers and multiple representations. Questions ask students to compare and analyze problem situations from the previous lesson, and to estimate before performing actual calculations. X.7 Introduction to Non- Linear Functions 8.F.1 8.F. 8.F.5 1 This lesson explores non-linear functions in contrast to the linear functions that have been previously presented in earlier lessons. Questions ask students to complete tables, graphs, and analyze each. X Chapter Linear Functions 1B
Skills Practice Correlation for Chapter Lesson Problem Set Objective(s).1 Developing Sequences of Numbers from Diagrams and Contexts Vocabulary 1 6 Write or draw terms in sequences 7 1 Draw pictures and use sequences to solve problems 13 18 Write sequences to solve problems Vocabulary. Describing Characteristics of Graphs 1 6 7 1 Tell whether graphs are discrete or continuous, and increasing, decreasing, both, or neither Tell whether graphs are linear or nonlinear, and increasing, decreasing, both, or neither 13 18 Tell stories to describe graphs Vocabulary.3 Defining and Recognizing Functions 1 6 Write ordered pairs and tell whether the relations are functions 7 1 Determine whether graphs represent functions 13 0 Determine whether equations are functions 1 8 Determine whether situations are functions Vocabulary. Linear Functions 1 6 Identify dependent and independent quantities in problem situations 7 1 Complete tables and graphs for problem situations 1 Complete tables to model problem situations.5 Using Tables, Graphs, and Equations, Part 1 5 8 Create graphs of data in tables 9 1 Use given information to answer questions 13 16 Use given information to answer questions 17 Write equations in two variables to model situations 1C Chapter Linear Functions
Lesson Problem Set Objective(s).6.7 Using Tables, Graphs, and Equations, Part Introduction to Non- Linear Functions Vocabulary 1 6 Estimate and calculate expressions 7 1 Complete tables and answer questions 13 18 Use given equations to answer questions 19 Use given graphs to answer questions Vocabulary 1-8 Complete tables and graph given functions Chapter Linear Functions 1D
Chapter Linear Functions
Patterns, Patterns, Patterns Developing Sequences of Numbers from Diagrams and Contexts Learning Goals In this lesson, you will: Write sequences of numbers generated from the creation of diagrams and written contexts. State varying growth patterns of sequences. Key Terms sequence term ellipsis Essential Ideas A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term in a sequence is an individual number, figure, or letter in the sequence. A diagram can be used to show how each term changes as the sequence progresses. There are many different patterns that can generate a sequence of numbers. Some possible patterns are: adding or subtracting by the same number each time. multiplying or dividing by the same number each time. adding by a different number each time, with the numbers being part of a pattern. alternating between adding and subtracting. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output..1 Developing Sequences of Numbers from Diagrams and Contexts 3A
Overview Sequences and terms in a sequence are introduced. Sequences that involve numbers, figures, and letters are provided, and students determine the next term in each sequence. Different contexts and diagrams are provided for students to develop an understanding of sequences. It is important that all students discuss all problems, as each problem demonstrates a different type of pattern. Next, students will write sequences from written contexts only. In the last problem, students summarize the ten sequences generated in this lesson, first by documenting whether each sequence was increasing or decreasing, and then by defining the growth pattern of each sequence. 3B Chapter Linear Functions
Warm Up 1. List six consecutive numbers. Six consecutive numbers are 3,, 5, 6, 7 and 8.. List six consecutive even numbers. Six consecutive even numbers are,, 6, 8, 10 and 1. 3. List six consecutive multiples of seven. Six consecutive multiples of seven are 7, 1, 1, 8, 35 and.. List six consecutive multiples of five that are decreasing. Six consecutive multiples of five that are decreasing are 35, 30, 5, 0, 15 and 10. 5. List six consecutive prime numbers. Six consecutive prime numbers are, 3, 5, 7, 11 and 13..1 Developing Sequences of Numbers from Diagrams and Contexts 3C
3D Chapter Linear Functions
Patterns, Patterns, Patterns Developing Sequences of Numbers from Diagrams and Contexts Learning Goals In this lesson, you will: Write sequences of numbers generated from the creation of diagrams and written contexts. State varying growth patterns of sequences. Key Terms sequence term ellipsis Legend tells us that when the inventor of the game of chess showed his work to the emperor, the emperor was so pleased that he allowed the inventor to choose any prize he wished. So the very wise inventor asked for the following: 1 gold coin for the first square on the chess board, gold coins for the second square, coins for the third, and so on up to the 6th square. The emperor, not as wise as the inventor, quickly agreed to such a cheap prize. Unfortunately, the emperor could not afford to pay even the amount for just the 3nd square:,9,967,95 gold coins! How many gold coins would the emperor have to pay for just the 10th square? 0th square? What pattern did you use to calculate your answers?.1 Developing Sequences of Numbers from Diagrams and Contexts 3
Problem 1 Sequence, term in a sequence, and ellipsis are defined. Examples of sequences that involve numbers, figures, and letters are provided. Students determine the next term in each sequence. Grouping Ask a student to read the introduction to Problem 1 aloud. Discuss the definitions and worked example as a class. Have students complete Questions 1 through with a partner. Then share the responses as a class. Problem 1 Sequences The inventor from the story used his knowledge of sequences to his advantage to gain riches. A sequence is a pattern involving an ordered arrangement of numbers, geometric figures, letters, or other objects. A term in a sequence is an individual number, figure, or letter in the sequence. Here are some examples of sequences. Sequence A:,, 6, 8, 10, 1, Sequence B:,,,, Sequence C: A, B, C, D, E, F, G, Sequence D:,,,,, Discuss Phase, Problem 1 Explain what a sequence is in your own words. Do all sequences have terms? How many terms are in a sequence? Is the term in a sequence a number, a figure, or a letter? How have you heard the word sequence or term used outside of the math class? How is the usage of these words outside of the math class related to their meaning in math class? Share Phase, Questions 1 through How did you determine the next term in Sequence A? How could you determine the next term in Sequence B? Often, only the first few terms of a sequence are listed, followed by an ellipsis. An ellipsis is three periods, which stand for and so on. 1. What is the next term in Sequence A? The next term is 1.. What is the third term in Sequence B? The third term is a pentagon. 3. What is the twenty-fifth term in Sequence C? The twenty-fifth term is the letter Y.. What is the twelfth term in Sequence D? The twelfth term is an arrow pointing upward. How could you determine the next term in Sequence C? How could you determine the next term in Sequence D? How did you determine the third term in Sequence B? How did you determine the twenty-fifth term in Sequence C? How did you determine the twelfth term in Sequence D? Was it necessary to list all the terms leading up to the term you were trying to determine? Chapter Linear Functions
Note In mathematical usage, the terms sequence and series have different meanings. This chapter addresses sequences, not series; it would be inaccurate to use sequence and series interchangeably in the mathematical sense of the terms. Problem Students write a sequence to represent designing a bead necklace. The terms of the sequence follow the pattern of even numbers. The context helps students make sense of this pattern. Problem Designing a Bead Necklace Emily is designing a necklace by alternating black and green beads. To create her necklace, she performs the following steps. Step 1: She starts with one black bead. Step : Next, she places one green bead on each side of the black bead. Step 3: Then, she places two black beads on each side of the green beads. Step : Then, she places three green beads on each side of the black beads. Step 5 and 6: She continues this pattern two more times, alternating between black and green sets of beads. Grouping Have students complete Problems through 5 with a partner. Then share the responses as a class. Share Phase, Problem Explain how many black beads and green beads are added to the necklace during each of the first 6 steps. Explain how you can determine how many beads would be added to the necklace in the next step without drawing the beads. Will there be more green or black beads on the necklace when you are done? Explain? How did you determine the values in your sequence? 1. Write the first six terms in the sequence that represents this situation. Make sure each term indicates the total number of beads on the necklace after Emily completes that step. Finally, explain how you determined the sequence. The sequence is 1, 3, 7, 13, 1, 31. The number of beads increases by, then, then 6, then 8, and then 10. If you need help, draw the sequence on the necklace. Did you draw all the beads on the necklace to complete the sequence? If so, did you count all the beads or just the newly-added beads to determine the next term in the sequence? If not, explain your thinking process. How much did the numbers in your sequence grow by each time? What do those numbers represent in the context of the problem? Why are all those numbers even? Why are all the numbers in your sequence odd?.1 Developing Sequences of Numbers from Diagrams and Contexts 5
Problem 3 Students write a sequence to represent crafting toothpick houses. The terms of the sequence increase by a constant value; however, the initial term is one more than the amount of increase between consecutive terms. The context helps students make sense of this discrepancy between the initial value and the amount of increase between consecutive terms. Problem 3 Crafting Toothpick Houses Ross is crafting toothpick houses for the background of a diorama. He creates one house and then adds additional houses by adjoining them as shown. A diorama is a three-dimensional natural scene in which models of people, animals, or plants are seen against a background. Share Phase, Problem 3 How many houses require the use of 6 new toothpicks? How many houses require the use of 5 new toothpicks? What operation was used to determine the terms in your sequence? What do you notice about every other term in your sequence? Is there a pattern? Why is every other term in your sequence even? Why is every other term in your sequence odd? Did you draw all the houses to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process. If a house is comprised of six toothpicks, then why doesn t your sequence grow by six each time? 1. Write the first eight terms in the sequence that represents this situation. The first term should indicate the number of toothpicks used for one house. The second term should indicate the total number of toothpicks needed for two houses, and so on. Explain your reasoning. The sequence is 6, 11, 16, 1, 6, 31, 36, 1. The first house uses six toothpicks. The remaining houses use only an additional five toothpicks each because they each share a side with the previous house.. How is the number of toothpicks needed to build each house represented in the sequence? The sequence starts with a 6. That is the number of toothpicks needed to build the first house. The sequence increases by 5 each time because that is the number of toothpicks needed for each house built after the first house. 6 Chapter Linear Functions
Problem Students write a sequence to represent a card trick. This is the only problem in this lesson where the sequences are decreasing. The first sequence decreases by a constant value of two. The second sequence decreases by a pattern of odd numbers, with the odd numbers being the terms in the first sequence. The context helps students make sense of why the sequences are decreasing and their patterns of decrease. Problem Taking Apart a Card Trick Matthew is performing a card trick. It is important that he collect the cards shown in a particular order. Each turn, he collects all of the cards in the right-most column, and all the cards in the bottom row. Share Phase, Problem How did you determine the terms in the first sequence? How did you determine the terms in the second sequence? Did you use the diagram of cards to complete the sequence? If so, explain how you wrote the sequence from the diagram. If not, explain. If the original diagram has eight columns and six rows, why are only 13 cards removed instead of 1 cards to determine the first term in the first sequence? How is the decrease by in the first sequence represented in the diagram? How is the decrease in the second sequence represented in the diagram? 1. Write a sequence to show the number of cards removed during each of the first five turns. The sequence is 13, 11, 9, 7, 5.. Write a sequence to show the number of cards remaining after each of the first five turns. The sequence is 35,, 15, 8, 3. 3. What pattern is shown in each sequence? In the first sequence, the number of cards decreases by each time. In the second sequence, the number of cards decreases by the values shown in the first sequence starting with 11. Add the first terms of the two sequences together, the second terms, etc. and use the sum in each instance to create a new sequence. What familiar pattern do you notice in the new sequence?.1 Developing Sequences of Numbers from Diagrams and Contexts 7
Problem 5 Students write a sequence to represent triangular arrangements of pennies. The terms of the sequence increase by adding consecutive integers. The context helps students make sense of this pattern. Problem 5 Arranging Pennies Lenny is making arrangements with pennies. He has made three penny arrangements and now he wants to make five more arrangements. Each time he adds another arrangement, he needs to add one more row to the base than the previous row in the previous arrangement. Share Phase, Problem 5 How did you determine the first eight terms in your sequences? How many new rows of pennies are added in each term? Did you draw all of the pennies to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process. How is the increase by consecutive numbers represented in the diagram? 1. Write the first eight terms in the sequence that represents this situation. Each term should indicate the total number of pennies in each arrangement. Explain your reasoning. The sequence is 1, 3, 6, 10, 15, 1, 8, 36. The number of pennies increases by, 3,, 5, 6, 7, and 8. Every increase is by one more penny than the number contained in the previous arrangement s bottom row.. Explain why the pattern does not increase by the same amount each time. Every time a new row is added on to the base, the triangle gets wider and takes one more penny than the previous row. 8 Chapter Linear Functions
Problem 6 Students write a sequence to represent building stairs. This is the only problem in this lesson which requires students to interpret three-dimensional drawings. The terms of the sequence increase by adding consecutive odd integers. The context helps students make sense of this pattern. Grouping Have students complete Problems 6 through 11 with a partner. Then share the responses as a class. Misconception Students may have difficulty drawing the stacked cubes and counting the number of exposed faces. Some students may benefit from using actual cubes to build the models. Problem 6 Building Stairs A configuration is another way of saying an arrangement of things. Dawson is stacking cubes in configurations that look like stairs. Each new configuration has one additional step. 1. Write the first five terms in the sequence that represents this situation. Each term should indicate the number of faces shown from the cubes shown. The bottom faces are not shown. The first cube has 5 shown faces. Explain your reasoning. The sequence is 5, 1, 1, 3, 5. The number of exposed faces increases by 7, then 9, then 11, and then 13. Every increase is by two more than in the previous configuration.. Predict the number of shown faces in a stair configuration that is 7 cubes high. Show your work. The number of exposed faces would be 77. Continuing the pattern: 5 1 15 5 60, 60 1 17 5 77. Share Phase, Problem 6 How did you determine the first five terms in your sequence? How many additional exposed faces appear between the first and second diagram? How many additional exposed faces appear between the second and the third diagram? Did you draw or construct the stacked cubes to complete the sequence? If so, explain how you wrote the sequence from your diagram or model. If not, explain your thinking process. How were you able to organize your counting process when counting the number of exposed faces of the cubes? What patterns did you notice? How much does the increase between the terms grow each time? The increase between the terms grows by two every time. How is this two represented in the diagram?.1 Developing Sequences of Numbers from Diagrams and Contexts 9
Problem 7 Students write a sequence to represent arranging classroom tables. The terms of the sequence increase by a constant value; however, the initial term is two more than the amount of increase between consecutive terms. The context helps students make sense of this discrepancy between the initial value and the amount of increase between consecutive terms. This problem is similar to Problem 3 because adjacent figures are sharing sides thus reducing the total count for the term number. This problem is different from Problem 3 because the sequence represents the perimeter rather than the total number of sides in the diagram. Problem 7 Arranging Classroom Tables Some schools purchase classroom tables that have trapezoid-shaped tops rather than rectangular tops. The tables fit together nicely to arrange the classroom in a variety of ways. The number of students that can fit around a table is shown in the first diagram. The second diagram shows how the tables can be joined at the sides to make one longer table. 1 5 3 1. Write the first 5 terms in the sequence that represents this situation. Each term should indicate the total number of students that can sit around one, two, three, four, and five tables. Explain your reasoning. The sequence is 5, 8, 11, 1, 17. As the number of tables increases by 1, the number of students increases by 3. Share Phase, Problem 7 How did you determine first five terms in your sequence? Did you draw all the tables to complete the sequence? If so, explain how you wrote the sequence from your diagram. If not, explain your thinking process. If a table seats five students, why doesn t your sequence grow by five each time? How much does each term grow when a table is added?. The first trapezoid table seats five students. Explain why each additional table does not have seats for five students. There are seats for only three more students each time a table is added. That is because two seats are lost where the tables connect, one from the first table and one from the second table. How is this problem similar to Problem 3 when you crafted toothpick houses? How is this problem different from Problem 3 when you crafted toothpick houses? 50 Chapter Linear Functions
Problem 8 Students write a sequence to represent the numbers of petals in various stages when drawing a flower. Students follow the directions to draw the flower petals and generate the sequence of numbers. This is the only problem in this lesson where the sequence is increasing by a factor of. The context helps students make sense of this pattern. Misconception Students may have difficulty keeping track of what petals were drawn in which stage. If this is the case, have students use a different color of pencil for the petals generated in each stage. Also, encourage students to write the value for each term of the sequence as they complete each stage of the flower drawing, rather than writing all the values after the entire flower has been drawn. Problem 8 Drawing Flower Petals There are all kinds of sequences! Draw a flower in a series of stages. The figure shows a pair of flower petals as the starting point, Stage 0. In each stage, draw new petal pairs in the middle of every petal pair already drawn. In Stage 1, you will draw petals. In Stage, you will draw petals. In Stage 3, you will draw 8 petals. 3 1. Write the first 5 terms in the sequence that represents this situation. Each term should indicate the number of new petals drawn in that stage. Explain your reasoning. The sequence is,, 8, 16, 3. For each new stage, the new number of petals increases by multiplying the previous number of new petals by. 1 3 3 0 3 3 0 3 3 1 3 Share Phase, Problem 8 Without drawing, how could you determine the next term in the sequence? What is the growth pattern of this sequence? What other mathematical operation could you use to represent the growth of this sequence? In what way is the growth of the terms in this sequence different from all other problems?.1 Developing Sequences of Numbers from Diagrams and Contexts 51
Problem 9 Students write a sequence to represent babysitting earnings. This is the only problem in this lesson where the sequence alternates between decreasing by one values and increasing by a different value. The context helps students make sense of this pattern. Share Phase, Problem 9 What is the growth pattern of this sequence? Why does this growth pattern make sense in context of the problem? Problem 9 Babysitting Every Friday, Sarah earns $1 for babysitting. Every Saturday, Sarah spends $10 going out with her friends. 1. Write a sequence to show the amounts of money Sarah has every Friday after babysitting and every Saturday after going out with her friends for five consecutive weeks. The sequence should have 10 terms. Explain your reasoning. The sequence is 1,, 18, 8,, 1, 6, 16, 30, 0. This pattern is generated by beginning with 1, subtracting 10, adding 1, subtracting 10, and continuing. Problem 10 Recycling Problem 10 Students write a sequence to represent recycling. This is the only problem in this lesson where the sequence begins with zero. Because the sequence begins with zero and the terms of the sequence increase by a constant value, the sequence represents a list of multiples. The first week of school, Ms. Sinopoli asked her class to participate in collecting cans for recycling. The students started bringing in cans the second week of school. They collected 10 cans per week. 1. Write a sequence to show the running total of cans collected through the first nine weeks of school. Explain your reasoning. The sequence is 0, 10, 0, 360, 80, 600, 70, 80, 960. Every term is calculated by adding 10 to the previous term. Since this sequence begins with 0, the terms are multiples of 10. Share Phase, Problem 10 What is the first term of this sequence? Why is the first term of the sequence zero? Each term is a multiple of what number? Problems 3 and 7 also increase by a constant value. Why is it that the sequence in this problem is a list of multiples, but that is not the case with the sequences in Problems 3 and 7? 5 Chapter Linear Functions
Problem 11 Students write a sequence to represent selling tickets. In this problem, students must do some arithmetic to determine the initial value of the sequence. The sequence increases by a constant value. Share Phase, Problem 11 How did you determine the starting value of the sequence? What is the growth pattern of this sequence? Explain how the growth pattern of this sequence connects to the context of the problem. Talk the Talk Students summarize the patterns of each sequence in Problems through 11, and describe the similarities. Grouping Have students complete the Questions 1 and with a partner. Then share the responses as a class. Problem 11 Selling Tickets Sam is working at the ticket booth during a basketball game. His cash box has two $10 bills, five $5 bills, and twenty $1 bills. Tickets cost $3. 1. How much money does Sam have at the beginning of the basketball game? Sam has $65 at the start of the game.. Write a sequence to show the amount of cash Sam has available to start selling tickets, and the amounts available after selling one ticket, two tickets, three tickets, four tickets, and five tickets. Explain your reasoning. The sequence is 65, 68, 71, 7, 77, 80. Every term is calculated by adding 3 to the previous term. Talk the Talk There are many different patterns that can generate a sequence. Some possible patterns are: adding or subtracting by the same number each time, multiplying or dividing by the same number each time, adding by a different number each time, with the numbers being part of a pattern, alternating between adding and subtracting. The next term in a sequence is calculated by determining the pattern of the sequence and then using that pattern on the last known term of the sequence..1 Developing Sequences of Numbers from Diagrams and Contexts 53
Share Phase, Questions 1 and Which sequences increase? Which sequences decrease? Which sequences have growth patterns that generate new terms by adding or subtracting by the same number? Which sequences have growth patterns that generate new terms by multiplying or dividing by the same number? Which sequences have growth patterns that generate new terms by adding a different number each time? Which sequences have growth patterns that generate new terms by alternating between adding and subtracting? How are the sequences from Problems 3, 7, 10 and 11 alike? How are the sequences from Problems, 5 and 6 alike? Look back at Problems through 11. 1. Describe the pattern of each sequence by completing the table shown. Sequence Name Increases or Decreases Describe the Pattern Designing a Bead Necklace Increases 1 1 1 6... Crafting a Toothpick House Increases Starts at 6, adding by 5 Taking Apart a Card Trick (1) Decreases Subtracting Arranging Pennies Increases 1 1 1 1 3... Building Stairs Increases 1 7 1 9 1 11... Arranging Classroom Tables Increases Starts at 5, adding 3 Drawing Flower Petals Increases Multiplying by Babysitting Increases and Decreases 1 1 10 1 1 10... Recycling Increases Adding 10, starts at 0 Selling Tickets Increases Adding 3, starts at 65. Which sequences are similar? Explain your reasoning. Answers will vary. Arranging classroom tables, recycling, and selling tickets are similar because each sequence is generated by adding the same number each time. Be prepared to share your solutions and methods. 5 Chapter Linear Functions
Follow Up Assignment Use the Assignment for Lesson.1 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson.1 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter. Check for Students Understanding Determine the next three terms in each sequence of numbers. Show the pattern used to determine the next terms. 1. 17, 38, 33, 5, 9,,,,... The next terms are 70, 65 and 86. The pattern was 11, 5, 1 1, 5,.... 109, 103, 97, 91,,,,... The next terms are 85, 79 and 73. The pattern was decreasing by a constant value of 6. 3. 3, 37,, 9,,,,... The next terms are 58, 69 and 8. The pattern was 13, 15, 17, or increasing by consecutive odd numbers.. 1, 3, 9, 7,,,, The next terms are 81, 3 and 79. The pattern was increasing by multiplying by 3 each time. 5. 68, 60, 51, 1,,,, The next terms are 30, 18 and 5. The pattern was 8, 9, 10, or decreasing by consecutive decreasing numbers..1 Developing Sequences of Numbers from Diagrams and Contexts 5A
5B Chapter Linear Functions
Every Graph Tells a Story Describing Characteristics of Graphs Learning Goals In this lesson, you will: Describe characteristics of graphs using mathematical terminology. Describe a real-world situation that could be represented by a given graph. Key Terms discrete graph continuous graph linear graph collinear points non-linear graph Essential Ideas Graphs can be described by characteristics such as: discrete or continuous, linear or nonlinear and increasing, decreasing, neither increasing or decreasing, or both increasing and decreasing. A discrete graph is a graph of isolated points. A continuous graph is a graph with no breaks in it. A linear graph is a graph that is a line or series of collinear points. A nonlinear graph is a graph that is not a line and therefore not a series of collinear points. The graphs of all sequences are discrete graphs. Real world contexts can be represented as graphs or piecewise functions. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Use functions to model relationships between quantities. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Materials Scissors. Describing Characteristics of Graphs 55A
Overview Students describe and interpret graphs that are discrete or continuous, linear or nonlinear, and increasing, decreasing, neither increasing or decreasing, or both increasing and decreasing. They will distinguish among the characteristics of graphs by completing a sorting activity. The activity guides students to the realization that the graphs of all of the sequences are discrete graphs, and more specifically, the graphs of sequences are discrete linear graphs. Next, students are presented with two contexts and two piecewise graphs that represent those contexts. Because the graphs are composed of line segments only, students will use the fact that linear graphs represent constant rates of change in order to calculate the rate of change for each piece of the graph and describe how it relates to the given context. Students are then presented with a basic context and two different numberless graphs representing the context. Students will write stories to describe the context in more detail by interpreting the information from the graphs. 55B Chapter Linear Functions
Warm Up Consider the sequence, 6, 8, 10. 1. Use the table to list each term of the sequence. Term Number 1 3 Term 6 8 10. Use the chart to write each of the terms as an ordered pair. (1, ) (, 6) (3, 8) (, 10) 3. Graph the sequence on the coordinate plane. y 10 8 6 0 6 8 10 x. Describing Characteristics of Graphs 55C
55D Chapter Linear Functions
Every Graph Tells a Story Describing Characteristics of Graphs Learning Goals In this lesson, you will: Describe characteristics of graphs using mathematical terminology. Describe a real-world situation that could be represented by a given graph. Key Terms discrete graph continuous graph linear graph collinear points non-linear graph Have you ever followed a trail of animal tracks? For expert animal trackers, there are many more signs to look for instead of just paw prints. Expert trackers look for rub, like when a deer scrapes velvet off its antlers. They look for chews where a twig or section of grass has been eaten. If there is a clean cut on the plant, it may likely have been caused by an animal with incisors (like a rodent). If the plants have teeth marks all over them, those plants may likely have been eaten by a predator. And of course, trackers look for scat, or droppings. From scat, trackers can tell an animal s shape and size and what the animal eats. Tubular scat may come from raccoons, bears, and skunks. Teardrop-shaped scat may come from an animal in the cat family. How do you follow clues in mathematics to solve problems? Incisors are the sharp teeth in humans and animals!. Describing Characteristics of Graphs 55
Problem 1 Students describe the characteristics of graphs using the terms: discrete or continuous, linear or nonlinear, and increasing, decreasing, neither increasing or decreasing, or both increasing and decreasing. They will distinguish among the characteristics of graphs by completing a sorting activity. The activity guides students to the realization that the graphs of all sequences are discrete graphs, and linear graphs. Problem 1 Characteristics of Graphs There are many ways that data can be represented through graphical displays. In this lesson, you will explore many characteristics of graphs. 1. Graph the first four terms of Sequence A: 0,,, 6. Let the term number represent the x-coordinate, and let the term value represent the y-coordinate. Then, list the coordinates of the points on your graph. Term Value y 9 8 7 6 5 3 Note that the term value is the number itself. The term number indicates where the term falls in the sequence (1st, nd, 3rd, and so on). Materials Scissors 1 0 0 1 3 5 6 Term Number 7 8 9 x Grouping Have students complete Questions 1 through with a partner. Then share the responses as a class. Have students complete Question 5 by cutting out the graphs. Share Phase, Questions 1 through What type of sequence is Sequence A? How can you tell? How were you able to graph the sequence without coordinate pairs being given? How could you turn your discrete graph into a continuous graph? If you connected the points, a point on the line would be (.5, 3). Why does this point not make sense in the sequence? The points are (1, 0), (, ), (3, ), and (, 6).. Would it make sense to connect the points on your graph? Why or why not? It would not make sense to connect the points on my graph because the terms are separate points. A discrete graph is a graph of isolated points. Often, those points are counting numbers and do not consist of fractional numbers. A continuous graph is a graph with no breaks in it. The points in a continuous graph can have whole numbers and fractions to represent data points. 3. Is your graph from Question 1 discrete or continuous? Explain your reasoning. My graph is discrete because I did not connect the points.. Are the graphs of any sequence discrete or continuous? Explain your reasoning. The graphs of sequences are discrete because the graphs always have points that are separated, or isolated. 5. Carefully cut out Graphs A through L on the following pages. Time to get out your scissors. 56 Chapter Linear Functions
. Describing Characteristics of Graphs 57 Note These graphs are also used in later lessons to develop the concept of functions. A x 8 6 8 6 0 0 y B x 8 6 8 6 y 0 0 C x 8 6 8 6 y 0 0 D x 8 6 8 6 y 0 0 E x 8 6 8 6 y 0 0 F x 8 6 8 6 y 0 0
58 Chapter Linear Functions
. Describing Characteristics of Graphs 59 Note These graphs are also used in later lessons to develop the concept of functions. G x 8 6 8 6 y 0 0 H x 8 6 8 6 y 0 0 I x 8 6 8 6 y 0 0 J x 8 6 8 6 y 0 0 K x 8 6 8 6 y 0 0 L x 8 6 8 6 y 0 0
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Grouping Have students complete Questions 6 through 8 with a partner. Then share the responses as a class. Share Phase, Questions 6 through 8 How did you determine whether the graphs were discrete or continuous? Explain why both lines and curves are continuous graphs. Would this graph be continuous or discrete? Explain. 6. Determine if the graphs you cut out are discrete or continuous. a. Sort the graphs into two groups: those graphs that are discrete and those graphs that are continuous. b. Record your findings in the table by writing the letter of each graph. Discrete Graphs Continuous Graphs A, F, J B, C, D, E, G, H, I, K, L 7. Determine if the graphs are increasing, decreasing, both increasing and decreasing, or neither increasing nor decreasing. a. Analyze each graph from left to right. b. Sort the graphs into four groups: those that are increasing, those that are decreasing, those that are both increasing and decreasing, and those that are neither increasing nor decreasing. c. Record your findings in the table by writing the letter of each graph. Increasing Decreasing Both Increasing and Decreasing Neither Increasing nor Decreasing Explain how Graph D, the circle, is different from the other graphs that are both increasing and decreasing. Draw a discrete graph that is decreasing only. What is the difference between a line and a series of collinear points? Explain why Graph L is not linear. Which of the linear graphs is also discrete and increasing only? Which of the linear graphs is also continuous and decreasing only? Which of the nonlinear graphs is also continuous and decreasing only? A, B E, K C, D, H, I, L F, G, J A linear graph is a graph that is a line or a series of collinear points. Collinear points are points that lie in the same straight line. A non-linear graph is a graph that is not a line and therefore not a series of collinear points. 8. Determine whether Graphs A L are linear or non-linear graphs. a. Sort the graphs into two groups: those that are linear and those that are non-linear. b. Record your findings in the table by writing the letter of each graph. Linear Graph Non-linear Graph A, B, F, G, K C, D, E, H, I, J, L. Describing Characteristics of Graphs 61
9. Clip Graphs A L together, and keep them for Lessons and 3. You will use these graphs in another lesson. So, put them in a safe place. Problem Students are presented with two contexts and two piecewise graphs that represent those contexts. Because the graphs are composed of line segments only, students will use the fact that linear graphs represent constant rates of change in order to calculate the rate of change for each piece of the graph and describe how it relates to the given context. Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Problem Making Sense of Graphs The graph shown represents Greg s distance from home after driving for x hours. Distance (mi) y 180 160 10 10 100 80 60 0 0 0 0 1 3 5 6 Time (hr) 1. Analyze the graph between 0 and hours. a. How far from home was Greg after driving for hours? Greg was 10 miles from home after hours of driving. b. How fast did Greg drive during this time? Explain your reasoning. Greg was traveling at 60 mph. The speed 10 miles in hours can be simplified to 60 miles per hour. 7 8 9 x How can you tell by looking at the graph when Greg was traveling the fastest? Share Phase, Questions 1 and Is this graph discrete or continuous? Explain why your response makes sense with the given context. Why is this graph considered to be nonlinear? Is the graph increasing or decreasing? Explain. What does the point (3.5, 150) represent in the context of the problem? c. How do you know that Greg traveled at the same rate for the first two hours? Describe in terms of the graph. The graph is increasing at the same rate. The graph is a straight line.. Analyze the graph between and.5 hours. a. How far did Greg travel from home between and.5 hours? Greg did not travel any miles from home between and.5 hours. 6 Chapter Linear Functions
Share Phase, Question 3 How far did Greg travel before he headed back home? How can you tell? When did Greg arrive back home? How can you tell? How can you tell from the graph that Greg did not travel at the same rate during his entire trip? What do the increasing segments of the graph represent in the problem context? What do the decreasing segments of the graph represent in the problem context? What do the horizontal segments of the graph represent in the problem context? b. How fast did he travel during this time? Explain your reasoning. Greg traveled at 0 miles per hour. Because the graph is a straight horizontal line, I know that his distance did not increase or decrease from home during that time. c. Describe the shape of the graph between and.5 hours. The graph is a horizontal line. 3. Complete the table. Label each segment of the graph with letters A through G, beginning from the left. Record the time interval for each segment. Then, describe what happened in the problem situation represented by that segment of the graph. State how fast Greg traveled and in what direction (either from home or to home). Segment Time Interval (hours) A 0 to B to.5 C.5 to.5 Description of Greg s Trip Greg traveled 10 miles from home at a rate of 60 mph. Greg took a half-hour break when he was 10 miles from home. Greg traveled 60 more miles from home at a rate of 30 mph. D.5 to 5.5 Greg took a one-hour break when he was 180 miles from home. E 5.5 to 8 Greg traveled 50 miles toward home at 0 mph. F 8 to 9.5 Greg took a 1.5-hour break when he was 130 miles from home. G 9.5 to 10 Greg traveled 30 miles toward home at 60 mph (or 30 miles per half hour), ending 100 miles from home.. Describing Characteristics of Graphs 63
Grouping Have students complete Question with a partner. Then share the responses as a class. Share Phase, Question Is this graph discrete or continuous? Explain why your response makes sense with the given context. Why is this graph considered to be nonlinear? Is the graph increasing or decreasing? Explain. What does the point (, 6) represent in the context of the problem? Where does the graph represent when the pool was empty? How can you tell? What do the increasing segments of the graph represent in the problem context? What do the decreasing segments of the graph represent in the problem context? What do the horizontal segments of the graph represent in the problem context?. The crew at the community swimming pool prepared the pool for opening day. The graph shows the depth of water in the swimming pool after x hours. Depth of water (ft) y 9 8 7 6 5 3 1 0 0 6 8 10 1 Time (hr) 1 16 18 x a. Why do you think the pool was emptied and then refilled? Answers will vary. The bottom of the pool needed to be cleaned, repairs had to be made, or the crew wanted to fill the pool with clean water. b. Complete the table. Label each segment of the graph with letters A through E, beginning from the left. Record the time interval for each segment. Then, describe what occurred in the problem situation represented by that segment in the graph. State how fast the water level in the pool changed and whether it was being drained or filled. 6 Chapter Linear Functions
Segment Time Interval (hours) Description of the Water in the Pool A 0 to The water in the pool remained at 7 feet deep. B to 6 The pool was being drained. The depth of the water in the pool decreased at a rate of feet per hours, or 0.5 ft per hour. At the end of the sixth hour, the depth of the water was 5 feet. C 6 to 8 The pool was still being drained. The depth of the water in the pool decreased at a rate of 5 feet per hours, or.5 ft per hour. At the end of the eighth hour, the depth was at 0 feet, or the pool was empty. D 8 to 1 The pool remained empty for 6 hours. E 1 to 0 The pool was being filled at a rate of 9 feet in 6 hours, or 1.5 feet per hour. At the end of the twentieth hour, the depth was 9 feet. c. Was the pool being emptied at the same rate the entire time? Explain using mathematics and the graph. First, the pool was being emptied at 0.5 feet per hour, and then it was being emptied at.5 feet per hour. The graph is not a straight line because the water level did not decrease by the same amount every hour. d. Why does it make sense for the graph of this situation to be continuous rather than discrete? For any given time, there is an appropriate water level. It is logical to monitor the entire draining and filling process of the pool, not just one certain time.. Describing Characteristics of Graphs 65
Problem 3 Students are presented with a basic context and two different numberless graphs to represent the context. They write stories to describe the context in more detail by interpreting the information from the graphs. Grouping Have students complete Questions 1 and with a partner. Then share the responses as a class. Share Phase, Questions 1 and How are these graphs different from the other graphs in this lesson? Are the graphs increasing or decreasing? Explain why your response makes sense with the given context. How can you tell from the graph whether the amount of popcorn is decreasing slowly or quickly? What does the horizontal portion of the graph represent in the context of the problem? What does the vertical portion of the graph represent in the context of the problem? Problem 3 Tell a Story You and a friend go to the movies and decide to share a large bucket of popcorn. Write a story to describe each graph. 1.. Amount of Popcorn y Time We buy the popcorn and just take a taste when we are heading to our seats. Then, we eat it very quickly once we sit down. Amount of Popcorn y Time We did not eat any popcorn until we got to our seat. We had a few handfuls and then spilled the entire container on the floor. Be prepared to share your solutions and methods. x x As time increases, what happens to the amount of popcorn? 66 Chapter Linear Functions
Follow Up Assignment Use the Assignment for Lesson. in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson. in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter. Check for Students Understanding The table provides information regarding Erin s trip. Segment Time Interval Description of Erin s Trip A 5 PM Erin was at work 30 miles from her home. B C D 5 PM 5:5 PM 5:5 PM 7 PM 7 PM 8:30 PM Because of the rush hour traffic, she traveled at a steady rate, and arrived home 5 minutes later. Once home, Erin took an hour and fifteen minutes to cook dinner and eat with her family. Immediately after dinner, Erin left her house and traveled at the speed limit of 60 mph to a town 1.5 hours away. 1. Complete the table using the time of day for the time intervals.. Draw a graph to represent the situation. Label each segment of the graph. Miles from Home 100 50 A D B C 5 PM 6 PM 7 PM 8 PM 9 PM Time of Day. Describing Characteristics of Graphs 66A
3. Is the graph discrete or continuous? Explain why your response makes sense with the given context. The graph is continuous and has no breaks because at every time Erin is always located somewhere in relation to her home.. Explain why the entire graph is nonlinear but the segments of the graph are linear. The entire graph is not linear because Erin is not always traveling at the same rate. The segments are linear because Erin is traveling at a steady rate (0 mph) going home from work and at a constant rate (60 mph) leaving her home in the evening. 5. Explain what the increasing and decreasing segments of the graph represent in the context of the problem. The increasing segment is when Erin is traveling away from home, so her distance from her house is increasing. The decreasing segment is when Erin is traveling home from work, so her distance to her house is decreasing. 66B Chapter Linear Functions
To Be or Not To Be a Function? Defining and Recognizing Functions Learning Goal In this lesson, you will: Define relation and function. Determine whether a relation (represented as a mapping, set of ordered pairs, table, sequence, graph, equation, or context) is a function. Key Terms mapping set relation input output function domain range scatter plot vertical line test Essential Ideas A relation is any set of ordered pairs or the mapping between a set of inputs and a set of outputs. The first coordinate in an ordered pair in a relation is the input, and the second coordinate is the output. A relation can be represented as a mapping, set of ordered pairs, table, sequence, graph, equation, or context. A function is a relation which maps each input to one and only one output. Relations that are not functions will have more than one output for each input. A scatter plot is a graph of a collection of ordered pairs that allows an exploration of the relationship between the points. The vertical line test is a visual method of determining whether a relation represented as a graph is a function. To test whether a relation represented as an equation is a function, substitute values for x into the equation and determine if any x-value can be mapped to more than one y-value. It is important to consider both positive and negative values. All sequences are functions. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 3. Interpret the equation y 5 mx 1 b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Use functions to model relationships between quantities. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally..3 Defining and Recognizing Functions 67A
Materials Scissors Overview The terms relation and function are defined. Relations are represented as mappings, sets of ordered pairs, tables, sequences, graphs, equations, and contexts. Students analyze mappings, ordered pairs, tables, and sequences. They will determine whether these relations are functions according to the definition of a function. Next, students determine whether different real-world contexts represent functions. Relations are presented in graphical displays and the vertical line test is introduced. Students will analyze graphs and use the vertical line test to determine whether the various displays are functions. In the next activity, students determine whether equations are functions by substituting values for x into the equation, and then determining if any x-values can be mapped to more than one y-value. Finally, students solidify their understanding of functions by completing a sorting activity; relations are represented seven different ways, and students must determine whether or not they are functions. 67B Chapter Linear Functions
Warm Up Complete each table. 1. y = x + 1 x y 0 1 1 3-1 - 3. x = y + 1 x y 0 none 1 0 1, -1 3, - 3, -3.3 Defining and Recognizing Functions 67C
67D Chapter Linear Functions
To Be or Not To Be a Function? Defining and Recognizing Functions Learning Goals In this lesson, you will: Define relation and function. Determine whether a relation (represented as a mapping, set of ordered pairs, table, sequence, graph, equation, or context) is a function. Key Terms mapping set relation input output function domain range scatter plot vertical line test In September 009, museum volunteers in England began work on restoring the WITCH machine regarded as the first modern computer still able to work. This huge computer, as long as an entire wall in a large room, was built starting in 199 and was functional until 1957. WITCH was used to perform mathematical calculations, but instead of typed input, the computer had to be fed paper tape for inputs. Then, the computer would produce its output on paper as well. Even though it was so huge, WITCH could only perform calculations as fast as a human with a modern calculator. What types of inputs and outputs do modern computers use and produce? How does a modern computer turn inputs into outputs?.3 Defining and Recognizing Functions 67
Problem 1 Students represent mappings and tables of values as sets of ordered pairs. They are introduced to the terms relation, input, output, and function. Students will analyze sets of ordered pairs (or mappings, tables, and sequences) to determine if they are functions according to the definition. The problem concludes by making the connection to sequences and the fact that all sequences are functions. Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Problem 1 Analyzing Ordered Pairs Use brackets, { }, to denote a set. As you learned previously, ordered pairs consist of an x-coordinate and a y-coordinate. You also learned that a series of ordered pairs on a coordinate plane can represent a pattern. You can also use a mapping to show ordered pairs. Mapping represents two sets of objects or items. An arrow connects the items together to represent a relationship between the two items. 1. Write the set of ordered pairs that represent a relationship in each mapping. a. b. 1 1 1 1 3 3 3 3 5 5 7 7 5 {(1, 7), (, 1), (3, 5), (, 3)} {(1, 1), (, 1), (3, 5), (, 3), (5, 7)} c. 1 3 5 1 3 5 7 {(1, 1), (, 3), (, 5), (3, 5), (, 7), (5, 7)} {(, 7), (, 0), (, 9), (6, 9), (8, )} d. 6 8 7 9 0 Share Phase, Questions 1 through 3 What is a mapping? How did you determine the ordered pairs? How did you determine how many ordered pairs to list? What notation must you use so that the ordered pairs are represented as a set? How did you determine which values to place in each oval? How do you handle the situation where a number is repeated in an oval? Can the same value be placed in both ovals?. Create a mapping from the set of ordered pairs. a. {(5, 8), (11, 9), (6, 8), (8, 5)} b. {(3, ), (9, 8), (3, 7), (, 0)} 5 5 3 6 7 8 8 8 9 9 11 0 When you write out the ordered pairs for a mapping, you are writing a set of ordered pairs. A set is a collection of numbers, geometric figures, letters, or other objects that have some characteristic in common. 68 Chapter Linear Functions
3. Write the set of ordered pairs to represent each table. a. Input Output b. x y 10 0 5 10 0 0 5 10 10 0 0 10 10 5 0 0 10 5 0 10 {(10, 0), (5, 10), (0, 0), {(0, 10), (10, 5), (0, 0), (5, 10), (10, 0)} (10, 5), (0, 10)} Grouping Ask a student to read the information following Question 3 aloud. Discuss the definitions, worked examples, and complete Question as a class. The mappings or ordered pairs shown in Questions 1 through 3 form relations. A relation is any set of ordered pairs or the mapping between a set of inputs and a set of outputs. The first coordinate of an ordered pair in a relation is the input, and the second coordinate is the output. A function maps each input to one and only one output. In other words, a function has no input with more than one output. The domain of a function is the set of all inputs of the function. The range of a function is the set of all outputs of the function. Discuss Phase, Definitions How is the mathematical term relation related to the everyday term relationship? Explain why a table of values represents a relation? Explain why Questions 1 to 3 all represent relations? What makes a function a special type of relation? Explain what a function is in your own words. Explain what is meant by this statement, A function is a special case of a relation. 1 1 3 3 5 7 In the mapping shown the domain is {1,, 3, } and the range is {1, 3, 5, 7}. This mapping represents a function because each input, or domain value, is mapped to only one output, or range value. Notice the use of set notation when writing the domain and range..3 Defining and Recognizing Functions 69
Share Phase, Question How did you determine whether the relation was a function? Can a function have two inputs with the same output? Can a function have one input with two different outputs? Can a function have one x with two different y s? Can a function have two x s with the same one y? Grouping Have students complete Questions 5 and 6 with a partner. Then share the responses as a class. Share Phase, Question 5 Can the relation in Question 1, part (b) be a function considering inputs of 1 and both have an output of 1? Explain. In the relation in Question, part (b), what is the output for an input of 3? How can this type of question help you determine if a relation is a function? Can the relation in Question 3, part (b) be a function considering the input of 10 had outputs of both 5 and 5? Explain. 1 3 5 In the mapping shown the domain is {1,, 3,, 5} and the range is {1, 3, 5, 7}. This mapping does not represent a function.. State why the relation in the example shown is not a function. The relation is not a function because the domain value of has two outputs, 3 and 5. 5. State the domain and range for each relation in Questions and 3. Then, determine which relations represent functions. If the relation is not a function, state why not. Question (a): Domain: {5, 6, 8, 11} Range: {5, 8, 9} The ordered pairs represent a function. Every input has one and only one output value. Question (b): Domain: {3,, 9} Range: {, 7, 8, 0} The ordered pairs do not represent a function. The input 3 has two different outputs, and 7. Question 3(a): Domain: {10, 5, 0, 5, 10} Range: {0, 10, 0} This table represents a function. Every input has one and only one output. Question 3(b): Domain: {0, 10, 0} Range: {10, 5, 0, 5, 10} This table does not represent a function. The input of 0 has two outputs, 10 and 10. The input of 10 has two outputs, 5 and 5. To be a function, no input can have more than one output. Is it easier to determine if a relation is a function by viewing a mapping, a set of ordered pairs, or a table? Explain. 1 3 5 7 70 Chapter Linear Functions
6. Review and analyze Emil s work. Emil My mapping represents a function. 1 7 3 5 6 9 1 Explain why Emil s mapping is not an example of a function. Emil shows the input having two outputs. Therefore, this cannot be an example of a function. Problem Relations are represented as contexts. Students test whether the contexts are functions by analyzing the contexts in light of the definition of a function. Drawing mappings to represent the contexts may be a useful tool for students to make sense of the various contexts. Grouping Have students complete Questions 1 through 9 with a partner. Then share the responses as a class. Problem Analyzing Contexts Read each context and decide whether it fits the definition of a function. Explain your reasoning. 1. Input: Sue writes a thank-you note to her best friend. Output: Her best friend receives the thank-you note in the mail. Yes. Sue s one note goes to one place.. Input: A football game is being telecast. Output: It appears on televisions in millions of homes. No. The football game is mapped to more than one home..3 Defining and Recognizing Functions 71
Share Phase, Questions 1 through 9 Did a mapping help you decide whether this context was a function or not? Explain. How many people received Sue s thank you note? How many homes are able to view the telecast? How many homes does each puppy have? Does more than one person have this zip code? Is there more than one person in the audience? How many jobs does Tara perform in one day? Regarding Question 7, explain how you used the phrasing in the Output to create the mapping. Regarding Question 9, explain any confusion when someone else tried to solve your problem. How could your problem be reworded to clarify the context? 3. Input: There are four puppies in a litter. Output: One puppy was adopted by the Smiths, another by the Jacksons, and the remaining two by the Fullers. Yes. Each puppy has one and only one home.. Input: The basketball team has numbered uniforms. Output: Each player wears a uniform with her assigned number. Yes. Each player wears one uniform with one specific number. 5. Input: Beverly Hills, California, has the zip code 9010. Output: There are 3,675 people living in Beverly Hills. No. One zip code is mapped to many people. 6. Input: A sneak preview of a new movie is being shown in a local theater. Output: 65 people are in the audience. No. One movie is mapped to 65 people. 7. Input: Tara works at a fast food restaurant on weekdays and a card store on weekends. Output: Tara s job on any one day. Yes. On any given day of the week, Tara has just one job. 8. Input: Janelle sends a text message to everyone in her contact list on her cell phone. Output: There are 1 friends and family on Janelle s contact list. No. One text message is mapped to 1 people. 9. Create your own context problem, and decide whether it represents a function. Trade with a partner, and solve your partner s problem. Then, discuss your responses. Input: Output: Answers will vary. 7 Chapter Linear Functions
Problem 3 Relations are represented as sequences. Students test whether the sequences are functions by writing out the ordered pairs. Students conclude that all sequences are functions. Grouping Have students complete Questions 1 and with a partner. Then share the responses as a class. Share Phase, Questions 1 and Are all sequences linear functions? Explain. Can you create a sequence that is not a function? Why or why not. Problem 3 Analyzing Sequences Think about the mappings as ordered pairs. 1. Determine if each sequence represents a function. Explain why or why not. If it is a function, identify its domain and range. a.,, 6, 8, 10, (1, ) (, ) (3, 6) (, 8) (5, 10) This sequence represents a function. Each input, or term number in the sequence, has one output, the term. Domain: {1,, 3,, 5} Range: {,, 6, 8, 10} b. 1, 0, 1, 0, 1, (1, 1) (, 0) (3, 1) (, 0) (5, 1) This sequence represents a function. Each input, or term number in the sequence, has one output, the term. Domain: {1,, 3,, 5} Range: {0, 1} c. 0, 5, 10, 15, 0, (1, 0) (, 5) (3, 10) (, 15) (5, 0) This sequence represents a function. Each input, or term number in the sequence, has one output, the term. Domain: {1,, 3,, 5} Range: {0, 5, 10, 15, 0} Remember a sequence has a term number and a term value.. What do you notice about each answer in Question 1? What conclusion can you make about sequences? The term numbers are the inputs, and the term values are the outputs. Each input can only have one output; therefore, all sequences are functions..3 Defining and Recognizing Functions 73
Problem Relations are represented as graphs; the graphs presented are discrete graphs (defined as scatter plots) and continuous graphs. Students will use the vertical line test to determine whether the relations represented as graphs are functions. Problem Analyzing Graphs A relation can be represented as a graph. Graphs A L from Lesson. provide examples of graphical representations of relations. A scatter plot is a graph of a collection of ordered pairs that allows an exploration of the relationship between the points. 1. Determine if these scatter plots represent functions. Explain your reasoning. a. y 9 Grouping Have students complete Question 1 with a partner. Then share the responses as a class. Output 8 7 6 5 3 Keep in mind, a function maps each input to one and only one output. Share Phase, Question 1 Why are graphs considered to be relations? Is a scatter plot (always, sometimes, never) a discrete graph? Is a scatter plot (always, sometimes, never) a linear graph? Is a scatter plot (always, sometimes, never) an increasing graph? How did you determine whether the scatter plot was a function? Is a scatter plot (always, sometimes, never) a function? b. 1 0 0 1 3 5 6 7 8 9 x Input This scatter plot represents a function. Each input value has only one output value. Output y 9 8 7 6 5 3 1 0 0 1 3 5 6 7 8 9 x Input This scatter plot does not represent a function. The input value x 5 has two output values, 1 and. 7 Chapter Linear Functions
Grouping Ask a student to read the definition aloud. Discuss the definition and worked example as a class. 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. Discuss Phase, Worked Example How is the vertical line test related to the definition of a function? Is it easier to determine if a graph is a function by using the vertical line test or by listing the graphed points as ordered pairs and checking them against the definition of a function? Explain. Review the scatter plot shown. y 9 8 7 6 5 3 1 0 0 1 3 5 6 7 8 9 x In this scatter plot, the relation is not a function. The input value can be mapped to two different outputs, 1 and. Those two outputs are shown as intersections to the vertical line segment drawn at x 5..3 Defining and Recognizing Functions 75
Grouping Have students complete Question with a partner. Then share the responses as a class.. Use the vertical line test to determine if each graph represents a function. Explain your reasoning. a. y 9 Share Phase, Question How did you determine whether the graph was a function? If the graph is not a function, what are the ordered pairs for two points on the graph that verify this conclusion? Can you glance at any graph and determine it is not a function? Explain. b. 8 7 6 5 3 1 0 0 1 3 5 6 7 8 9 x This graph does not represent a function. A vertical line can be drawn that passes through points in all places except where x 5 1. y 9 8 7 6 5 3 1 0 0 1 3 5 6 7 8 9 x This graph represents a function. When a vertical line is drawn through any portion of the graph, it never passes through more than one point. 76 Chapter Linear Functions
Problem 5 Relations are represented as equations. Students test whether the equations are functions by substituting values for x into the equation and then determining if any x-value can be mapped to more than one y-value. The example provided helps students make sense of the fact that this equation represents a function. In order for students to correctly determine whether the equations represent functions, they will need to consider negative values for x. Grouping Ask a student to read the worked example aloud. Discuss the example and information as a class. Have students complete Question 1 with a partner. Then share the responses as a class. Share Phase, Question 1 How do you know when you have substituted enough different values for x to determine whether the equation is a function? Could a given number of yards represent two different numbers of feet? How can this question help you verify that this equation is a function? Problem 5 Analyzing Equations So far, you have determined whether a set of data points in a scatter plot represents a function. You can also determine whether an equation is a function. The given equation can be used to convert yards into feet. Let x represent the number of yards, and y represent the number of feet. y 5 3x To test whether this equation is a function, first, substitute values for x into the equation, and then determine if any x-value can be mapped to more than one y-value. If each x-value has exactly one y-value, then it is a function; otherwise, it is not a function. In this case, every x-value can be mapped to only one y-value. Each x-value is multiplied by 3. Some examples of ordered pairs are (, 6), (10, 30), and (5, 15). So, this equation is a function. 1. Determine whether each equation is a function. List three ordered pairs that are solutions to each. Explain your reasoning. a. y 5 5x 1 3 This equation is a function. No x-value can be mapped to more than one y-value. (0, 3) (1, 8) (, 13) b. y 5 x This equation is a function. No x-value can be mapped to more than one y-value. (0, 0) (1, 1) (, ) c. y 5 x This equation is a function. No x-value can be mapped to more than one y-value. (0, 0) (1, 1) (1, 1) So, if two different inputs go to the same output, that's still a function. Why is this method of substituting values for x in an equation not as conclusive as the vertical line test for graphs? What are the different types of numbers that you should consider when substituting values for x? What is meant by three ordered pairs that are solutions in the directions? Why is the method of substituting values for x tricky when determining whether an equation is a function?.3 Defining and Recognizing Functions 77
Problem 6 Students solidify their understanding of functions by completing a sorting activity. Students determine whether relations represented as mappings, sets of ordered pairs, tables, sequences, graphs, equations, and contexts are functions or are not functions. Materials Scissors Problem 6 Function or Not? 1. Sorting Activity a. Carefully cut out Relations M through X on the following pages. b. Refer to Graphs A through L from Lesson. c. Sort Relations A through X into two groups: those that are functions and those that are not functions. d. Record your findings in the table by writing the letter of each relation. Functions Time to get your graphs from the first lesson back out. Not Functions A, B, C, E, G, H, K, L, M, O, Q, R, T, U, V, X D, F, I, J, N, P, S, W Grouping Have students complete this sorting activity with a partner. Then share the responses as a class. Share Phase, Problem 6 How did you determine whether or not the graphs (A - L) represented functions? How did you determine whether the mappings (M - N) represented functions? How did you determine whether the tables (O - P) represented functions? How did you determine whether the sequences (Q - R) represented functions? How did you determine whether the ordered pairs (S - T) represented functions? How did you determine whether the equations (U - V) represented functions? How did you determine whether the contexts (W - X) represented functions? Did you convert any representation to another representation to determine whether it was a function or not? For example, did you convert the table content to a graph to decide if it was a function? If so, explain why you decided upon that method and if it was useful. 78 Chapter Linear Functions
M N 10 11 1 13 1000 000 3000 10 11 1 13 1000 000 3000 O P Input Output x y 1 1 1 1 0 0 0 0 1 1 1 1 Q R The terms of a sequence: The terms of a sequence: 7, 10, 13, 16, 19, 10, 30, 10, 30, 10,.3 Defining and Recognizing Functions 79
80 Chapter Linear Functions
S The set of ordered pairs {(, 3), (, ), (, 5), (, 6), (, 7)} T The set of ordered pairs {(, 1), (3, 1), (, 1), (5, 1), (6, 1)} U y 5 x 1 1 V This equation is used to calculate the number of inches in a foot: y 5 1 x Let x represent the number of feet and y represent the number of inches. W Input: The morning announcements are read over the school intercom system during homeroom period. Output: All students report to homeroom at the start of the school day to listen to the announcements. X Input: Each student goes through the cafeteria line. Output: Each student selects a lunch from the menu..3 Defining and Recognizing Functions 81
8 Chapter Linear Functions
Talk the Talk Students complete sentences to demonstrate their understanding of functions. Grouping Have students complete Questions 1 and independently. Then share the responses as a class. Talk the Talk Choose the appropriate description to complete each sentence. 1. A relation is (always, sometimes, never) a function. sometimes. A function is (always, sometimes, never) a relation. always Be prepared to share your solutions and methods..3 Defining and Recognizing Functions 83
Follow Up Assignment Use the Assignment for Lesson.3 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson.3 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter. Check for Students Understanding 1. Create a mapping that is a function. Answers will vary. One possible response is: 1 3 5. Draw a graph that is not a function. Answers will vary. One possible response is: 3. Create a table of values that is a function. Answers will vary. One possible response is: x y 0 1 1 3 3. Write a sequence that is not a function. This is impossible. Every sequence is a function. 8 Chapter Linear Functions
Scaling A Cliff Linear Functions Learning Goals In this lesson, you will: Make input-output tables for linear functions. Graph linear functions. Determine characteristics of linear functions. Key Terms linear function Essential Ideas Linear functions are represented using input-output tables. Linear functions are represented using graphs. 3. Interpret the equation y 5 mx 1 b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Use functions to model relationships between quantities. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.. Linear Functions 85A
Overview A situation is described and students answer related questions, complete a table of values using the answers to the questions, define a variable, and write an expression that can be used to generate additional values. The expression is used to answer questions and students will then graph the situation using the table of values. They will conclude that the graph of the relation is a function and any function whose graph is a straight line is a linear function. The term independent variable and dependent variable are introduced. Several questions focus on the relationship between the dependent and independent variables. 85B Chapter Linear Functions
Warm Up Complete each table. 1. Write each row in the table as an ordered pair. x y 0 1 1 3-1 - 3 (0, 1) (1, ) (, 3) (-1, ) (-, 3). Graph the ordered pairs in the coordinate plane. y 0 x 3. Is the graph considered a function? Explain your reasoning. The graph is a function. It passes the vertical line test.. Linear Functions 85C
85D Chapter Linear Functions
Scaling a Cliff Linear Functions Learning Goals In this lesson, you will: Make input-output tables for linear functions. Graph linear functions. Determine characteristics of linear functions. Key Term linear function Can you draw a perfectly straight line without using a ruler or other straightedge? What about over a long distance? Carpenters and other construction workers use what is called a chalk line to mark straight lines over long distances. A chalk line tool looks a bit like a tape measure. A cord that is coated in chalk is wound inside the tool. One person pulls the cord to the end of where the line will be, and the other person holds the tool at the beginning of the line. When they have the line where they want, both people pull the cord tight, and one person pulls up on the cord and lets go so that the cord snaps a straight line of chalk onto the surface below the cord. Have you ever seen or used a chalk line tool? Does anyone in your class have one they could show?. Linear Functions 85
Problem 1 Two rock climbers are 36 feet above ground and continue to climb up at a rate of 1 feet per hour. Students answer questions related to the situation, and use the answers to create a table of values. They will define a variable and write an expression to represent the situation. After graphing the table of values, students conclude the graph is a linear function. They will then identify the dependent and independent variables and answer questions that focus on the relationship between them. Grouping Have students complete Question 1 with a partner. Then share the responses as a class. Problem 1 Climbing to the Top! 1. You and your friends are rock climbing a vertical cliff that is 108 feet tall along a beach. You have been climbing for a while and are currently 36 feet above the beach when you stop on a ledge to have a snack and then begin climbing again. You can climb about 1 feet in height each hour. If you maintain your pace after your break, how high will you have climbed in: a. 1 hour? After 1 hour, I will have climbed 8 feet. b. hours? After hours, I will have climbed 60 feet. c. 180 minutes? After 180 minutes, I will have climbed 7 feet. d. 10 minutes? After 10 minutes, I will have climbed 78 feet. e. Which quantities are changing? Which quantities remain constant? The elapsed time and distance are the quantities that are changing. The pace of climbing and the starting height are the quantities that remain the same. Share Phase, Question 1 How far did you and a friend climb before you stopped for a snack? What is the average rate at which you are climbing? How did you determine how high you would be 1 hour into the climb? How did you determine how high you would be hours into the climb? How many hours is 180 minutes? How many hours is 10 minutes? f. Which quantity depends on the other quantity? The distance climbed is the quantity that depends on the elapsed time. Is the rate at which you are climbing a constant? Explain. Is the starting height a constant? Explain. Does the time elapsed depend on the distance climbed, or does the distance climbed depend on the time elapsed? 86 Chapter Linear Functions
Grouping Have students complete Questions through 5 with a partner. Then share the responses as a class. Share Phase, Questions through 5 How did you determine the name of the quantities that are changing? Will the expression you wrote generate the values that are already in your table? What equation did you use to determine the time it will take to climb 8 feet? What equation did you use to determine the time it will take to climb 96 feet? What equation did you use to determine the time it will take to climb 108 feet?. Complete the table shown by first writing the name and the unit of measure for each quantity. Then, write your answers from Problem 1 in the table. Please note that you will complete the table at a later time. To set up your labels for the table, think about what quantities are being measured and how you are counting them. Input Output Quantity Name time height Unit of Measure hours feet Question 1, Part (a) 1 8 Question 1, Part (b) 60 Question 1, Part (c) 3 7 Question 1, Part (d) 3.5 78 Question 5, Part (a) 8 Question 5, Part (b) 5 96 Question 5, Part (c) 6 108 Expression t 1t 1 36 3. Define a variable for the input quantity. Enter this variable in the Expression row at the bottom of Column 1.. Write an expression that you can use to represent the output quantity in terms of the input quantity. Enter this expression in the Expression row under output. 5. Use your expression to write an equation that you can solve to determine each answer. Then, write your answer in the appropriate place in table. a. How long will it be until you have climbed to 8 feet above the beach? b. How long will it be until you have climbed to 96 feet above the beach? c. How long will it be until you have reached the top of the cliff?. Linear Functions 87
Grouping Have students complete Questions 6 through 10 with a partner. Then share the responses as a class. Ask a student to read the information following Question 10 aloud. Discuss this information as a class. Share Phase, Questions 6 through 10 What is the unit of measure on the x-axis? What is the unit of measure on the y-axis? Do the points on the graph form a straight line? Are there points located between the points plotted on the graph? What is their relevance to the problem situation? Does the relation shown on the graph pass the vertical line test? What does this imply? 6. Create a graph to represent the values in your table. Label the horizontal axis with the input quantity and the vertical axis with the output quantity. The axes are already numbered. Finally, plot the points on the coordinate plane. Height (feet) y 110 100 90 80 70 60 50 0 30 0 10 0 0 1 3 5 6 7 8 9 10 11 1 13 1 15 Time (hours) 7. Connect the points on your graphs. See the coordinate plane. 8. Determine the domain and range of this situation. Domain: {all numbers between and including 1 and 6} Range: {all numbers between and including 8 and 108} 9. Does your table represent the same domain and range? Why or why not? The table does not show all the possible climbing times and corresponding heights. 10. Is the relation shown in the graph a function? Explain why or why not. Yes. The relation shown in the graph is a function. For each value of the input, there is only one value of the output. Drawing a line through the data set of a graph is a way to model or represent relationships. The points on your graph represent equivalent ratios because the climbing time per height remained constant. In some problem situations, when you draw a line all the points will make sense. In other problem situations, not all the points on the line will make sense. For example, if a graph displayed the cost per ticket, you cannot purchase a fractional part of a ticket, but the line you would draw on the graph would help you model the relationship and see how the cost changes as more tickets are purchased. So, when you graph relations and model that relationship with a line, it is up to you to consider each situation and interpret the meaning of the data values from a line drawn on a graph. x Remember, the independent quantities are always shown on the horizontal axis and the dependent quantities on the vertical axis. 88 Chapter Linear Functions
Talk the Talk Linear function is defined and students answer questions to help them understand that the graph in the previous problem is a linear function. Grouping Ask a student to read the introduction to the Talk the Talk aloud. Discuss the definition as a class. Have students complete Questions 1 through 6 with a partner. Then share the responses as a class. Share Phase, Questions 1 through How is a linear function different from other functions? How is a linear function similar to other functions? Does the height depend on the time, or does the time depend on the height? Talk the Talk When you graph the input and output values of some functions, the graph forms a straight line. A function whose graph is a straight line is a linear function. The relation shown in the graph in this lesson is a linear function. The graph is a line segment. Let s think about the problem situation, your table, and your graph. 1. Which variable is the dependent variable? The height is the dependent variable.. Which variable is the independent variable? Time is the independent variable. 3. Describe what happens to the value of the dependent variable each time the independent variable increases by 1. Each time the value of the independent variable increases by 1, the value of the dependent variable increases by 1.. Describe what happens to the value of the dependent variable when the independent variable increases by. Each time the value of the independent variable increases by, the value of the dependent variable increases by.. Linear Functions 89
Share Phase, Questions 5 and 6 As the value of the independent variable increases, does the value of the dependent variable increase or decrease? As the value of the independent variable decreases, does the value of the dependent variable increase or decrease? 5. Compare the values of the dependent variable when the independent variable is 1 and 6. Describe how the dependent variable changes in relation to the independent variable. When the independent variable is 1, the dependent variable is 8. When the independent variable is 6, the dependent variable is 108. The independent variable increased by 1 times the change in the independent variable: 1 3 (6 1) 5 60. 6. Describe how the independent and dependent values change in linear functions. The independent and dependent values change at a constant rate. Be prepared to share your solutions and methods. 90 Chapter Linear Functions
Follow Up Assignment Use the Assignment for Lesson. in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson. in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter. Check for Students Understanding Create a table of values for each situation. Explain how you determined the values. Identify the domain and range. 1. Create a table that describes a linear function. Answers will vary. x y 0 0 1 3 6 3 9 I used the linear equation y 5 3x to generate all of the values in the table. Domain: 0, 1,, 3 Range: 0, 3, 6, 9. Linear Functions 90A
. Create a table that describes a function that is not linear. Answers will vary. x y 0 0 1 1 3 9 I used the equation y 5 x to generate all of the values in the table. Domain: 0, 1,, 3 Range: 0, 1,, 9 3. Create a table that describes a relation that is not a function. Answers will vary. x y 0 1 0 0 3 0 I chose one x-value and multiple y-values to generate all of the values in the table. Domain: 0 Range: 1,, 3, 90B Chapter Linear Functions
U.S. Shirts Using Tables, Graphs, and Equations, Part 1 Learning Goals In this lesson, you will: Use different models to represent a problem situation. Determine an initial value when given a final result. Identify the advantages and disadvantages of using a particular representation. Essential Ideas Different methods can be used to represent a problem situation. An initial value can be determined when given a final result. There are advantages and disadvantages of each form of representing the problem situation: a sentence, a graph, a table, and an equation. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 3. Interpret the equation y 5 mx 1 b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Use functions to model relationships between quantities. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)..5 Using Tables, Graphs, and Equations, Part 1 91A
Overview Tables, graphs, and equations can provide different representations of the same problem situation. Students will create equations, tables, and graphs to solve this problem situation. Students then summarize the advantages and disadvantages of using each representation in a graphic organizer. 91B Chapter Linear Functions
Warm Up A paper store sells each ream of high quality paper for $0. They will deliver it for a one-time fee of $5. An office records their total paper costs, c, in dollars, and the total number of reams delivered, r. 1. Complete the table of values. Number of Reams Total Paper Costs Delivered (dollars) 6 65 8 35 10 5 1 505 1 585 16 665. Write the values in the table as ordered pairs. (6, 65) (8, 35) (10, 5) (1, 505) (1, 585) (16, 665) 3. Graph the ordered pairs on the coordinate plane. Total Paper Costs (dollars) 1000 c 800 600 00 00 r 0 8 1 16 0 Number of Reams Delivered. Are the points on the graph collinear? Yes, the points on the graph are collinear. 5. Connect the points on the graph to create a linear model. 6. Why is this linear model misleading for this situation? This linear model is misleading because you cannot purchase partial reams of paper. 7. Is the data set in this situation discrete or continuous? The data set is discrete..5 Using Tables, Graphs, and Equations, Part 1 91C
91D Chapter Linear Functions
U.S. Shirts Using Tables, Graphs, and Equations, Part 1 Learning Goals In this lesson, you will: Use different models to represent a problem situation. Determine an initial value when given a final result. Identify the advantages and disadvantages of using a particular representation. Have ever wondered where your clothes come from? Who actually makes the clothes you wear? For the most part, clothes are made in countries like Vietnam, India, Pakistan, and Mexico, just to name a few. However, only 0 to 50 years ago, clothes were created here in the United States. It was common for people to seek employment creating clothes. Well, the trend of creating clothes in the United States is slowly on the rise. The opening of boutiques and American clothes designers have stressed creating unique and cutting edge fashion, but also not to mass produce clothing and this idea of creating clothes in the United States has reinvented itself. Why do you think clothing began being made in other countries? Do you think the United States will one day become a clothing creating powerhouse that it once was?.5 Using Tables, Graphs, and Equations, Part 1 91
Problem 1 Students calculate the cost for an order of T-shirts for various given values. They will create a table of values for the problem situation, classify the variables as dependent or independent, and create a graph. Students then analyze the various representations that they used to model the problem situation. Grouping Have students complete Questions 1 through 5 with a partner. Then share the responses as a class. Problem 1 Cost Analysis If the order doubles, does the total cost double? This past summer you were hired to work at a custom T-shirt shop, U.S. Shirts. One of your responsibilities is to calculate the total cost of customers orders. The shop charges $8 per shirt plus a one-time charge of $15 to set up a T-shirt design. 1. Describe the problem situation and your responsibility in your own words. I will calculate the total cost of orders. The total cost of an order is the cost of each shirt ordered plus a setup fee. The cost of one shirt is $8, and the setup fee is $15.. What is the total cost of an order for: a. 3 shirts? Total cost in dollars: 3(8) 1 15 5 39 An order of 3 shirts will cost $39. b. 10 shirts? Total cost in dollars: 10(8) 1 15 5 95 An order of 10 shirts will cost $95. Your answers should include the number of shirts and the total cost. Share Phase, Questions 1 through 5 What information is given in this problem? What does the number 8 represent in this situation? What does the number 15 represent in this situation? What are you solving for in Question? What are you solving for in Question? How is Question different from Question? How can you calculate the number of shirts that can be purchased for a given amount of money? c. 100 shirts? Total cost in dollars: 100(8) 1 15 5 815 An order of 100 shirts will cost $815. 3. Explain how you calculated each total cost. I calculated the total costs by first multiplying the number of shirts by the cost of one shirt, and then adding the setup fee to the product.. How many shirts can a customer buy if they have: a. $50 to spend? Number of shirts: 50 15 5.375 8 Because the customer cannot receive a partial shirt, the customer can buy shirts. 9 Chapter Linear Functions
b. $60 to spend? Number of shirts: 60 15 5 5.65 8 Because the customer cannot receive a partial shirt, the customer can buy 5 shirts. What operations do you need to perform to answer each? c. $0 to spend? Number of shirts: 0 15 5 5.65 8 Because the customer cannot receive a partial shirt, the customer can buy 5 shirts. 5. Explain how you calculated the number of shirts that each customer can buy. I determined the numbers of shirts a customer can buy by first subtracting the setup fee from the amount of money available, and then dividing the difference by the cost for one shirt. The decimal portion of each result is dropped because a customer cannot buy a partial shirt. Grouping Have students complete Questions 6 through 10 with a partner. Then share the responses as a class. 6. Complete the table of values for the problem situation. Number of Shirts Ordered Total Cost (dollars) 3 39 Note In Question 6, students will create a table of values for this problem situation. It is important that they be required to create a table of values that is reasonable for the situation and also models examples of different sized orders. Always require that students tables and graphs encompass the entire situation. 7 10 95 5 15 100 815 10 975 150 115.5 Using Tables, Graphs, and Equations, Part 1 93
Share Phase, Questions 6 through 10 What is the smallest possible number of shirts that could be ordered? Why might a group pay the set- up fee for a shirt design, but not buy any shirts? Do you think that it is very common for a group to pay the set- up fee but order no shirts? What is the largest reasonable number of shirts of one design that you think may be ordered? When might a person or group order such a large number of shirts of the same design? 7. What are the variable quantities in this problem situation? Define the variables that can represent these quantities including each quantity s units. The variable quantities are the number of shirts ordered, and the total cost in dollars. I will use s to represent the number of shirts ordered, and I will use C to represent the total cost. 8. What are the constant quantities in this problem situation? Include the units that are used to measure these quantities. The constant quantities are the cost per shirt in dollars, and the setup fee in dollars. 9. Which variable quantity depends on the other variable quantity? The total cost depends on the number of shirts ordered. 10. Which of the variables from Question 7 is the independent variable, and which is the dependent variable? The variable s is the independent variable, and the variable C is the dependent variable. Variable quantities are quantities that change, and constant quantities are quantities that don't change. 9 Chapter Linear Functions
Grouping Have students complete Questions 11 through 15 with a partner. Then share the responses as a class. 11. Create a graph of the data from your table in Question 6 on the grid shown. First, choose your bounds and intervals by completing the table shown. Remember to label your graph clearly and name your graph. Share Phase, Questions 11 through 15 What bounds would be appropriate for the number of shirts? Explain. What bounds would be appropriate for the total cost of the order? Explain. Why aren t negative bounds used in this problem situation? What interval is appropriate for the number of shirts? Is there more than one answer? What interval is appropriate for the total cost of the order? Is there more than one answer? Are the points in the graph collinear? A line best models this problem situation, but why aren t all of the points on the line relevant to the problem situation? Are there other points located between any two points on the graph? Explain. Is the data discrete or continuous? Explain. Why does the model of this problem situation include points that are not relevant to the problem situation? Consider all the data values when choosing your lower and upper bounds. Variable Quantity Lower Bound Upper Bound Interval Number of shirts 0 150 10 Total cost 0 1500 100 Total Cost (dollars) y U.S. Shirts 1500 100 1300 100 1100 1000 900 800 C = 8s + 15 700 600 500 00 300 00 100 0 x 0 10 0 30 0 50 60 70 80 90 100110 10 130 10 150 Number of Shirts 1. Draw a line to model the relationship between the number of shirts and the total cost of the shirts. See coordinate plane. 13. Do all the points on the line make sense in terms of this problem situation? Why or why not? Not all the points make sense because you can not have fractional parts of shirts. 1. Define the variables and write an algebraic equation for the problem situation. The algebraic equation is C 5 8s 1 15. The variable C represents the total cost in dollars, and s represents the number of shirts ordered. Use variables that make sense to you in terms of the problem situation..5 Using Tables, Graphs, and Equations, Part 1 95
15. Define the domain and range for this problem situation. Answers will vary. Some students will state that the domain is all whole numbers, while other students will state that the domain is all natural numbers. The range will also vary. Some students may state that the range is all whole numbers greater than 15. Talk the Talk Students complete a graphic organizer to explain the advantages and disadvantages of representing a situation using a sentence, a table, a graph, and an equation. Grouping Have students complete the graphic organizer independently. Then share the responses as a class. Think about the type of information each representation displays. Talk the Talk So far in this chapter, you have represented problem situations in four different ways: as a sentence, as a table, as a graph, and as an equation. 1. Complete the graphic organizer to explain the advantages and disadvantages of each representation. Sentences allow me to understand what information I need to determine. However, sentences don t give a visual representation of a problem situation. A table gives me specific values for the problem situation. However, it does not show values between those given. A graph allows me to determine different values for the problem situation and to visually see how the data in the problem situation are related. However, the values may not be exact. An equation allows me to generalize the problem situation and calculate any value exactly. However, an equation doesn t give a visual representation of the problem situation. Also think about the types of questions you can answer using each representation. Be prepared to share your solutions and methods. 96 Chapter Linear Functions
Advantages Sentence Advantages Table Disadvantages Disadvantages Multiple Representations Advantages Advantages Disadvantages Disadvantages Graph Equation.5 Using Tables, Graphs, and Equations, Part 1 97
Follow Up Assignment Use the Assignment for Lesson.5 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson.5 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter. Check for Students Understanding A truck rental company charges $150 plus an additional fee for the miles driven. The additional fee for miles driven is shown. 0 9 miles no charge 50 99 miles $5 100 19 miles $50 150 199 miles $75 00 5 miles $100 1. Complete the table. Miles Driven Truck Rental Fee 0 150 50 175 100 00 150 5 00 50. Write the table of values as ordered pairs. (0, 150) (50, 175) (100, 00) (150, 5) (00, 50) 98 Chapter Linear Functions
3. Graph the ordered pairs on the coordinate plane. Truck Rental Fee 50 00 150 100 50 0 100 00 Miles Driven. Is the graph considered a function? Explain your reasoning. The graph is a function. It passes the vertical line test. 5. Are the points on the graph collinear? Yes, the points on the graph are collinear. 6. Connect the points on the graph to create a model of this situation. 7. The linear model of this situation is misleading. Why? The linear model of this situation implies that there are partial charges for mileage between each 50 mile interval and there are no partial charges. 8. Another truck rental company charges an initial fee of $150 and an additional fee of $0.50 per mile driven. a. Write an expression representing this situation. 0.5x 1 150 b. Write an equation representing this situation, where x represents the miles driven, and y represent the total rental fee. y 5 0.5x 1 150 c. How does this equation compare the graph in Question 3? This equation is graphed in Question 3. The graph of this equation accurately models this situation..5 Using Tables, Graphs, and Equations, Part 1 98A
98B Chapter Linear Functions
Hot Shirts Using Tables, Graphs, and Equations, Part Learning Goals In this lesson, you will: Use different methods to represent a problem situation. Estimate values of expressions that involve decimals. Determine an initial value when given a final result. Key Terms estimation point of intersection Essential Ideas Different methods can be used to represent a problem situation. Values in expressions that involve decimals can be estimated. The initial value can be determined when given a final result. Two problem situations can be compared algebraically. Two problem situations can be compared graphically. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. 3. Interpret the equation y 5 mx 1 b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. Use functions to model relationships between quantities. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions)..6 Using Tables, Graphs, and Equations, Part 99A
Overview Problem situations can be compared both algebraically and graphically. Students compare and analyze the U.S. Shirts and Hot Shirts Problem situations algebraically and graphically. They will then write a response that compares the pricing plans for the two companies and predict how the pricing by Hot Shirts will affect the business of U.S. Shirts. Problem situations are represented using sentences, a table of values, equations in two variables, and a graph. Estimation is used to determine the value of expressions that involve decimals which are otherwise difficult to calculate. Students will estimate the value of expressions that involve decimals. Students also determine the initial value for a number when given the final result. 99B Chapter Linear Functions
Warm Up Two catering companies host children s sports banquets. Company A charges a fixed fee of $100 plus $3 per person. Company B charges a fixed fee of $75 plus $ per person. The total charge for each company for any number of persons, p, can be calculated using the equations shown. Company A: c 5 3p 1 100 Company B: c 5 p 1 75 Which company would charge less to cater for 00 people? Company A: 3(00) 1 100 5 700 Company B: (00) 1 75 5 875 Company A would charge $175 less to cater for 00 people..6 Using Tables, Graphs, and Equations, Part 99C
99D Chapter Linear Functions
Hot Shirts Using Tables, Graphs, and Equations, Part Learning Goals In this lesson, you will: Use different methods to represent a problem situation. Estimate values of expressions that involve decimals. Determine an initial value when given a final result. Key Terms estimation point of intersection You might be surprised to know that the word T-shirt wasn t really used until the 190s. And, until the 1950s, people thought of T-shirts as underwear. Popular actors like John Wayne and James Dean surprised audiences in the mid-1950s by wearing this underwear on screen! Since then, T-shirts have become one of the most popular items of clothing in the world..6 Using Tables, Graphs, and Equations, Part 99
Problem 1 Students calculate the cost for orders of T-shirts for various given values from a competitor with a different cost value. The number of shirts that can be purchased for various amounts of money is estimated. Students will create a table and graph to represent the problem situation. Grouping Have students complete Questions 1 through 3 with a partner. Then share the responses as a class. Share Phase, Questions 1 through 3 What information is given in this problem? What is a competitor? What does the number 5.50 represent in this situation? What does the number 9.95 represent in this situation? How does the set- up fee for Hit Shirts compare to the setup fee for U.S. Shirts from the previous lesson? How does the per shirt rate for Hot Shirts compare to the per shirt rate for U.S. Shirts from the previous lesson? How can you calculate the cost of an order of shirts from Hot Shirts? Problem 1 Analyzing the Competition Previously, you explored a job at U.S. Shirts. One of U.S. Shirts competitors, Hot Shirts, advertises that it makes custom T-shirts for $5.50 each with a one-time setup fee of $9.95. Your boss brings you the advertisement from Hot Shirts and asks you to figure out how the competition might affect business. 1. Describe the problem situation and how it will affect business in your own words. I should determine the competitor s total cost of orders. The total cost of an order is the cost of each shirt ordered plus a setup fee. The cost of one shirt is $5.50, and the setup fee is $9.95.. What is the total customer cost of an order for: a. 3 shirts from Hot Shirts? Total customer cost in dollars: 3(5.50) 1 9.95 5 66.5 An order of 3 shirts will cost a customer $66.5 from Hot Shirts. b. 10 shirts from Hot Shirts? Total customer cost in dollars: 10(5.50) 1 9.95 5 10.95 An order of 10 shirts will cost a customer $10.95 from Hot Shirts. c. 50 shirts from Hot Shirts? Total customer cost in dollars: 50(5.50) 1 9.95 5 3.95 An order of 50 shirts will cost a customer $3.95 from Hot Shirts. d. 100 shirts from Hot Shirts? Total customer cost in dollars: 100(5.50) 1 9.95 5 599.95 An order of 100 shirts will cost a customer $599.95 from Hot Shirts. What is your initial prediction? Is Hot Shirts a strong competitor for US Shirts? 3. Explain how you calculated the total customer costs. I determined the total customer cost by first multiplying the number of shirts by the cost of one shirt and then adding the setup fee to the product. 100 Chapter Linear Functions
Grouping Ask a student to read the information prior to Question aloud. Discuss the worked example and complete Question as a class. Have students complete Questions 5 and 6 with a partner. Then share the responses as a class. Remember, you can use estimation to determine the approximate values before you do actual calculations to get a sense of the answer. For example, you can estimate the difference of 15.35 and 8.95. So, you could round 15.35 down to 15, and round 8.95 up to 85. Then, calculate the difference of 15 and 85, to 15 85 5 0. You can write this as 15.35 8.95 0. The symbol means is approximately equal to. Discuss Phase, Question What does it mean to estimate a value? What symbol can be used to show that the answer is an estimate? Have you used any other symbols to show that the answer is an estimate? Why does it make sense to estimate the answers to these questions? Share Phase, Questions 5 and 6 If someone could pay for 10. shirts, what is the approximate total cost? Can someone buy more shirts than they can pay for? When you estimate the number of shirts, do you round in the same way as when you estimate the other values? If someone could pay for 10.98 shirts, approximately how many shirts could they buy?. Estimate the value of each expression. a. 78.75 1 60. 750 1 60 5 810 78.75 1 60. < 810 b. 35 1 350 10 5 10 35 1 < 10 c. 5.13(0.) 5(0) 5 900 5.13(0.) < 900 5. Estimate the number of shirts that a customer can purchase from Hot Shirts for: a. $50. Number of shirts: 50 9.95 < 0 5.50 6 5 0 The customer can buy no shirts from Hot Shirts for $50. b. $60. Number of shirts: 60 9.95 < 10 5.50 5 1. 6 6 The customer can buy approximately shirts from Hot Shirts for $60. c. $0. Number of shirts: 0 9.95 < 170 5.50 5 8. 3 6 The customer can buy approximately 8 shirts from Hot Shirts for $0..6 Using Tables, Graphs, and Equations, Part 101
6. Explain how you used estimation to efficiently determine the number of shirts that can be purchased. I determined the number of shirts by first subtracting the rounded-up setup fee from the amount of money available, and then dividing the difference by the rounded-up cost for one shirt. Grouping Have students complete Questions 7 through 9 with a partner. Then share the responses as a class. Share Phase, Question 7 What is the smallest number of shirts possible for an order? What is the largest reasonable number of shirts for an order? What are the smallest possible and largest reasonable total costs? 7. Complete the table of values for the problem situation. Number of Shirts Ordered Total Cost (dollars) 3 66.5 71.95 10 10.95 5 187.5 100 599.95 10 709.95 150 87.95 Round to the nearest penny. 10 Chapter Linear Functions
Share Phase, Questions 8 and 9 What bounds are appropriate for the number of shirts? What bounds are appropriate for the total cost of the order? Explain. What interval is appropriate for the number of shirts? What interval is appropriate for the total cost? Explain. Was this problem easier or more difficult to complete than the U.S. Shirts Problem? What representations did you use to model the situation in this lesson? How did the advantages and disadvantages for each representation used in this lesson compare to the advantages and disadvantages you found in the U.S. Shirts Problem? 8. Create a graph of the data from the table on the grid shown. First, choose your bounds and intervals by completing the table shown. Remember to label your graph clearly and name your graph. Variable Quantity Lower Bound Upper Bound Interval Number of shirts 0 150 10 Total cost 0 1500 100 Total Cost (dollars) y Hot Shirts 1500 100 1300 100 1100 1000 900 800 C = 5.5s + 9.95 700 600 500 00 300 00 100 0 0 10 0 30 0 50 60 70 80 90 100110 10 130 10 150 Number of Shirts x 9. Define the variables and write an algebraic equation for this problem situation. The variable C represents the total cost in dollars, and s represents the number of shirts ordered. The equation is C 5 5.5s 1 9.95. How did you define the variables in the U.S. Shirt problem?.6 Using Tables, Graphs, and Equations, Part 103
Problem Students compare the cost for orders of T-shirts from U.S. Shirts and from Hot Shirts. They will graph the total cost of an order from both U.S. Shirts and Hot Shirts on the same grid. Finally, students write a report that compares and analyzes the two different shirt companies. Grouping Have students complete Questions 1 through 9 with a partner. Then share the responses as a class. Problem Which Is the Better Buy? You have explored the costs of ordering T-shirts from two companies, U.S. Shirts and Hot Shirts. Your boss asked you to determine which company has the better price for T-shirts in different situations. 1. Would you recommend U.S. Shirts or Hot Shirts as the better buy for an order of five or fewer T-shirts? What would each company charge for exactly five shirts? Describe how you calculated your answer. Total cost from U.S. Shirts in dollars: 5(8) 1 15 5 55 Total cost from Hot Shirts in dollars: 5(5.50) 1 9.95 5 77.5 U.S. Shirts would charge $55 for five shirts, and Hot Shirts would charge $77.5 for five shirts. Because of the setup fee, U.S. Shirts gives a better price for five shirts or fewer. To calculate the total cost of ordering, I multiplied the number of shirts by the cost of one shirt, and then I added the setup fee to the product. Share Phase, Questions 1 through 3 What information is given in this problem? What is being asked in this problem situation? What was the equation for C, the total cost of an order of s shirts from U.S. Shirts? What was the equation for C, the total cost of an order of s shorts from Hot Shirts? How are the equations different? How are the equations similar?. For an order of 18 shirts, which company s price is the better buy? How much better is the price? Explain your reasoning. Total cost from U.S. Shirts in dollars: 18(8) 1 15 5 159 Total cost from Hot Shirts in dollars: 18(5.50) 1 9.95 5 18.95 Difference in cost in dollars: 159 18.95 5 10.05 For an order of 18 shirts, the Hot Shirts price is the better buy by $10.05. To determine the better buy, I first calculated the total cost of ordering from each company. Then, I subtracted the total costs to calculate the difference between them. 10 Chapter Linear Functions
Share Phase, Question What two variable quantities are important from the U.S. Shirts Problem? What two variable quantities are important from the Hot Shirts Problem? What did you use as the label for the horizontal axis on your graph for the U.S. Shirts Problem? What did you use as the label for the horizontal axis on your graph for the Hot Shirts Problem? What did you use as the label for the vertical axis on your graph for the U.S. Shirts Problem? What did you use as the label for the vertical axis on your graph for the Hot Shirts Problem? What is the smallest possible number of shirts that each company can sell? What is the largest reasonable size of order for each company? What bounds should you use for your combined graph for the number of shirts? What is the smallest possible total cost and largest reasonable total cost of an order from each of the companies? What bounds should you use for your combined graph for the total cost? How can you graph two different lines on the same graph? What do you think the graph will look like? Make sure you label each graph. 3. For an order of 80 shirts, which company s price is better? How much better is the price? Explain your reasoning. Total cost from U.S. Shirts in dollars: 80(8) 1 15 5 655 Total cost from Hot Shirts in dollars: 80(5.50) 1 9.95 5 89.95 Difference in cost in dollars: 655 89.95 5 165.05 For an order of 80 shirts, the Hot Shirts price is the better buy by $165.05. I first calculated the total cost of ordering from each company. Then, I subtracted the total costs to calculate the difference between them.. Create the graphs for the total cost for U.S. Shirts and Hot Shirts on the grid shown. First, determine the bounds and intervals for the grid by completing the table shown. Variable Quantity Lower Bound Upper Bound Interval Number of shirts 0 150 10 Total cost 0 1500 100 Total Cost (in dollars) y Comparing U.S. Shirts and Hot Shirts 1500 100 1300 100 1100 C = 8s + 15 1000 900 800 700 600 C = 5.5s + 9.95 500 00 300 00 100 0 x 0 10 0 30 0 50 60 70 80 90 1001101013010150 Number of Shirts.6 Using Tables, Graphs, and Equations, Part 105
5. Estimate the number of T-shirts for which the total costs are the same. Explain how you determined the number of T-shirts. The total costs are the same when about 1 shirts are ordered. The total costs are the same where the graphs cross each other, so I started at this point and moved down and read the number of shirts for this total cost. 6. For how many T-shirts is it more expensive to order from U.S. Shirts? U.S. Shirts is more expensive to order from when the order is more than 1 T-shirts. 7. For how many T-shirts is it more expensive to order from Hot Shirts? Hot Shirts is more expensive to order from when the order is fewer than 1 T-shirts. 8. Look at your graph. Describe the graphs of the lines in your own words. The graphs are straight lines that start at the bottom left of the graph and move to the upper right of the graph. The graph for U.S. Shirts is steeper than the graph for Hot Shirts. The graphs cross one another at about (1, 17). Notice that the graphs intersect at about (1, 17). This point of intersection indicates where the total cost for each company is the same. So, when U.S. Shirts sells 1 shirts, the total cost is $17, and when Hot Shirts sells 1 shirts, the total cost is $17. 9. Write a response to your boss that compares the costs of ordering from each company. Try to answer your boss s question, Will Hot Shirts prices affect the business at U.S. Shirts? Students should write a report that includes the following key points. Hot Shirts will affect the business at U.S. Shirts for those customers who order large numbers of T-shirts. The main customers for U.S. Shirts should continue to be customers who order small numbers of T-shirts. Be prepared to share your solutions and methods. 106 Chapter Linear Functions
Follow Up Assignment Use the Assignment for Lesson.6 in the Student Assignments book. See the Teacher s Resources and Assessments book for answers. Skills Practice Refer to the Skills Practice worksheet for Lesson.6 in the Student Assignments book for additional resources. See the Teacher s Resources and Assessments book for answers. Assessment See the Assessments provided in the Teacher s Resources and Assessments book for Chapter. Check for Students Understanding Two catering companies have different charges. Company A charges a fixed fee of $75 plus $5 per person. Company B charges a fixed fee of $100 plus $ per person. The total charge for each company for any number of persons, p, can be calculated using the equations shown. Company A: c 5 5p 1 75 Company B: c 5 p 1 100 1. Use the equations to complete a table of values for each company and use the table of values to list the ordered pairs. Company A Number of People Cost 0 75 10 15 0 175 30 5 (0, 75) (10, 15) (0, 175) (30, 5) Company B Number of People Cost 0 100 10 10 0 180 30 0 (0, 100) (10, 10) (0, 180) (30, 0).6 Using Tables, Graphs, and Equations, Part 106A
. Create graphs for both Company A and Company B on the coordinate plane. 50 00 Cost 150 100 50 0 0 0 Number of People 3. Where do the two lines intersect? The two lines intersect at the point (5, 00). What is the significance of the point of intersection with respect to the problem situation? The point of intersection tells us that if we cater an event for 5 guests, both Company A and Company B charge $00. 5. If you were planning an event for less than 5 guests, which company would you hire? Explain your reasoning. I would hire Company A if there were less than 5 guests because I would get a better price. 6. If you were planning an event for more than 5 guests, which company would you hire? Explain your reasoning. I would hire Company B if there were more than 5 guests because I would get a better price. 106B Chapter Linear Functions
What, Not Lines? Introduction to Non-Linear Functions Learning Goals In this lesson, you will: Define, graph, and analyze non-linear functions, including: absolute value area of a square volume of a cube Key Terms absolute value function square or quadratic function cube or cubic function Essential Ideas Non-linear functions such as absolute value functions, quadratic functions, and cubic functions are defined, graphed, and analyzed. An absolute value function is a function that can be written in the form f(x) 5 x and has a V shaped graph. A quadratic function is a function that can be written in the form f(x) 5 ax 1 bx 1 c, where a, b, and c are any numbers and a is not equal to zero. A quadratic function has a U shaped graph. A cubic function is a function that can be written in the form f(x) = a 3 x 3 1 a x 1 a 1 x 1 a 0. Common Core State Standards for Mathematics 8.F Functions Define, evaluate, and compare functions. 1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Use functions to model relationships between quantities. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally..7 Introduction to Non-Linear Functions 107A
Overview Students explore functions that are not linear such as absolute value functions, quadratic functions, and cubic functions. They will first evaluate each function for given values. Next, they use an equation to generate a table of values, and then use the table of values to graph the function. Students will determine that each graph is function different than a linear function and answer related questions. 107B Chapter Linear Functions
Warm Up 1. Use the vertical line test to determine which graphs are functions. A. B. y 8 6 y 8 6 8 6 6 8 x 8 6 6 8 x 6 6 8 Function 8 Function C. D. y 8 y 8 6 6 8 6 6 8 x 8 6 6 8 x 6 6 8 Function 8 Function. Are any of the graphs considered linear functions? Explain your reasoning. Graph B is the only linear function. All of the points on the other graphs are not collinear..7 Introduction to Non-Linear Functions 107C
107D Chapter Linear Functions
What, Not Lines? Introduction to Non-Linear Functions Learning Goals In this lesson, you will: Define, graph, and analyze non-linear functions, including: absolute value area of a square volume of a cube Key Terms absolute value function square or quadratic function cube or cubic function Have someone in your class think of a whole number from 1 to 0. Ask each other student in the class to guess what the number is. Record all the guesses without revealing the mystery number. On the graph shown, have the recorder determine each guess on the x-axis and plot its distance (shown on the y-axis) from the mystery number. What is the mystery number? Did you graph a function? Distance from Mystery Number y 0 19 18 17 16 15 1 13 1 11 10 9 8 7 6 5 3 1 0 1 3 5 6 7 8 9 10 11 1 13 1 15 16 17 18 19 0 Guess x.7 Introduction to Non-Linear Functions 107
Problem 1 Students evaluate absolute value expressions. They will complete a table of values for an absolute value equation, and then graph the table of values. Students conclude the graph is a function but not a linear function. Grouping Have students complete Questions 1 through 8 with a partner. Then share the responses as a class. Share Phase, Questions 1 and What is the distance from -3 to zero on a number line? What is the distance from 11 to zero on a number line? What is the distance from -5 to zero on a number 3 line? What is the distance from 110.89 to zero on a number line? Problem 1 The V Recall that the absolute value of a number is defined as the distance from the number to zero on a number line. The symbol for absolute value is x. 1. Evaluate each expression shown. a. 3 5 3 b. 11 5 11 c. 5 3 5 5 3 d. 110.89 5 110.89. Use the function y 5 x, to complete the table. If it is stated that you are working with a function, what does that tell you about the relationship between the input and output values? x y 5 x 7 7 3 3 1 1 0.5 0.5 0 0 7 7 108 Chapter Linear Functions
Share Phase, Questions 3 through 8 How would you describe the graph of this absolute value equation? Is the graph considered a function? Why or why not? Does the graph pass the vertical line test? Explain. What is the maximum or greatest value of y? How do you know? How is an absolute value function different than a linear function? How is an absolute value function similar to a linear function? 3. Graph the values from the table on the coordinate plane. 8 6 8 6. Connect the points to model the relationship of the equation y 5 x. See graph. 6 8 5. What is the domain of this function? Do all the points on the graph make sense in terms of the equation y 5 x. Explain your reasoning. The domain is the set of all numbers. It makes sense to connect these points because the absolute value can be determined for every number. y 6 8 x Keep in mind, the domain and range represent sets of numbers. 6. Does the graph of these points form a straight line? Explain your reasoning. No. The points go down and then back up. The distance from the number to zero gets smaller as the points get closer to zero and then larger as the points get farther away. 7. What is the minimum, or least value of y? How do you know? State the range of this function. The least value of y is zero. The range is all numbers greater than or equal to 0. 8. Is this a linear function? Explain your reasoning. No. This is not a linear function because its graph is not a straight line, but is more like two parts of two different lines. You have just graphed an absolute value function. An absolute value function is a function that can be written in the form f(x) 5 x, where x is any number. Function notation can be used to write functions such that the dependent variable is replaced with the name of the function, such as f(x)..7 Introduction to Non-Linear Functions 109
Problem Students calculate the area of squares, given the length of a side. They will complete a table of values for a quadratic equation, and then graph the table of values. Students conclude the graph is a function but not a linear function and not an absolute value function. Grouping Have students complete Questions 1 through 8 with a partner. Then share the responses as a class. Problem Not V b ut U Recall that the area of a square is equal to the side length, s, multiplied by itself and is written as A 5 s. 1. Calculate the area of squares with side lengths that are: a. 3 inches. A 5 s 5 (3) 5 9 square inches b. 5 feet. A 5 s 5 (5) 5 5 square feet c.. centimeters. A 5 s 5 (.) 5 5.76 cm d. 1 5 8 inches. A 5 s 5 ( 1 5 8 ) 5 ( 101 8 ) ( 101 8 ) 5 10,01 5 159 6 5 square inches 6 Share Phase, Questions 1 and What unit of measure is used to describe the length of a side of a square? What unit of measure is used to describe the area of a square? What is the sign of a negative number after it has been squared? In the table of values, is it possible for y to have a negative value? Why or why not? In the equation A 5 s the side length of a square, s, is the independent variable and the area of a square, A, is the dependent variable. This formula can also be modeled by the equation y 5 x, where x represents the side length of a square and y represents the area of a square.. Use the equation, y 5 x, to complete the table. x y 5 x 3 9 1 1 0.5 0.5 0 0.3 5.9 3 9 Does this equation represent a function? 110 Chapter Linear Functions
Share Phase, Questions 3 through 8 How would you describe the graph of this squared equation? Is the graph considered a function? Why or why not? Does the graph pass the vertical line test? Explain. What is the maximum or greatest value of y? How do you know? How is this quadratic function different than a linear function? How is this quadratic function similar to a linear function? How is this quadratic function different than an absolute value function? How is this quadratic function similar to an absolute value function? 3. Graph the values from the table on the coordinate plane. 8 6 8 6. Connect the points to model the relationship of the equation y 5 x. See graph. 6 8 y 5. What is the domain of this function? Do all the points on the graph make sense in terms of the equation y 5 x. Explain your reasoning. The domain is the set of all numbers. It makes sense to connect the points since the square of a number can be determined for every number. 6. What is the minimum, or least value of y? How do you know? State the range of this function. The minimum value of y is zero since 0 3 0 5 0 and the product of any other number and itself is greater than zero. The range is the set of all numbers greater than or equal to 0. 6 8 x 7. Does the graph of these points form a straight line? Explain your reasoning. No. The points go down and then back up. Because the square of a number is always positive, the points are in the first and second quadrants. 8. Is this a linear function? Explain your reasoning. No. This is not a linear function because the graph is not a straight line but looks like a U. You have just graphed a quadratic function. A quadratic function is a function that can be written in the form f(x) 5 ax 1 bx 1 c, where a, b, and c are any numbers and a is not equal to zero..7 Introduction to Non-Linear Functions 111
Problem 3 Students calculate the volume of cubes, given the edge length. They will complete a table of values for a cubic equation, and then graph the table of values. Students conclude the graph is a function but not a linear function, not an absolute value function, and not a quadratic function. Grouping Have students complete Questions 1 through 8 with a partner. Then share the responses as a class. Problem 3 Not V or U Recall that the volume of a cube is defined as the product of the length of one edge times itself 3 times and is written as V 5 s 3. 1. Calculate the volume of cubes with an edge length that is: a. inches. V 5 s 3 5 () 3 5 8 cubic inches b. 1.5 feet. V 5 s 3 5 (1.5) 3 5 3.375 cubic feet c..1 centimeters. V 5 s 3 5 (.1) 3 5 9.61 cubic centimeters d. 1 3 inches. V 5 s 3 5 ( 1 3 ) 3 5 ( 7 ) 3 5 33 6 5 5 3 cubic inches 6 Share Phase, Questions 1 and What unit of measure is used to describe the edge length of a cube? What unit of measure is used to describe the volume of a cube? What is the sign of a negative number after it has been cubed? In the table of values, is it possible for y to have a negative value? Why or why not? In the equation V 5 s 3, the side length of a cube, s, is the independent variable and the volume of the cube, V is the dependent variable. This formula can also be modeled by the equation y 5 x 3, where x represents the side length of a cube and y represents the volume of a cube.. Use the equation, y 5 x 3, to complete the table. x y 5 x 3 8 1.5 3.375 1 1 0.5 0.15 0 0 1.5 3.375 8.1 9.61 Does this equation represent a function? 11 Chapter Linear Functions