Proportional and Nonproportional Relationships and Functions

Transcription

1 UNIT Proportional and Nonproportional Relationships and Functions MDULE 313 Proportional Relationships MDULE 8.EE..5, 8.EE..6, 8.F.1., 8.F.. Nonproportional Relationships MDULE 5 8.EE..6, 8.F.1., 8.F.1.3, 8.F.. Writing Linear Equations 8.F.., 8.SP.1.1, 8.SP.1., 8.SP.1.3 MDULE 6 Functions 8.EE..5, 8.F.1.1, 8.F.1., 8.F.1.3, 8.F.., 8.F..5 Image Credits: Adrian Bradshaw/EPA/Corbis Unit Performance Task At the end of the unit, check out how cost estimators use math. CAREERS IN MATH Cost Estimator A cost estimator determines the cost of a product or project, which helps businesses decide whether or not to manufacture a product or build a structure. Cost estimators analze the costs of labor, materials, and use of equipment, among other things. Cost estimators use math when the assemble and analze data. If ou are interested in a career as a cost estimator, ou should stud these mathematical subjects: Algebra Trigonometr Calculus Research other careers that require analzing costs. Unit 65

2 UNIT Vocabular Preview Use the puzzle to preview ke vocabular from this unit. Unscramble the circled letters within the found words to answer the riddle at the bottom of the page. D I N T F K F Z Y X I R H L T I D N T N I R A V Z B I U U R Z I E K N M C J E H Z N P N X E V T T H W C V W S E E T A E B U E C E K Q P D U A U I V A R N P N M K Q W N Z R X G C B I Q U M Y X P L E C P E B L K B T J F H L A K Q J P V S L Q M J R G H R U N U T E F N I Z B M C A D A X A A T T S V D T H W B T D B E T K I I E Y W U Q E P K I P G T G L S J Q R F D I Q D T A M H Q T S T A J Q X N B I V A R I A T E D A T A Y N The -coordinate of the point where the graph crosses the -ais. (Lesson.) A rule that assigns eactl one output to each input. (Lesson 6.1) The result after appling the function machine s rule. (Lesson 6.1) A rate in which the second quantit in the comparison is one unit. (Lesson 3.3) The ratio of change in rise to the corresponding change in run on a graph. (Lesson 3.) A set of data that is made up of two paired variables. (Lesson 5.3) An equation whose solutions form a straight line on a coordinate plane. (Lesson.1) Q: How much of the mone earned does a professional sports team pa its star athlete? A: An! 66 Vocabular Preview

3 Proportional Relationships MDULE 3? ESSENTIAL QUESTIN How can ou use proportional relationships to solve real-world problems? LESSN 3.1 Representing Proportional Relationships 8.EE..6, 8.F.. LESSN 3. Rate of Change and Slope 8.F.. LESSN 3.3 Interpreting the Unit Rate as Slope 8.EE..5, 8.F.1., 8.F.. Image Credits: Angelo Giampiccolo/Shutterstock Real-World Video Speedboats can travel at fast rates while sailboats travel more slowl. If ou graphed distance versus time for both tpes of boats, ou could tell b the steepness of the graph which boat was faster. Math n the Spot Animated Math Personal Math Trainer Go digital with our write-in student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 67

4 Are YU Read? Complete these eercises to review skills ou will need for this module. Write Fractions as Decimals Personal Math Trainer nline Assessment and Intervention EXAMPLE =? Multipl the numerator and the denominator b a power of 10 so that the denominator is a whole number. Write the fraction as a division problem. Write a decimal point and zeros in the dividend. Place a decimal point in the quotient. Divide as with whole numbers = Write each fraction as a decimal. 1. 3_ _ Solve Proportions EXAMPLE 5_ 7 = = = 1 = 10 Solve each proportion for = = = _ =1, so multipl the numerator and denominator b. 5 =10 1 = = _ = 6 _ _ = 16 _ 9 = _ = Unit

5 Reading Start-Up Visualize Vocabular Use the words to complete the diagram. :6, 3 to Reviewing Proportions Understand Vocabular 5 10, 5 50, Match the term on the left to the definition on the right. 1 inches, 1 foot $1.5 per ounce Vocabular Review Words constant (constante) equivalent ratios (razones equivalentes) proportion (proporción) rate (tasa) ratios (razón) unit rates (tasas unitarias) Preview Words constant of proportionalit (constante de proporcionalidad) proportional relationship (relación proporcional) rate of change (tasa de cambio) slope (pendiente) 1. unit rate A. A constant ratio of two variables related proportionall.. constant of B. A rate in which the second quantit proportionalit in the comparison is one unit. 3. proportional C. A relationship between two relationship quantities in which the ratio of one quantit to the other quantit is constant. Active Reading Ke-Term Fold Before beginning the module, create a ke-term fold to help ou learn the vocabular in this module. Write the highlighted vocabular words on one side of the flap. Write the definition for each word on the other side of the flap. Use the ke-term fold to quiz ourself on the definitions used in this module. Module 3 69

6 MDULE 3 Unpacking the Standards Understanding the standards and the vocabular terms in the standards will help ou know eactl what ou are epected to learn in this module. 8.EE..5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different was. Ke Vocabular proportional relationship (relación proporcional) A relationship between two quantities in which the ratio of one quantit to the other quantit is constant. slope (pendiente) A measure of the steepness of a line on a graph; the rise divided b the run. unit rate (tasa unitaria) A rate in which the second quantit in the comparison is one unit. Visit to see all Florida Common Core Standards unpacked. What It Means to You You will use data from a table and a graph to appl our understanding of rates to analzing real-world situations. UNPACKING EXAMPLE 8.EE..5 The table shows the volume of water released b Hoover Dam over a certain period of time. Use the data to make a graph. Find the slope of the line and eplain what it shows. Water Released from Hoover Dam Time (s) Volume of water (m 3 ) 5 75, , , ,000 Volume of water (m 3 ) 350, ,000 50,000 00, , ,000 50,000 Water Released from Hoover Dam Time (s) The slope of the line is 15,000. This means that for ever second that passed, 15,000 m 3 of water was released from Hoover Dam. Suppose another dam releases water over the same period of time at a rate of 50 m 3 per minute. How do the two rates compare? 50 m 3 per minute is equal to 3,000 m 3 per second. This rate is one fifth the rate released b the Hoover Dam over the same time period. Image Credits: Gett Images 70 Unit

7 ? LESSN 3.1 Representing Proportional ESSENTIAL QUESTIN Relationships 8.EE..6 derive the equation = m for a line through the origin Also 8.F.. How can ou use tables, graphs, and equations to represent proportional situations? EXPLRE ACTIVITY Prep for 8.EE..6 Representing Proportional Relationships with Tables In 1870, the French writer Jules Verne published 0,000 Leagues Under the Sea, one of the most popular science fiction novels ever written. ne definition of a league is a unit of measure equaling 3 miles. A Complete the table. Distance (leagues) Distance (miles) 1 6 0, B What relationships do ou see among the numbers in the table? C D For each column of the table, find the ratio of the distance in miles to the distance in leagues. Write each ratio in simplest form. 3 1 = = = 6 What do ou notice about the ratios? Reflect 1. If ou know the distance between two points in leagues, how can ou find the distance in miles? 36 =. If ou know the distance between two points in miles, how can ou find = 0,000 the distance in leagues? Lesson

8 Math n the Spot Representing Proportional Relationships with Equations The ratio of the distance in miles to the distance in leagues is constant. This relationship is said to be proportional. A proportional relationship is a relationship between two quantities in which the ratio of one quantit to the other quantit is constant. A proportional relationship can be described b an equation of the form = k, where k is a number called the constant of proportionalit. Sometimes it is useful to use another form of the equation, k = _. EXAMPLE 1 8.EE..6 Meghan earns $1 an hour at her part-time job. Show that the relationship between the amount she earned and the number of hours she worked is a proportional relationship. Then write an equation for the relationship. STEP 1 Make a table relating amount earned to number of hours. For ever hour Meghan works, she earns $1. So, for 8 hours of work, she earns 8 $1 = $96. Number of hours 1 8 Amount earned ($) Math Talk Personal Math Trainer nline Assessment and Intervention STEP Mathematical Practices STEP 3 Describe two real-world quantities with a proportional relationship that can be described b the equation = 5. YUR TURN For each number of hours, write the relationship of the amount earned and the number of hours as a ratio in simplest form. amount earned number of hours Since the ratios between the two quantities are all equal to 1, the relationship is proportional. Write an equation. 1 1 = 1 Let represent the number of hours. Let represent the amount earned. Use the ratio as the constant of proportionalit in the equation = k. The equation is = 1. = 1 3. Fifteen biccles are produced each hour at the Speed Bike Works. Show that the relationship between the number of bikes produced and the number of hours is a proportional relationship. Then write an equation for the relationship. 8 = = 1 First tell what the variables represent. 7 Unit

9 Representing Proportional Relationships with Graphs You can represent a proportional relationship with a graph. The graph will be a line that passes through the origin (0, 0). The graph shows the relationship between distance measured in miles to distance measured in leagues. Miles 10 5 (,6) (1,3) (3,9) Math n the Spot EXAMPLE 5 10 Leagues 8.EE..6 The graph shows the relationship between the weight of an object on the Moon and its weight on Earth. Write an equation for this relationship. STEP 1 Use the points on the graph to make a table. Earth weight (lb) Moon weight (lb) Moon weight (lb) Earth weight (lb) STEP Find the constant of proportionalit. Image Credits: David Epperson/ PhotoDisc/Gett Images STEP 3 YUR TURN Moon weight Earth weight The constant of proportionalit is 1_ 6. Write an equation. 1_ 6 = 1_ 6 Let represent weight on Earth. Let represent weight on the Moon. The equation is = 1_ 6. 1 = 1_ 6 The graph shows the relationship between the amount of time that a backpacker hikes and the distance traveled.. What does the point (5, 6) represent? 5. What is the equation of the relationship? 3 18 = 1_ 6 Distance (mi) = 1_ 6 Replace k with 1 in = k Time (h) Personal Math Trainer nline Assessment and Intervention Lesson

10 Guided Practice 1. Vocabular A proportional relationship is a relationship between two quantities in which the ratio of one quantit to the other quantit is / is not constant.. Vocabular When writing an equation of a proportional relationship in the form = k, k is replaced with the. 3. Write an equation that describes the proportional relationship between the number of das and the number of weeks in a given length of time. (Eplore Activit and Eample 1) a. Complete the table. Time (weeks) 1 10 Time (das) 7 56 b. Let represent. Let represent. The equation that describes the relationship is. Each table or graph represents a proportional relationship. Write an equation that describes the relationship. (Eample 1 and Eample )?. Phsical Science The relationship between the numbers of ogen atoms and hdrogen atoms in water gen atoms Hdrogen atoms ESSENTIAL QUESTIN CHECK-IN If ou know the equation of a proportional relationship, how can ou draw the graph of the equation? Actual distance (mi) Map of Iowa Distance (in.) 7 Unit

11 Name Class Date 3.1 Independent Practice 8.EE..6, 8.F.. The table shows the relationship between temperatures measured on the Celsius and Fahrenheit scales. Celsius temperature Fahrenheit temperature Is the relationship between the temperature scales proportional? Wh or wh not? Personal Math Trainer nline Assessment and Intervention 8. Describe the graph of the Celsius-Fahrenheit relationship. 9. Analze Relationships Ralph opened a savings account with a deposit of $100. Ever month after that, he deposited $0 more. a. Wh is the relationship described not proportional? b. How could the situation be changed to make the situation proportional? 10. Represent Real-World Problems Describe a real-world situation that can be modeled b the equation = 1 0. Be sure to describe what each variable represents. Look for a Pattern The variables and are related proportionall. 11. When = 8, = 0. Find when =. 1. When = 1, = 8. Find when = 1. Lesson

12 13. The graph shows the relationship between the distance that a snail crawls and the time that it crawls. a. Use the points on the graph to make a table. Distance (in.) Time (min) b. Write the equation for the relationship and tell what each variable represents. Time (min) Snail Crawling Distance (in.) c. How long does it take the snail to crawl 85 inches? FCUS N HIGHER RDER THINKING Work Area 1. Communicate Mathematical Ideas Eplain wh all of the graphs in this lesson show the first quadrant but omit the other three quadrants. 15. Analze Relationships Complete the table. Length of side of square Perimeter of square Area of square a. Are the length of the side of a square and the perimeter of the square related proportionall? Wh or wh not? b. Are the length of the side of a square and the area of the square related proportionall? Wh or wh not? 16. Make a Conjecture A table shows a proportional relationship where k is the constant of proportionalit. The rows are then switched. How does the new constant of proportionalit relate to the original one? 76 Unit

13 ? LESSN 3. Rate of Change and Slope ESSENTIAL QUESTIN How do ou find a rate of change or a slope? 8.F.. Determine the rate of change of the function from two (, ) values, including reading these from a table or from a graph. Investigating Rates of Change A rate of change is a ratio of the amount of change in the output to the amount of change in the input. EXAMPLE 1 8.F.. Math n the Spot Eve keeps a record of the number of lawns she has mowed and the mone she has earned. Tell whether the rates of change are constant or variable. Da 1 Da Da 3 Da Number of lawns Amount earned ($) STEP 1 Identif the input and output variables. Input variable: number of lawns utput variable: amount earned STEP YUR TURN Find the rates of change. Da 1 to Da : Da to Da 3: change in $ change in lawns change in $ change in lawns = = = 30 = 15 = 5 3 = 15 change in $ Da 3 to Da : change in lawns = = = 15 The rates of change are constant: $15 per lawn. 1. The table shows the approimate height of a football after it is kicked. Tell whether the rates of change are constant or variable. Find the rates of change: The rates of change are constant / variable. Time (s) Height (ft) Math Talk Mathematical Practices Would ou epect the rates of change of a car s speed during a drive through a cit to be constant or variable? Eplain. Personal Math Trainer nline Assessment and Intervention Lesson 3. 77

14 EXPLRE ACTIVITY 8.F.. Using Graphs to Find Rates of Change You can also use a graph to find rates of change. The graph shows the distance Nathan biccled over time. What is Nathan s rate of change? A Find the rate of change from 1 hour to hours. change in distance 30 - = = change in time = miles per hour Distance (mi) (,60) (3,5) (,30) (1,15) B Find the rate of change from 1 hour to hours. 6 Time (h) change in distance 60 - = change in time - = = miles per hour C Find the rate of change from hour to hours. change in distance change in time = = = miles per hour D Recall that the graph of a proportional relationship is a line through the origin. Eplain whether the relationship between Nathan s time and distance is a proportional relationship. Reflect. Make a Conjecture Does a proportional relationship have a constant rate of change? 3. Does it matter what interval ou use when ou find the rate of change of a proportional relationship? Eplain. 78 Unit

15 Calculating Slope When the rate of change of a relationship is constant, ever segment of its graph has the same steepness, and the segments together form a line. The constant rate of change is called the slope of the line. The slope of a line is the ratio of the change in -values (rise) for a segment of the graph to the corresponding change in -values (run). Rise Run Math n the Spot EXAMPLE 8.F.. Find the slope of the line. STEP 1 STEP Choose two points on the line. Find the change in -values (rise) and the change in -values (run) as ou move from one point to the other. rise = + run = -3 If ou move up or right, the change is positive. If ou move down or left, the change is negative. Run Rise 5 STEP 3 Slope = rise run = = - _ 3-5 YUR TURN. The graph shows the rate at which water is leaking from a tank. The slope of the line gives the leaking rate in gallons per minute. Rise = Amount (gal) 10 5 Leaking tank Run = Rate of leaking = minute gallon(s) per 5 10 Time (min) Personal Math Trainer nline Assessment and Intervention Lesson 3. 79

16 Guided Practice Tell whether the rates of change are constant or variable. (Eample 1) 1. building measurements Feet Yards distance an object falls Distance (ft) Time (s) 1 3. computers sold Week 9 0 Number Sold cost of sweaters Number 7 9 Cost ($) Erica walks to her friend Philip s house. The graph shows Erica s distance from home over time. (Eplore Activit) 5. Find the rate of change from 1 minute to minutes change in distance = = change in time = ft per min - 6. Find the rate of change from 1 minute to minutes. Find the slope of each line. (Eample ) Distance (ft) Time (min) ? slope = -5 ESSENTIAL QUESTIN CHECK-IN slope = 9. If ou know two points on a line, how can ou find the rate of change of the variables being graphed? Unit

17 Name Class Date 3. Independent Practice 8.F.. Personal Math Trainer nline Assessment and Intervention 10. Rectangle EFGH is graphed on a coordinate plane with vertices at E(-3, 5), F(6, ), G(, -), and H(-5, -1). a. Find the slopes of each side. b. What do ou notice about the slopes of opposite sides? c. What do ou notice about the slopes of adjacent sides? 11. A bicclist started riding at 8:00 A.M. The diagram below shows the distance the bicclist had traveled at different times. What was the bicclist s average rate of speed in miles per hour? 8:00 A.M..5 miles 8:18 A.M. 7.5 miles 8:8 A.M. 1. Multistep A line passes through (6, 3), (8, ), and (n, -). Find the value of n. 13. A large container holds 5 gallons of water. It begins leaking at a constant rate. After 10 minutes, the container has 3 gallons of water left. a. At what rate is the water leaking? b. After how man minutes will the container be empt? 1. Critique Reasoning Bill found the slope of the line through the points (, 5) and (-, -5) using the equation - (-) 5 - (-5) = _ 5. What mistake did he make? Lesson 3. 81

18 15. Multiple Representations Graph parallelogram ABCD on a coordinate plane with vertices at A(3, ), B(6, 1), C(0, -), and D(-3, 1). a. Find the slope of each side b. What do ou notice about the slopes? c. Draw another parallelogram on the coordinate plane. Do the slopes have the same characteristics? -10 FCUS N HIGHER RDER THINKING Work Area 16. Communicate Mathematical Ideas Ben and Phoebe are finding the slope of a line. Ben chose two points on the line and used them to find the slope. Phoebe used two different points to find the slope. Did the get the same answer? Eplain. 17. Analze Relationships Two lines pass through the origin. The lines have slopes that are opposites. Compare and contrast the lines. 18. Reason Abstractl What is the slope of the -ais? Eplain. 8 Unit

19 ? LESSN 3.3 Interpreting the Unit Rate as Slope ESSENTIAL QUESTIN How do ou interpret the unit rate as slope? 8.EE..5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different was. Also 8.F.1., 8.F.. EXPLRE ACTIVITY 8.EE..5, 8.F.. Relating the Unit Rate to Slope A rate is a comparison of two quantities that have different units, such as miles and hours. A unit rate is a rate in which the second quantit in the comparison is one unit. Image Credits: Cavan Images/ Gett Images A storm is raging on Mist Mountain. The graph shows the constant rate of change of the snow level on the mountain. A B C D Find the slope of the graph using the points (1, ) and (5, 10). Remember that the slope is the constant rate of change. Snowfall (in.) Mist Mountain Storm Time (h) Find the unit rate of snowfall in inches per hour. Eplain our method. Compare the slope of the graph and the unit rate of change in the snow level. What do ou notice? Which point on the graph tells ou the slope of the graph and the unit rate of change in the snow level? Eplain how ou found the point. Lesson

20 Math n the Spot Graphing Proportional Relationships You can use a table and a graph to find the unit rate and slope that describe a real-world proportional relationship. The constant of proportionalit for a proportional relationship is the same as the slope. EXAMPLE 1 8.EE..5 Ever 3 seconds, cubic feet of water pass over a dam. Draw a graph of the situation. Find the unit rate of this proportional relationship. Animated Math Math Talk Mathematical Practices In a proportional relationship, how are the constant of proportionalit, the unit rate, and the slope of the graph of the relationship related? STEP 1 STEP STEP 3 Make a table. Time (s) Volume (ft 3 ) Draw a graph. Find the slope. slope = rise run = _ 8 6 = _ 3 Amount (cu ft) Water ver the Dam Time (sec) The unit rate of water passing over the dam and the slope of the graph of the relationship are equal, _ cubic feet per second. 3 Reflect 1. What If? Without referring to the graph, how do ou know that the point ( 1, _ 3 ) is on the graph? Personal Math Trainer nline Assessment and Intervention YUR TURN. Tomas rides his bike at a stead rate of miles ever 10 minutes. Graph the situation. Find the unit rate of this proportional relationship. Distance (mi) 10 5 Tomas s Ride 5 10 Time (min) 8 Unit

21 Using Slopes to Compare Unit Rates You can compare proportional relationships presented in different was. EXAMPLE The equation =.75 represents the rate, in barrels per hour, that oil is pumped from Well A. The graph represents the rate that oil is pumped from Well B. Which well pumped oil at a faster rate? STEP 1 Use the equation =.75 to make a table for Well A s pumping rate, in barrels per hour. Time (h) 1 3 Quantit (barrels) Amount (barrels) 0 10 Well B Pumping Rate 8.EE..5, 8.F Time (h) Math n the Spot STEP STEP 3 STEP Use the table to find the slope of the graph of Well A s rate. slope = unit rate =.75 1 = 5.5 = = 11 =.75 barrels/hour Use the graph to find the slope of the graph of Well B s rate. slope = 10 =.5 barrels/hour Compare the slopes. slope = rise run Image Credits: Tom McHugh/ Photo Researchers, Inc..75 >.5, so Well A s rate,.75 barrels/hour, is faster. Reflect 3. Describe the relationships among the slope of the graph of Well A s rate, the equation representing Well A s rate, and the constant of proportionalit. YUR TURN. The equation = 375 represents the relationship between, the time that a plane flies in hours, and, the distance the plane flies in miles for Plane A. The table represents the relationship for Plane B. Find the slope of the graph for each plane and the plane s rate of speed. Determine which plane is fling at a faster rate of speed. Time (h) 1 3 Distance (mi) Personal Math Trainer nline Assessment and Intervention Lesson

22 Guided Practice Give the slope of the graph and the unit rate. (Eplore Activit and Eample 1) 1. Jorge: 5 miles ever 6 hours. Akiko Distance (mi) 0 10 Jorge 10 0 Time (h) Time (h) Distance (mi) Distance (mi) 0 10 Akiko 10 0 Time (h) 3. The equation = 0.5 represents the distance Henr hikes in miles over time in hours. The graph represents the rate that Clark hikes. Determine which hiker is faster. Eplain. (Eample ) Distance (mi) 0 10 Clark Write an equation relating the variables in each table. (Eample ) 10 0 Time (h). Time () 1 6 Distance () Time () Distance () ? ESSENTIAL QUESTIN CHECK-IN 6. Describe methods ou can use to show a proportional relationship between two variables, and. For each method, eplain how ou can find the unit rate and the slope. 86 Unit

23 Name Class Date 3.3 Independent Practice 8.EE..5, 8.F.1., 8.F.. Personal Math Trainer nline Assessment and Intervention 7. A Canadian goose migrated at a stead rate of 3 miles ever minutes. a. Fill in the table to describe the relationship. Time (min) 8 0 Distance (mi) 9 1 b. Graph the relationship. c. Find the slope of the graph and describe what it means in the contet of this Migration Flight problem. 0 Distance (mi) Time (min) 8. Vocabular A unit rate is a rate in which the first quantit / second quantit in the comparison is one unit. 9. The table and the graph represent the rate at which two machines are bottling milk in gallons per second. Machine Machine 1 Time (s) 1 3 Amount (gal) Amount (gal) a. Determine the slope and unit rate of each machine Time (sec) b. Determine which machine is working at a faster rate. Lesson

24 10. Ccling The equation = 1_ represents the distance, in kilometers, that 9 Patrick traveled in minutes while training for the ccling portion of a triathlon. The table shows the distance Jennifer traveled in minutes in her training. Who has the faster training rate? Time (min) Distance (km) FCUS N HIGHER RDER THINKING Work Area 11. Analze Relationships There is a proportional relationship between minutes and cost per minute in dollars. The graph passes through the point (1,.75). What is the slope of the graph? What is the unit rate? Eplain. 1. Draw Conclusions Two cars start at the same time and travel at different constant rates. The graph of the distance in miles given the time in hours for Car A passes through the point (0.5, 7.5), and the graph for Car B passes through the point (, 0). Which car is traveling faster? Eplain. 13. Critical Thinking The table shows Time (min) the rate at which water is being pumped into a swimming pool. Amount (gal) Use the unit rate and the amount of water pumped after 1 minutes to find how much water will have been pumped into the pool after 13 1_ minutes. Eplain our reasoning. 88 Unit

25 MDULE QUIZ Read 3.1 Representing Proportional Relationships 1. Find the constant of proportionalit for the table of values Personal Math Trainer nline Assessment and Intervention. Phil is riding his bike. He rides 5 miles in hours, 37.5 miles in 3 hours, and 50 miles in hours. Find the constant of proportionalit and write an equation to describe the situation. 3. Rate of Change and Slope Find the slope of each line Interpreting the Unit Rate as Slope 5. The distance Train A travels is represented b d = 70t, where d is the distance in kilometers and t is the time in hours. The distance Train B travels at various times is shown in the table. What is the rate of each train? Which train is going faster? ESSENTIAL QUESTIN Time (hours) Distance (km) What is the relationship among proportional relationships, lines, rates of change, and slope? Module 3 89

26 MDULE 3 MIXED REVIEW PARCC Assessment Readiness Personal Math Trainer nline Assessment and Intervention Selected Response 1. Which of the following is equivalent to 5 1? A C - 1_ 5 B 1_ D Prasert earns $9 an hour. Which table represents this proportional relationship? A B C D Hours 6 8 Earnings ($) Hours 6 8 Earnings ($) Hours 3 Earnings ($) Hours 3 Earnings ($) A factor produces widgets at a constant rate. After hours, 3,10 widgets have been produced. At what rate are the widgets being produced? A 630 widgets per hour B 708 widgets per hour C 780 widgets per hour D 1,365 widgets per hour. A full lake begins dropping at a constant rate. After weeks it has dropped 3 feet. What is the unit rate of change in the lake s level compared to its full level? 90 Unit A 0.75 feet per week B 1.33 feet per week C 0.75 feet per week D 1.33 feet per week 5. What is the slope of the line below? - Pages - - A - C 1_ B - 1_ D 6. Jim earns $1.5 in 5 hours. Susan earns $30.00 in hours. Pierre s hourl rate is less than Jim s, but more than Susan s. What is his hourl rate? A $6.50 C $7.35 B $7.75 D $8.5 Mini-Task 7. Joelle can read 3 pages in minutes,.5 pages in 6 minutes, and 6 pages in 9 minutes. a. Make a table of the data. Minutes Pages b. Use the values in the table to find the unit rate. c. Graph the relationship between minutes and pages read. 6 6 Minutes

27 Nonproportional MDULE Relationships? ESSENTIAL QUESTIN How can ou use nonproportional relationships to solve real-world problems? LESSN.1 Representing Linear Nonproportional Relationships 8.F.1.3 LESSN. Determining Slope and -intercept 8.EE..6, 8.F.. LESSN.3 Graphing Linear Nonproportional Relationships using Slope and -intercept 8.F.1.3, 8.F.. Image Credits: viapp/ Shutterstock Real-World Video The distance a car can travel on a tank of gas or a full batter charge in an electric car depends on factors such as fuel capacit and the car s efficienc. This is described b a nonproportional relationship. LESSN. Proportional and Nonproportional Situations 8.F.1., 8.F.1.3, 8.F.. Math n the Spot Animated Math Personal Math Trainer Go digital with our write-in student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 91

28 Are YU Read? Complete these eercises to review skills ou will need for this module. Integer perations Personal Math Trainer nline Assessment and Intervention EXAMPLE 7 ( ) = , or 3 = 3 To subtract an integer, add its opposite. The signs are different, so find the difference of the absolute values. Use the sign of the number with the greater absolute value. Find each difference ( 5) ( 3) 5. 8 ( 8) ( 6) (-9) (-) Graph rdered Pairs (First Quadrant) EXAMPLE 8 6 A To graph a point at (6, ), start at the origin. Move 6 units right. Then move units up. Graph point A(6, ). 6 8 Graph each point on the coordinate grid. 13. B (0, 5) 1. C (8, 0) 15. D (5, 7) E (, 3) Unit

29 Reading Start-Up Visualize Vocabular Use the words to complete the diagram. You can put more than one word in each bo. Rise is the change in Run is the change in Reviewing Slope Understand Vocabular Complete the sentences using the preview words. rise run is Vocabular Review Words ordered pair (par ordenado) proportional relationship (relación proporcional) rate of change (tasa de cambio) slope (pendiente) -coordinate (coordenada ) -coordinate (coordenada ) Preview Words linear equation (ecuación lineal) slope-intercept form of an equation (forma de pendiente-intersección) -intercept (intersección con el eje ) 1. The second number in an ordered pair is the.. A is an equation whose solutions form a straight line on a coordinate plane. 3. A linear equation written in the form = m + b is the Active Reading Booklet Before beginning the module, create a booklet to help ou learn the concepts. Write the main idea of each lesson on each page of the booklet. As ou stud each lesson, write important details that support the main idea, such as vocabular and formulas. Refer to our finished booklet as ou work on assignments and stud for tests.. Module 93

30 MDULE Unpacking the Standards Understanding the standards and the vocabular terms in the standards will help ou know eactl what ou are epected to learn in this module. 8.F.1.3 Interpret the equation = m + b as defining a linear function whose graph is a straight line. What It Means to You You will identif the slope and the -intercept of a line b looking at its equation and use them to graph the line. Ke Vocabular slope (pendiente) A measure of the steepness of a line on a graph; the rise divided b the run. -intercept (intersección con el eje ) The -coordinate of the point where the graph of a line crosses the -ais. UNPACKING EXAMPLE 8.F.1.3 Graph = 3 - using the slope and the -intercept. = m + b slope -intercept The slope m is 3, and the -intercept is -. Plot the point (0, -). Use the slope 3 = 3 _ 1 to find another point b moving up 3 and to the right 1. Connect the points F.1.3 Give eamples of functions that are not linear. Ke Vocabular function (función) An input-output relationship that has eactl one output for each input. linear function (función lineal) A function whose graph is a straight line. Visit to see all Florida Common Core Standards unpacked. What It Means to You You will distinguish linear relationships from nonlinear relationships b looking at graphs. UNPACKING EXAMPLE 8.F.1.3 Which relationship is linear and which is nonlinear? P = s A = s P P = s is linear because its graph is a line. s A A = s is not linear because its graph is not a line. s 9 Unit

31 ? LESSN.1 Representing Linear Nonproportional Relationships ESSENTIAL QUESTIN 8.F.1.3 Interpret the equation = m + b as defining a linear function, whose graph is a straight line; How can ou use tables, graphs, and equations to represent linear nonproportional situations? Representing Linear Relationships Using Tables You can use an equation to describe the relationship between two quantities in a real-world situation. You can use a table to show some values that make the equation true. Math n the Spot EXAMPLE 1 Prep for 8.F.1.3 The equation = 3 + gives the total charge,, for bowling games at Bater Bowling Lanes based on the prices shown. Make a table of values for this situation. STEP 1 Choose several values for that make sense in contet. (number of games) 1 3 (total cost in dollars) STEP Use the equation = 3 + to find for each value of. YUR TURN (number of games) 1 3 (total cost in dollars) Francisco makes $1 per hour doing part-time work on Saturdas. He spends $ on transportation to and from work. The equation = 1 - gives his earnings, after transportation costs, for working hours. Make a table of values for this situation. (number of hours) (earnings in dollars) Substitute 1 for : = 3(1) + = 5. Personal Math Trainer nline Assessment and Intervention Lesson.1 95

32 EXPLRE ACTIVITY 8.F.1.3 Eamining Linear Relationships Recall that a proportional relationship is a relationship between two quantities in which the ratio of one quantit to the other quantit is constant. The graph of a proportional relationship is a line through the origin. Relationships can have a constant rate of change but not be proportional. The entrance fee for Mountain World theme park is $0. Visitors purchase additional $ tickets for rides, games, and food. The equation = + 0 gives the total cost,, to visit the park, including purchasing tickets. STEP 1 Complete the table. (number of tickets) (total cost in dollars) 0 STEP Plot the ordered pairs from the table. Describe the shape of the graph. 0 Theme Park Costs 3 STEP 3 Find the rate of change between each point and the net. Is the rate constant? Cost ($) 16 8 STEP Calculate _ for the values in the table. Eplain wh the relationship between number of tickets and total cost is not proportional Number of tickets Reflect. Analze Relationships Would it make sense to add more points to the graph from = 0 to = 10? Would it make sense to connect the points with a line? Eplain. 96 Unit

33 Representing Linear Relationships Using Graphs A linear equation is an equation whose solutions are ordered pairs that form a line when graphed on a coordinate plane. Linear equations can be written in the form = m + b. When b 0, the relationship between and is nonproportional. Math n the Spot EXAMPLE 8.F.1.3 The diameter of a Douglas fir tree is currentl 10 inches when measured at chest height. ver the net 50 ears, the diameter is epected to increase b an average growth rate of _ 5 inch per ear. The equation = _ gives, the diameter of the tree in inches, after ears. Draw a graph of the equation. Describe the relationship. M Notes STEP 1 Make a table. Choose several values for that make sense in contet. To make calculations easier, choose multiples of 5. (ears) (diameter in inches) STEP Plot the ordered pairs from the table. Then draw a line connecting the points to represent all the possible solutions. Image Credits: Don Mason/Corbis STEP 3 Diameter (in.) YUR TURN The relationship is linear but nonproportional. The graph is a line but it does not go through the origin. 8 Fir Tree Growth Time (r) 3. Make a table and graph the solutions of the equation = Personal Math Trainer nline Assessment and Intervention Lesson.1 97

34 Guided Practice Make a table of values for each equation. (Eample 1) 1. = = 3 _ Eplain wh each relationship is not proportional. (Eplore Activit) First calculate _ for the values in the table Complete the table for the equation. Then use the table to graph the equation. (Eample )? 5. = ESSENTIAL QUESTIN CHECK-IN 6. How can ou choose values for when making a table of values representing a real world situation? Unit

35 Name Class Date.1 Independent Practice 8.F.1.3 State whether the graph of each linear relationship is a solid line or a set of unconnected points. Eplain our reasoning. Personal Math Trainer nline Assessment and Intervention 7. The relationship between the number of $ lunches ou bu with a $100 school lunch card and the mone remaining on the card 8. The relationship between time and the distance remaining on a 3-mile walk for someone walking at a stead rate of miles per hour 9. Analze Relationships Simone paid $1 for an initial ear s subscription to a magazine. The renewal rate is $8 per ear. This situation can be represented b the equation = 8 + 1, where represents the number of ears the subscription is renewed and represents the total cost. a. Make a table of values for this situation. b. Draw a graph to represent the situation. Include a title and ais labels. c. Eplain wh this relationship is not proportional. d. Does it make sense to connect the points on the graph with a solid line? Eplain Lesson.1 99

36 10. Analze Relationships A direct variation is a linear relationship because the rate of change is constant (and equal to the constant of variation). What is required of a direct variation relationship that is not required of a general linear relationship? 11. Communicate Mathematical Ideas Eplain how ou can identif a linear non-proportional relationship from a table, a graph, and an equation. FCUS N HIGHER RDER THINKING Work Area 1. Critique Reasoning George observes that for ever increase of 1 in the value of, there is an increase of 60 in the corresponding value of. He claims that the relationship represented b the table is proportional. Critique George s reasoning Make a Conjecture Two parallel lines are graphed on a coordinate plane. How man of the lines could represent proportional relationships? Eplain. 100 Unit

37 ? LESSN. Determining Slope and -intercept ESSENTIAL QUESTIN 8.EE..6 ; derive the equation = m for a line through the origin and the equation = m + b for a line intercepting the vertical ais at b. Also 8.F.. How can ou determine the slope and the -intercept of a line? EXPLRE ACTIVITY 1 8.EE..6 Investigating Slope and -intercept The graph of ever nonvertical line crosses the -ais. The -intercept is the -coordinate of the point where the graph intersects the -ais. The -coordinate of this point is alwas 0. The graph represents the linear equation = - _ 3 +. STEP 1 Find the slope of the line using the points (0, ) and (-3, 6). 8 (-3, 6) 6 STEP 6 - m = = = - 0 The line also contains the point (6, 0). What is the slope using (0, ) and (6, 0)? Using (-3, 6) and (6, 0). What do ou notice? (0, ) 6 8 STEP 3 STEP 3 STEP 5 Compare our answers in Steps 1 and with the equation of the graphed line. Find the value of when = 0 using the equation = - _ 3 +. Describe the point on the graph that corresponds to this solution. Compare our answer in Step 3 with the equation of the line. Lesson. 101

38 Math n the Spot Determining Rate of Change and Initial Value The linear equation shown is written in the slope-intercept form of an equation. Its graph is a line with slope m and -intercept b. = m + b slope -intercept A linear relationship has a constant rate of change. You can find the rate of change m and the initial value b for a linear situation from a table of values. EXAMPLE 1 8.F.. A phone salesperson is paid a minimum weekl salar and a commission for each phone sold, as shown in the table. Confirm that the relationship is linear and give the constant rate of change and the initial value. STEP 1 Confirm that the rate of change is constant. change in income change in phones sold = = = 15 change in income change in phones sold = = = 15 change in income change in phones sold = = = 15 Number of Phones Sold Weekl Income ($) 10 $80 0 $ $780 0 $930 Math Talk Mathematical Practices How do ou use the rate of change to work backward to find the initial value? STEP The rate of change is a constant, 15. The salesperson receives a $15 commission for each phone sold. Find the initial value when the number of phones sold is YUR TURN Number of phones sold Weekl income ($) Work backward from = 10 to = 0 to find the initial value. The initial value is $330. The salesperson receives a salar of $330 each week before commissions. Find the slope and -intercept of the line represented b each table. Personal Math Trainer nline Assessment and Intervention Unit

39 EXPLRE ACTIVITY 8.EE..6 Deriving the Slope-intercept Form of an Equation In the following Eplore Activit, ou will derive the slope-intercept form of an equation. STEP 1 Let L be a line with slope m and -intercept b. Circle the point that must be on the line. Justif our choice. (b, 0) (0, b) (0, m) (m, 0) STEP Recall that slope is the ratio of change in to change in. Complete the equation for the slope m of the line using the -intercept (0, b) and another point (, ) on the line. m = STEP 3 In an equation of a line, we often want b itself on one side of the equation. Solve the equation from Step for. m = - b Simplif the denominator. m = - b Multipl both sides of the equation b. m = - b m + = - b + m + = = m + Reflect 3. Critical Thinking Write the equation of a line with slope m that passes through the origin. Eplain our reasoning. Add to both sides of the equation. Write the equation with on the left side. Lesson. 103

40 Guided Practice Find the slope and -intercept of the line in each graph. (Eplore Activit 1) (0, 1) (3, 0) - - (, -3) (0, -15) slope m = -intercept b = slope m = -intercept b = slope m = -intercept b = slope m = -intercept b = Find the slope and -intercept of the line represented b each table. (Eample 1)? slope m = -intercept b = ESSENTIAL QUESTIN CHECK-IN 7. How can ou determine the slope and the -intercept of a line from a graph? slope m = -intercept b = 10 Unit

41 Name Class Date. Independent Practice 8.EE..6, 8.F.. 8. Some carpet cleaning costs are shown in the table. The relationship is linear. Find and interpret the rate of change and the initial value for this situation. Personal Math Trainer nline Assessment and Intervention Rooms cleaned 1 3 Cost ($) Make Predictions The total cost to pa for parking at a state park for the da and rent a paddleboat are shown. a. Find the cost to park for a da and the hourl rate to rent a paddleboat. b. What will Lin pa if she rents a paddleboat for 3.5 hours and splits the total cost with a friend? Eplain. Number of Hours Cost ($) 1 $17 $9 3 $1 $ Multi-Step Ramond s parents will pa for him to take sailboard lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear. Lessons 1 3 Group ($) Private ($) a. Find the rate of change and the initial value for the group lessons. b. Find the rate of change and the initial value for the private lessons. c. Compare and contrast the rates of change and the initial values. Lesson. 105

42 Vocabular Eplain wh each relationship is not linear Communicate Mathematical Ideas Describe the procedure ou performed to derive the slope-intercept form of a linear equation. FCUS N HIGHER RDER THINKING Work Area 1. Critique Reasoning Your teacher asked our class to describe a realworld situation in which a -intercept is 100 and the slope is 5. Your partner gave the following description: M ounger brother originall had 100 small building blocks, but he has lost 5 of them ever month since. a. What mistake did our partner make? b. Describe a real-world situation that does match the situation. 15. Justif Reasoning John has a job parking cars. He earns a fied weekl salar of $300 plus a fee of $5 for each car he parks. His potential earnings for a week are shown in the graph. At what point does John begin to earn more from fees than his fied salar? Justif our answer. Weekl earnings Earnings Cars Cars parked 106 Unit

43 ? LESSN.3 Graphing Linear Nonproportional Relationships Using Slope and -intercept ESSENTIAL QUESTIN 8.F.. 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. Also 8.F.1.3 How can ou graph a line using the slope and -intercept? Using Slope-intercept Form to Graph a Line Recall that = m + b is the slope-intercept form of the equation of a line. In this form, it is eas to see the slope m and the -intercept b. So ou can use this form to quickl graph a line b plotting the point (0, b) and using the slope to find a second point. Math n the Spot EXAMPLE 1 8.F.1.3 A Graph = _ 3-1. B STEP 1 STEP STEP 3 Graph = - 5 _ + 3. STEP 1 STEP The -intercept is b = -1. Plot the point that contains the -intercept: (0, -1). The slope is m = _. Use the 3 slope to find a second point. From (0, -1), count up and right 3. The new point is (3, 1). Draw a line through the points. The -intercept is b = 3. Plot the point that contains the -intercept: (0, 3). The slope is m = - 5 _. Use the slope to find a second point. From (0, 3), count down 5 and right, or up 5 and left. The new point is (, -) or (-, 8) (3, 1) - - (0, -1) (-, 8) +5 (0, 3) (, -) - Animated Math Math Talk Mathematical Practices Is a line with a positive slope alwas steeper than a line with a negative slope? Eplain. STEP 3 Draw a line through the points. Note that the line passes through all three points: (-, 8), (0, 3), and (, -). Lesson.3 107

44 Reflect 1. Draw Conclusions How can ou use the slope of a line to predict the wa the line will be slanted? Eplain. YUR TURN Personal Math Trainer nline Assessment and Intervention Graph each equation.. = 1_ = Analzing a Graph Man real-world situations can be represented b linear relationships. You can use graphs of linear relationships to visualize situations and solve problems. Math n the Spot EXAMPLE 8.F.. Ken has a weekl goal for the number of calories he will burn b taking brisk walks. The equation = represents the number of calories Ken has left to burn after hours of walking. TIME HURS 0:30 1:30 CALRIES 150 A Graph the equation = STEP 1 STEP STEP 3 STEP Write the slope as a fraction. m = = -600 = Plot the point for the -intercept: (0, 00). Use the slope to locate a second point. From (0, 00), count down 900 and right 3. The new point is (3, 1500). Draw a line through the two points. Calories remaining Time (h) 108 Unit

45 B After how man hours of walking will Ken have 600 calories left to burn? After how man hours will he reach his weekl goal? STEP 1 STEP Locate 600 calories on the -ais. Read across and down to the -ais. Ken will have 600 calories left to burn after 6 hours. Ken will reach his weekl goal when the number of calories left to burn is 0. Because ever point on the -ais has a -value of 0, find the point where the line crosses the -ais. Calories remaining Time (h) Ken will reach his goal after 8 hours of brisk walking. YUR TURN What If? Ken decides to modif his eercise plans from Eample b slowing the speed at which he walks. The equation for the modified plan is = Math Talk Mathematical Practices What do the slope and the -intercept of the line represent in this situation?. Graph the equation. 5. How does the graph of the new equation compare with the graph in Eample? Calories remaining Time (h) 6. Will Ken have to eercise more or less to meet his goal? Eplain. 7. Suppose that Ken decides that instead of walking, he will jog, and that jogging burns 600 calories per hour. How do ou think that this would change the graph? Personal Math Trainer nline Assessment and Intervention Lesson.3 109

46 Guided Practice Graph each equation using the slope and the -intercept. (Eample 1) 1. = 1_ - 3 slope = -intercept =. = -3 + slope = -intercept = A friend gives ou two baseball cards for our birthda. Afterward, ou begin collecting them. You bu the same number of cards once each week. The equation = + describes the number of cards,, ou have after weeks. (Eample ) a. Find and interpret the slope and the -intercept of the line that represents this situation. Graph = +. Include ais labels b. Discuss which points on the line do not make sense in this situation. Then plot three more points on the line that do make sense. 6 8? ESSENTIAL QUESTIN CHECK-IN. Wh might someone choose to use the -intercept and the slope to graph a line? 110 Unit

47 Name Class Date.3 Independent Practice 8.F.1.3, 8.F.. Personal Math Trainer nline Assessment and Intervention 5. Science A spring stretches in relation to the weight hanging from it according to the equation = where is the weight in pounds and is the length of the spring in inches. a. Graph the equation. Include ais labels. 3 b. Interpret the slope and the -intercept of the line c. How long will the spring be if a -pound weight is hung on it? Will the length double if ou double the weight? Eplain Image Credits: Steve Williams/ Houghton Mifflin Harcourt Look for a Pattern Identif the coordinates of four points on the line with each given slope and -intercept. 6. slope = 5, -intercept = slope = -1, -intercept = 8 8. slope = 0., -intercept = slope = 1.5, -intercept = slope = - 1_, -intercept = 11. slope = _, -intercept = A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation = represents the total cost of lessons. Find and interpret the slope and -intercept of the line that represents this situation. Then find four points on the line. Lesson.3 111

48 13. A public pool charges a membership fee and a fee for each visit. The equation = represents the cost for visits. a. After locating the -intercept on the coordinate plane shown, can ou move up three gridlines and right one gridline to find a second point? Eplain. b. Graph the equation = Include ais labels. Then interpret the slope and -intercept c. How man visits to the pool can a member get for $00? FCUS N HIGHER RDER THINKING 1. Eplain the Error A student sas that the slope of the line for the equation = 0-15 is 0 and the -intercept is 15. Find and correct the error. Work Area 15. Critical Thinking Suppose ou know the slope of a linear relationship and a point that its graph passes through. Can ou graph the line even if the point provided does not represent the -intercept? Eplain. 16. Make a Conjecture Graph the lines = 3, = 3-3, and = What do ou notice about the lines? Make a conjecture based on our observation Unit

49 ? LESSN. Proportional and Nonproportional Situations ESSENTIAL QUESTIN How can ou distinguish between proportional and nonproportional situations? Distinguish Between Proportional and Nonproportional Situations Using a Graph If a relationship is nonlinear, it is nonproportional. If it is linear, it ma be either proportional or nonproportional. When the graph of the linear relationship contains the origin, the relationship is proportional. 8.F.1. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Also 8.F.1.3, 8.F.. Math n the Spot EXAMPLE 1 The graph shows the sales ta charged based on the amount spent at a video game store in a particular cit. Does the graph show a linear relationship? Is the relationship proportional or nonproportional? The graph shows a linear proportional relationship because it is a line that contains the origin. Sales ta ($) F Amount spent ($) YUR TURN Determine if each of the following graphs represents a proportional or nonproportional relationship Math Talk Mathematical Practices What do the slope and the -intercept of the graph represent in this situation? Personal Math Trainer nline Assessment and Intervention Lesson. 113

50 Math n the Spot Distinguish Between Proportional and Nonproportional Situations Using an Equation If an equation is not a linear equation, it represents a nonproportional relationship. A linear equation of the form = m + b ma represent either a proportional (b = 0) or nonproportional (b 0) relationship. EXAMPLE 8.F.. The number of ears since Keith graduated from middle school can be represented b the equation = a - 1, where is the number of ears and a is his age. Is the relationship between the number of ears since Keith graduated and his age proportional or nonproportional? = a - 1 Animated Math The equation is in the form = m + b, with a being used as the variable instead of. The value of m is 1, and the value of b is -1. Since b is not 0, the relationship between the number of ears since Keith graduated and his age is nonproportional. Reflect 3. Communicate Mathematical Ideas In a proportional relationship, the ratio _ is constant. Show that this ratio is not constant for the equation = a What If? Suppose another equation represents Keith s age in months given his age in ears a. Is this relationship proportional? Eplain. YUR TURN Determine if each of the following equations represents a proportional or nonproportional relationship. 5. d = 65t 6. p = 0.1s Personal Math Trainer nline Assessment and Intervention 7. n = 50-3p = 1d 11 Unit

51 Distinguish Between Proportional and Nonproportional Situations Using a Table If there is not a constant rate of change in the data displaed in a table, then the table represents a nonlinear nonproportional relationship. Math n the Spot A linear relationship represented b a table is a proportional relationship when the quotient of each pair of numbers is constant. therwise, the linear relationship is nonproportional. EXAMPLE 3 8.F.. Image Credits: Jupiter Images/ Hemera Technologies/Gett Images The values in the table represent the numbers of U.S. dollars three tourists traded for Meican pesos. The relationship is linear. Is the relationship proportional or nonproportional? U.S. Dollars Traded Meican Pesos Received 130 1, , ,565 1, = = 13 3, = 1 17 = 13 6, = = 13 The ratio of pesos received to dollars traded is constant at 13 Meican pesos per U.S. dollar. This is a proportional relationship. YUR TURN Determine if the linear relationship represented b each table is a proportional or nonproportional relationship Simplif the ratios to compare the pesos received to the dollars traded Animated Math Math Talk Mathematical Practices How could ou confirm that the values in the table have a linear relationship? Personal Math Trainer nline Assessment and Intervention Lesson. 115

52 Math n the Spot Comparing Proportional and Nonproportional Situations You can use what ou have learned about proportional and nonproportional relationships to compare similar real-world situations that are given using different representations. EXAMPLE 8.F.1. Math Talk Mathematical Practices How might graphing the equation for Arena A help ou to compare the situations? A A laser tag league has the choice of two arenas for a tournament. In both cases, is the number of hours and is the total charge. Compare and contrast these two situations. Arena A = 5 Arena A s equation has the form = m + b, where b = 0. So, Arena A s charges are a proportional relationship. The hourl rate, $5, is greater than Arena B s, but there is no additional fee. Total cost ($) Arena B Hours Arena B s graph is a line that does not include the origin. So, Arena B s charges are a nonproportional relationship. Arena B has a $50 initial fee but its hourl rate, $00, is lower. B Jessika is remodeling and has the choice of two painters. In both cases, is the number of hours and is the total charge. Compare and contrast these two situations. Painter A = $5 Painter A s equation has the form = m + b, where b = 0. So, Painter A s charges are proportional. The hourl rate, $5, is greater than Painter B s, but there is no additional fee. Painter B Painter B s table is a nonproportional relationship because the ratio of to is not constant. Because the table contains the ordered pair (0, 0), Painter B charges an initial fee of $0, but the hourl rate, $35, is less than Painter A s. 116 Unit

53 YUR TURN 11. Compare and contrast the following two situations. Test-Prep Center A The cost for Test-Prep Center A is given b c = 0h, where c is the cost in dollars and h is the number of hours ou attend. Test-Prep Center B Test-Prep Center B charges $5 per hour to attend, but ou have a $100 coupon that ou can use to reduce the cost. Personal Math Trainer nline Assessment and Intervention Guided Practice Determine if each relationship is a proportional or nonproportional situation. Eplain our reasoning. (Eample 1, Eample, Eample ) Look at the origin. 3. q = p + 1_ Compare the equation with = m + b.. v = 1 10 u Lesson. 117

54 The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation. (Eample 3, Eample ) Find the quotient of and. 7. The values in the table represent the numbers of households that watched three TV shows and the ratings of the shows. The relationship is linear. Describe the relationship in other was. (Eample ) Number of Households that Watched TV Show TV Show Rating 15,000, ,000, ,000,000 0? ESSENTIAL QUESTIN CHECK-IN 8. How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional situations? 118 Unit

55 Name Class Date. Independent Practice 8.F.1., 8.F.1.3, 8.F.. Personal Math Trainer nline Assessment and Intervention 9. The graph shows the weight of a cross-countr team s beverage cooler based on how much sports drink it contains. a. Is the relationship proportional or nonproportional? Eplain. b. Identif and interpret the slope and the -intercept. Weight (lb) Sports drink (cups) In 10 11, tell if the relationship between a rider s height above the first floor and the time since the rider stepped on the elevator or escalator is proportional or nonproportional. Eplain our reasoning. 10. The elevator paused for 10 seconds after ou stepped on before beginning to rise at a constant rate of 8 feet per second. height above floor height above floor 11. Your height, h, in feet above the first floor on the escalator is given b h = 0.75t, where t is the time in seconds. 1. Analze Relationships Compare and contrast the two graphs. Graph A = 1_ Graph B = _ Lesson. 119

56 13. Represent Real-World Problems Describe a real-world situation where the relationship is linear and nonproportional. FCUS N HIGHER RDER THINKING Work Area 1. Mathematical Reasoning Suppose ou know the slope of a linear relationship and one of the points that its graph passes through. How can ou determine if the relationship is proportional or nonproportional? 15. Multiple Representations An entrant at a science fair has included information about temperature conversion in various forms, as shown. The variables F, C, and K represent temperatures in degrees Fahrenheit, degrees Celsius, and Kelvin, respectivel. Equation A F = _ 9 5 C + 3 Equation B K = C Table C Degrees Celsius kelvins a. Is the relationship between kelvins and degrees Celsius proportional? Justif our answer in two different was. b. Is the relationship between degrees Celsius and degrees Fahrenheit proportional? Wh or wh not? 10 Unit

57 MDULE QUIZ Read.1 Representing Linear Nonproportional Relationships 1. Complete the table using the equation = Personal Math Trainer nline Assessment and Intervention. Determining Slope and -intercept 5. Find the slope and -intercept of the line in the graph Graphing Linear Nonproportional Relationships 3. Graph the equation = - 3 using slope and -intercept Proportional and Nonproportional Situations. Does the table represent a proportional or a nonproportional linear relationship? Does the graph in Eercise represent a proportional or a nonproportional linear relationship? 6. Does the graph in Eercise 3 represent a proportional or a nonproportional relationship? ESSENTIAL QUESTIN 7. How can ou identif a linear nonproportional relationship from a table, a graph, and an equation? Module 11

58 MDULE MIXED REVIEW PARCC Assessment Readiness Personal Math Trainer nline Assessment and Intervention Selected Response 1. The table below represents which equation? A = C = - 6 B = -6 D = - +. The graph of which equation is shown below? 5 5. The table shows a proportional relationship. What is the missing -value? ? A 16 C 18 B 0 D 6. What is written in scientific notation? A C B D Mini-Task The graph shows a linear relationship. 5-5 A = C = B = D = The table below represents a linear relationship What is the -intercept? A - C B - D 3. Which equation represents a nonproportional relationship? A = C = B = -3 D = 1_ 3-5 a. Is the relationship proportional or nonproportional? b. What is the slope of the line? c. What is the -intercept of the line? d. What is the equation of the line? 1 Unit

59 Writing Linear 5 MDULE Equations? ESSENTIAL QUESTIN How can ou use linear equations to solve real-world problems? LESSN 5.1 Writing Linear Equations from Situations and Graphs 8.F.. LESSN 5. Writing Linear Equations from a Table 8.F.. LESSN 5.3 Linear Relationships and Bivariate Data 8.SP.1.1, 8.SP.1., 8.SP.1.3 Image Credits: Yellow Dog Productions/Gett Images Real-World Video Linear equations can be used to describe man situations related to shopping. If a store advertised four books for $3.00, ou could write and solve a linear equation to find the price of each book. Math n the Spot Animated Math Personal Math Trainer Go digital with our write-in student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 13

60 Are YU Read? Complete these eercises to review skills ou will need for this module. Write Fractions as Decimals EXAMPLE Personal Math Trainer nline Assessment and Intervention =? Multipl the numerator and the denominator = _ 5 8 b a power of 10 so that the denominator is a whole number Write the fraction as a division problem. 0 Write a decimal point and zeros in the dividend. -16 Place a decimal point in the quotient. 0 Divide as with whole numbers Write each fraction as a decimal. 1. 3_ Inverse perations EXAMPLE 5n = 0 5n 5 = 0 5 n = k + 7 = 9 k = 9-7 k = Solve each equation using the inverse operation. 5. 7p = 8 6. h - 13 = 5 7. _ = b + 9 = c - 8 = n = = m n is multiplied b 5. To solve the equation, use the inverse operation, division. 7 is added to k. To solve the equation, use the inverse operation, subtraction. t -5 = -5 1 Unit

61 Reading Start-Up Visualize Vocabular Use the words to complete the diagram. You can put more than one word in each bubble. m = m + b Understand Vocabular Complete the sentences using the preview words. b Vocabular Review Words linear equation (ecuación lineal) ordered pair (par ordenado) proportional relationship (relación proporcional) rate of change (tasa de cambio) slope (pendiente) slope-intercept form of an equation (forma de pendiente-intersección) -coordinate (coordenada ) -coordinate (coordenada ) -intercept (intersección con el eje ) Preview Words bivariate data (datos bivariados) nonlinear relationship (relación no lineal) 1. A set of data that is made up of two paired variables is.. When the rate of change varies from point to point, the relationship is a. Active Reading Tri-Fold Before beginning the module, create a tri-fold to help ou learn the concepts and vocabular in this module. Fold the paper into three sections. Label the columns What I Know, What I Need to Know, and What I Learned. Complete the first two columns before ou read. After studing the module, complete the third column. Module 5 15

62 MDULE 5 Unpacking the Standards Understanding the standards and the vocabular terms in the standards will help ou know eactl what ou are epected to learn in this module. 8.F.. 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. 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. Ke Vocabular rate of change (tasa de cambio) A ratio that compares the amount of change in a dependent variable to the amount of change in an independent variable. What It Means to You You will learn how to write an equation based on a situation that models a linear relationship. UNPACKING EXAMPLE 8.F.. In 006 the fare for a taicab was an initial charge of $.50 plus $0.30 per mile. Write an equation in slope-intercept form that can be used to calculate the total fare. The constant charge is $.50. The rate of change is $0.30 per mile. The input variable,, is the number of miles driven. So 0.3 is the cost for the miles driven. The equation for the total fare,, is as follows: = SP.1.3 Use the equation of a linear model to solve problems in the contet of bivariate measurement data, interpreting the slope and intercept. Ke Vocabular bivariate data (datos bivariados) A set of data that is made up of two paired variables. Visit to see all Florida Common Core Standards unpacked. What It Means to You You will see how to use a linear relationship between sets of data to make predictions. UNPACKING EXAMPLE 8.SP.1.3 The graph shows the temperatures in degrees Celsius inside the earth at certain depths in kilometers. Use the graph to write an equation and find the temperature at a depth of 1 km. The initial temperature is 0 C. It increases at a rate of 10 C/km. The equation is t = 10d + 0. At a depth of 1 km, the temperature is 10 C. Temperature ( C) Temperature Inside Earth Depth (km) 16 Unit

63 ? LESSN 5.1 Writing Linear Equations from Situations and Graphs ESSENTIAL QUESTIN 8.F.. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value. Interpret the rate of change and initial value. How do ou write an equation to model a linear relationship given a graph or a description? EXPLRE ACTIVITY Writing an Equation in Slope-Intercept Form 8.F.. Greta makes cla mugs and bowls as gifts at the Craft Studio. She pas a membership fee of $15 a month and an equipment fee of $3.00 an hour to use the potter s wheel, table, and kiln. Write an equation in the form = m + b that Greta can use to calculate her monthl costs. A What is the input variable,, for this situation? B C D What is the output variable,, for this situation? During April, Greta does not use the equipment at all. What will be her number of hours () for April? What will be her cost () for April? What will be the -intercept, b, in the equation? Greta spends 8 hours in Ma for a cost of $15 + 8($3) =. In June, she spends 11 hours for a cost of. From Ma to June, the change in -values is. From Ma to June, the change in -values is. What will be the slope, m, in the equation? Use the values for m and b to write an equation for Greta s costs in the form = m + b: Math Talk Mathematical Practices What change could the studio make that would make a difference to the -intercept of the equation? Lesson

64 Math n the Spot Writing an Equation from a Graph You can use information presented in a graph to write an equation in slope-intercept form. EXAMPLE 1 8.F.. Math Talk Mathematical Practices If the graph of an equation is a line that goes through the origin, what is the value of the -intercept? A video club charges a one-time membership fee plus a rental fee for each DVD borrowed. Use the graph to write an equation in slope-intercept form to represent the amount spent,, on DVD rentals. STEP 1 STEP Choose two points on the graph to find the slope. m = m = m = 10 8 = 1.5 Read the -intercept from the graph. The -intercept is 8. Use the slope formula. Substitute (0, 8) for ( 1, 1 ) and (8,18) for (, ). Simplif. Amount spent ($) Rentals STEP 3 Use our slope and -intercept values to write an equation in slope-intercept form. = m + b = Slope-intercept form Substitute 1.5 for m and 8 for. Reflect 1. What does the value of the slope represent in this contet? Personal Math Trainer nline Assessment and Intervention. Describe the meaning of the -intercept. YUR TURN 3. The cash register subtracts $.50 from a $5 Coffee Café gift card for ever medium coffee the customer bus. Use the graph to write an equation in slope-intercept form to represent this situation. Amount on Gift Card Dollars Number of coffees 18 Unit

65 Writing an Equation from a Description You can use information from a description of a linear relationship to find the slope and -intercept and to write an equation. EXAMPLE The rent charged for space in an office building is a linear relationship related to the size of the space rented. Write an equation in slope-intercept form for the rent at West Main Street ffice Rentals. 8.F.. Math n the Spot M Notes STEP 1 Identif the input and output variables. The input variable is the square footage of floor space. The output variable is the monthl rent. West Main St. ffice Rentals ffices for rent at convenient locations. Monthl Rates: 600 square feet for $ square feet for $1150 STEP Write the information given in the problem as ordered pairs. The rent for 600 square feet of floor space is $750: (600, 750) The rent for 900 square feet of floor space is $1150: (900, 1150) STEP 3 STEP Find the slope. m = = = = 3 Find the -intercept. Use the slope and one of the ordered pairs. = m + b 750 = _ b 750 = b Slope-intercept form Substitute for, m, and. Multipl. STEP 5-50 = b Substitute the slope and -intercept. = m + b = _ 3-50 Subtract 800 from both sides. Slope-intercept form Substitute for m and -50 for b. 3 Reflect. Without graphing, tell whether the graph of this equation rises or falls from left to right. What does the sign of the slope mean in this contet? Lesson

66 YUR TURN Personal Math Trainer nline Assessment and Intervention 5. Hari s weekl allowance varies depending on the number of chores he does. He received $16 in allowance the week he did 1 chores, and $1 in allowance the week he did 8 chores. Write an equation for his allowance in slope-intercept form. Guided Practice 1. Li is making beaded necklaces. For each necklace, she uses 7 spacers, plus 5 beads per inch of necklace length. Write an equation to find how man beads Li needs for each necklace. (Eplore Activit) a. input variable: b. output variable: c. equation:. Kate is planning a trip to the beach. She estimates her average speed to graph her epected progress on the trip. Write an equation in slope-intercept form that represents the situation. (Eample 1) Choose two points on the graph to find the slope. m = = Read the -intercept from the graph: b = Distance to beach (mi) Driving time (h) Use our slope and -intercept values to write an equation in slope-intercept form.? 3. At 59 F, crickets chirp at a rate of 76 times per minute, and at 65 F, the chirp 100 times per minute. Write an equation in slope-intercept form that represents the situation. (Eample ) Input variable: m = = ESSENTIAL QUESTIN CHECK-IN utput variable: Substitute in = m + b: + b; = b Write an equation in slope-intercept form.. Eplain what m and b in the equation = m + b tell ou about the graph of the line with that equation. 130 Unit

67 Name Class Date 5.1 Independent Practice 8.F.. Personal Math Trainer nline Assessment and Intervention 5. A dragonfl can beat its wings 30 times per second. Write an equation in slope-intercept form that shows the relationship between fling time in seconds and the number of times the dragonfl beats its wings. 6. A balloon is released from the top of a platform that is 50 meters tall. The balloon rises at the rate of meters per second. Write an equation in slope-intercept form that tells the height of the balloon above the ground after a given number of seconds. The graph shows a scuba diver s ascent over time. 7. Use the graph to find the slope of the line. Tell what the slope means in this contet. 8. Identif the -intercept. Tell what the -intercept means in this contet. Depth (m) Scuba Diver s Ascent Time (sec) 9. Write an equation in slope-intercept form that represents the diver s depth over time. 10. The formula for converting Celsius temperatures to Fahrenheit temperatures is a linear equation. Water freezes at 0 C, or 3 F, and it boils at 100 C, or 1 F. Find the slope and -intercept for a graph that gives degrees Celsius on the horizontal ais and degrees Fahrenheit on the vertical ais. Then write an equation in slope-intercept form that converts degrees Celsius into degrees Fahrenheit. 11. The cost of renting a sailboat at a lake is $0 per hour plus $1 for lifejackets. Write an equation in slope-intercept form that can be used to calculate the total amount ou would pa for using this sailboat. Lesson

68 The graph shows the activit in a savings account. 1. What was the amount of the initial deposit that started this savings account? Amount saved ($) Find the slope and -intercept of the graphed line. 6 Months in plan 1. Write an equation in slope-intercept form for the activit in this savings account. 15. Eplain the meaning of the slope in this graph. FCUS N HIGHER RDER THINKING Work Area 16. Communicate Mathematical Ideas Eplain how ou decide which part of a problem will be represented b the variable, and which part will be represented b the variable in a graph of the situation. 17. Represent Real-World Problems Describe what would be true about the rate of change in a situation that could not be represented b a graphed line and an equation in the form = m + b. 18. Draw Conclusions Must m, in the equation = m + b, alwas be a positive number? Eplain. 13 Unit

69 ? LESSN 5. Writing Linear Equations from a Table ESSENTIAL QUESTIN How do ou write an equation to model a linear relationship given a table? 8.F.. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value.... Interpret the rate of change and initial value. Graphing from a Table to Write an Equation You can use information from a table to draw a graph of a linear relationship and to write an equation for the graphed line. EXAMPLE 1 8.F.. Math n the Spot The table shows the temperature of a fish tank during an eperiment. Graph the data, and find the slope and -intercept from the graph. Then write the equation for the graph in slope-intercept form. Time (h) Temperature ( F) Image Credits: Imagebroker/ Alam Images STEP 1 STEP STEP 3 STEP STEP 5 Graph the ordered pairs from the table (time, temperature). Draw a line through the points. Choose two points on the graph to find the slope: for eample, choose (0, 8) and (1, 80). m = m = m = - 1 = - Read the -intercept from the graph. b = 8 Temperature ( F) Tank Temperature Time (h) Use these slope and -intercept values to write an equation in slope-intercept form. = m + b = Use the slope formula. Substitute (0, 8) for ( 1, 1 ) and (1, 80) for (, ). Simplif. Math Talk Mathematical Practices Which variable in the equation = m + b shows the initial temperature of the fish tank at the beginning of the eperiment? Lesson

70 YUR TURN Personal Math Trainer nline Assessment and Intervention Math Talk Mathematical Practices Which variable in the equation = m + b tells ou the volume of water released ever second from Hoover Dam? 1. The table shows the volume of water released b Hoover Dam over a certain period of time. Graph the data, and find the slope and -intercept from the graph. Then write the equation for the graph in slope-intercept form. Water Released Water Released from Hoover Dam from Hoover Dam Time (s) Volume of water (m 3 ) 300, ,000 Volume (m 3 ) 00, , Time (s) , , ,000 Writing an Equation from a Table The information from a table can also help ou to write the equation that represents a given situation without drawing the graph. Math n the Spot EXAMPLE 8.F.. Elizabeth s cell phone plan lets her choose how man minutes are included each month. The table shows the plan s monthl cost for a given number of included minutes. Write an equation in slope-intercept form to represent the situation. Animated Math Minutes included, Cost of plan ($), STEP 1 STEP STEP 3 Notice that the change in cost is the same for each increase of 100 minutes. So, the relationship is linear. Choose an two ordered pairs from the table to find the slope. m = = (0-1) 1 (00-100) = = 0.06 Find the -intercept. Use the slope and an point from the table. = m + b 1 = b 1 = 6 + b 8 = b Substitute the slope and -intercept. Slope-intercept form Substitute for, m, and. Multipl. Subtract 6 from both sides. 13 Unit = m + b = Slope-intercept form Substitute 0.06 for m and 8 for b.

71 Reflect. What is the base price for the cell phone plan, regardless of how man minutes are included? What is the cost per minute? Eplain. 3. What If? Elizabeth s cell phone compan changes the cost of her plan as shown below. Write an equation in slope-intercept form to represent the situation. How did the plan change? Minutes included, Cost of plan ($), YUR TURN. A salesperson receives a weekl salar plus a commission for each computer sold. The table shows the total pa, p, and the number of computers sold, n. Write an equation in slope-intercept form to represent this situation. Number of computers sold, n Total pa ($), p To rent a van, a moving compan charges $0.00 plus $0.50 per mile. The table shows the total cost, c, and the number of miles driven, d. Write an equation in slope-intercept form to represent this situation. Number of miles driven, d Total cost ($), c Math Talk Mathematical Practices Eplain the meaning of the slope and -intercept of the equation. Personal Math Trainer nline Assessment and Intervention Lesson

72 Guided Practice 1. Jaime purchased a $0 bus pass. Each time she rides the bus, a certain amount is deducted from the pass. The table shows the amount,, left on her pass after rides. Graph the data, and find the slope and -intercept from the graph. Then write the equation for the graph in slope-intercept form. (Eample 1) Number of rides, Amount left on pass ($), Amount left on pass ($) Bus Pass Balance Number of rides The table shows the temperature () at different altitudes (). This is a linear relationship. (Eample )? Altitude (ft), Temperature ( F), Find the slope for this relationship. 3. Find the -intercept for this relationship.. Write an equation in slope-intercept form that represents this relationship. ESSENTIAL QUESTIN CHECK-IN 6. Describe how ou can use the information in a table showing a linear relationship to find the slope and -intercept for the equation. 5. Use our equation to determine the temperature at an altitude of 5000 feet. Image Credits: Photodisc/ Gett Images 136 Unit

73 Name Class Date 5. Independent Practice 8.F.. Personal Math Trainer nline Assessment and Intervention 7. The table shows the costs of a large cheese pizza with toppings at a local pizzeria. Graph the data, and find the slope and -intercept from the graph. Then write the equation for the graph in slope-intercept form. Number of toppings, t Total cost ($), C The table shows how much an air-conditioning repair compan charges for different numbers of hours of work. Graph the data, and find the slope and -intercept from the graph. Then write the equation for the graph in slope-intercept form. Number of hours (h), t Amount charged ($), A Amount charged ($) Total cost ($) Cost of Large Pizza Number of toppings a Number of hours (h) h 9. A friend gave Ms. Morris a gift card for a local car wash. The table shows the linear relationship of how the value left on the card relates to the number of car washes. Number of car washes, Amount left on card ($), a. Write an equation that shows the number of dollars left on the card. b. Eplain the meaning of the negative slope in this situation. c. What is the maimum value of that makes sense in this contet? Eplain. The tables show linear relationships between and. Write an equation in slope-intercept form for each relationship Lesson

74 1. Finance Desiree starts a savings account with $ Ever month, she deposits $ a. Complete the table to model the situation. Month, Amount in Savings ($), b. Write an equation in slope-intercept form that shows how much mone Desiree has in her savings account after months. c. Use the equation to find how much mone Desiree will have in savings after 11 months. 13. Mont documented the amount of rain his farm received on a monthl basis, as shown in the table. Month, Rainfall (in.), a. Is the relationship linear? Wh or wh not? b. Can an equation be written to describe the amount of rain? Eplain. FCUS N HIGHER RDER THINKING Work Area 1. Analze Relationships If ou have a table that shows a linear relationship, when can ou read the value for b, in = m + b, directl from the table without drawing a graph or doing an calculations? Eplain. 15. What If? Jaíme graphed linear data given in the form (cost, number). The -intercept was 0. Jala graphed the same data given in the form (number, cost). What was the -intercept of her graph? Eplain. 138 Unit

75 ? LESSN 5.3 Linear Relationships and Bivariate Data ESSENTIAL QUESTIN 8.SP.1.1 Construct and interpret scatter plots for bivariate measurement data.... Describe patterns such as... linear association, and nonlinear association. Also 8.SP.1., 8.SP.1.3 How can ou contrast linear and nonlinear sets of bivariate data? Finding the Equation of a Linear Relationship You can use the points on a graph of a linear relationship to write an equation for the relationship. The equation of a linear relationship is = m + b, where m is the rate of change, or slope, and b is the value of when is 0. Math n the Spot EXAMPLE 1 8.SP.1. A handrail runs alongside a stairwa. As the horizontal distance from the bottom of the stairwa changes, the height of the handrail changes. Show that the relationship is linear, and then find the equation for the relationship. STEP 1 STEP Height (ft) Show that the relationship is linear (5, 3) (0, 19) (15, 15) (10, 11) (5, 7) Horizontal distance (ft) Height (ft) Write the equation of the linear relationship. Choose two points (5, 3) (0, 19) (15, 15) (10, 11) (5, 7) Horizontal distance (ft) Choose a point and use the slope to to find the slope. substitute values for,, and m. (5, 7) and (5, 3) = m + b m = = 16 0 = 0.8 All of the points (5, 7), (10, 11), (15, 15), (0, 19), and (5, 3) lie on the same line, so the relationship is linear. 7 = 0.8(5) + b 7 = + b 3 = b The equation of the linear relationship is = Animated Math Math Talk Mathematical Practices What does the slope of the equation represent in this situation? What does the -intercept represent? Lesson

76 YUR TURN Personal Math Trainer nline Assessment and Intervention Find the equation of each linear relationship. 1.. Cost ($) Time (min) Hours () Number of units () ,600 5, , , ,00 Making Predictions You can use an equation of a linear relationship to predict a value between data points that ou alread know. Math n the Spot EXAMPLE 8.SP.1.3 The graph shows the cost for tai rides of different distances. Predict the cost of a tai ride that covers a distance of 6.5 miles. STEP 1 Write the equation of the linear relationship. (, 7) and (6, 15) m = = 8 = = m + b 15 = (6) + b 15 = 1 + b Select two points. Calculate the rate of change. Simplif. Fill in values for,, and m. Simplif. Cost ($) Distance (mi) Image Credits: Fuse/ Gett Images 3 = b Solve for b. 10 Unit The equation of the linear relationship is = + 3. You can check our equation using another point on the graph. Tr (8, 19). Substituting gives 19 = (8) + 3. The right side simplifies to 19, so 19 = 19.

77 STEP Use our equation from Step 1 to predict the cost of a 6.5-mile tai ride. M Notes = + 3 = (6.5) + 3 = 16 Substitute = 6.5. Solve for. Reflect A tai ride that covers a distance of 6.5 miles will cost $ What If? Suppose a regulation changes the cost of the tai ride to $1.80 per mile, plus a fee of $.30. How does the price of the 6.5 mile ride compare to the original price?. How can ou use a graph of a linear relationship to predict a value for a new input? 5. How can ou use a table of linear data to predict a value? YUR TURN Paulina s income from a job that pas her a fied amount per hour is shown in the graph. Use the graph to find the predicted value. 6. Income earned for working hours 7. Income earned for working 3.5 hours Income ($) Time (h) 8. Total income earned for working for five 8-hour das all at the standard rate Personal Math Trainer nline Assessment and Intervention Lesson

78 EXPLRE ACTIVITY 8.SP.1.1 Contrasting Linear and Nonlinear Data Bivariate data is a set of data that is made up of two paired variables. If the relationship between the variables is linear, then the rate of change (slope) is constant. If the graph shows a nonlinear relationship, then the rate of change varies between pairs of points. Andrew has two options in which to invest $00. ption A earns simple interest of 5%, while ption B earns interest of 5% compounded annuall. The table shows the amount of the investment for both options over 0 ears. Graph the data and describe the differences between the two graphs. ption A ption B Year, Total ($) Total ($) STEP 1 STEP Graph the data from the table for ptions A and B on the same coordinate grid. Amount of investment Find the rate of change between pairs of points for ption A and classif the relationship. ption A (0, 00) and (5, 50) m = (5, 50) and (10, 300) Rate of Change = Year (10, 300) and (15, 350) The rate of change between the data values is, so the graph of ption A shows a relationship. 1 Unit

79 STEP 3 Find the rate of change between pairs of points for ption B and classif the relationship. ption B (0, 00) and (5, 55.6) m = Rate of Change (5, 55.6) and (10, 35.78) (10, 35.78) and (15, 15.79) The rate of change between the data values is, so the graph of ption B shows a relationship. Reflect 9. Wh are the graphs drawn as lines or curves and not discrete points? 10. Can ou determine b viewing the graph if the data have a linear or nonlinear relationship? Eplain. 11. Draw Conclusions Find the differences in the account balances to the nearest dollar at 5 ear intervals for ption B. How does the length of time that mone is in an account affect the advantage that compound interest has over simple interest? Lesson

80 Guided Practice Use the following graphs to find the equation of the linear relationship. (Eample 1) Distance (mi) Amount of gas (gal) Cost ($) Time (h) 3. The graph shows the relationship between the number of hours a kaak is rented and the total cost of the rental. Write an equation of the relationship. Then use the equation to predict the cost of a rental that lasts 5.5 hours. (Eample ) Cost ($) Time (h) Does each of the following graphs represent a linear relationship? Wh or wh not? (Eplore Activit)? ESSENTIAL QUESTIN CHECK-IN How can ou tell if a set of bivariate data shows a linear relationship? Image Credits: Comstock/ Gett Images 1 Unit

81 Name Class Date 5.3 Independent Practice 8.SP.1.1, 8.SP.1., 8.SP.1.3 Personal Math Trainer nline Assessment and Intervention Does each of the following tables represent a linear relationship? Wh or wh not? 7. Number 8. of boes Weight (kg) Da Height (cm) Eplain whether or not ou think each relationship is linear. 9. the cost of equal-priced DVDs and the number purchased 10. the height of a person and the person s age 11. the area of a square quilt and its side length 1. the number of miles to the net service station and the number of kilometers 13. Multistep The Mars Rover travels 0.75 feet in 6 seconds. Add the point to the graph. Then determine whether the relationship between distance and time is linear, and if so, predict the distance that the Mars Rover would travel in 1 minute. Distance (ft) Mars Rover Time (s) Lesson

82 1. Make a Conjecture Zefram analzed a linear relationship, found that the slope-intercept equation was = , and made a prediction for the value of for a given value of. He realized that he made an error calculating the -intercept and that it was actuall 1. Can he just subtract from his prediction if he knows that the slope is correct? Eplain. FCUS N HIGHER RDER THINKING Work Area 15. Communicate Mathematical Ideas The table shows a linear relationship. How can ou predict the value of when = 6 without finding the equation of the relationship? Critique Reasoning Louis sas that if the differences between the values of are constant between all the points on a graph, then the relationship is linear. Do ou agree? Eplain. 17. Make a Conjecture Suppose ou know the slope of a linear relationship and one of the points that its graph passes through. How could ou predict another point that falls on the graph of the line? 18. Eplain the Error Thomas used (7, 17.5) and (18, 5) from a graph to find the equation of a linear relationship as shown. What was his mistake? m = = = 79 = 79 + b 5 = b 5 = 1 + b, so b = 1377 The equation is = Unit

83 MDULE QUIZ Read 5.1 Writing Linear Equations from Situations and Graphs Write the equation of each line in slope-intercept form. Personal Math Trainer nline Assessment and Intervention Writing Linear Equations from a Table Write the equation of each linear relationship in slope-intercept form Linear Relationships and Bivariate Data Write the equation of the line that connects each set of data points ESSENTIAL QUESTIN Write a real-world situation that can be represented b a linear relationship. Module 5 17

84 MDULE 5 MIXED REVIEW PARCC Assessment Readiness Personal Math Trainer nline Assessment and Intervention Selected Response 1. An hourglass is turned over with the top part filled with sand. After 3 minutes, there are 855 ml of sand in the top half. After 10 minutes, there are 750 ml of sand in the top half. Which equation represents this situation? A = 85 B = C = D = 75. Which graph shows a linear relationship? A B C D What are the slope and -intercept of the relationship shown in the table? A slope = 0.05, -intercept = 1,500 B slope = 0.5, -intercept = 1,500 C slope = 0.05, -intercept =,000 D slope = 0.5, -intercept =,000. Which is the sum of ? Write our answer in scientific notation. A Mini-Task 10,000 0,000 30,000,500 3,000 3,500 B C D Franklin s faucet was leaking, so he put a bucket underneath to catch the water. After a while, Franklin started keeping track of how much water was in the bucket. His data is in the table below. Hours 3 5 Quarts a. Is the relationship linear or nonlinear? b. Write the equation for the relationship. c. Predict how much water will be in the bucket after 1 hours if Franklin doesn t stop the leak. 18 Unit

85 ? Functions 6 MDULE ESSENTIAL QUESTIN How can ou use functions to solve real-world problems? LESSN 6.1 Identifing and Representing Functions 8.F.1.1 LESSN 6. Describing Functions 8.F.1.1, 8.F.1.3 LESSN 6.3 Comparing Functions 8.EE..5, 8.F.1., 8.F.. LESSN 6. Analzing Graphs 8.F..5 Image Credits: Huntstock/ Gett Images Real-World Video Computerized machines can assist doctors in surgeries such as laser vision correction. Each action the surgeon takes results in one end action b the machine. In math, functions also have a one-in-oneout relationship. Math n the Spot Animated Math Personal Math Trainer Go digital with our write-in student edition, accessible on an device. Scan with our smart phone to jump directl to the online edition, video tutor, and more. Interactivel eplore ke concepts to see how math works. Get immediate feedback and help as ou work through practice sets. 19

86 Are YU Read? Complete these eercises to review skills ou will need for this module. Evaluate Epressions Personal Math Trainer nline Assessment and Intervention EXAMPLE Evaluate 3-5 for = = 3(-) -5 = -6-5 = -11 Substitute the given value of for. Multipl. Subtract. Evaluate each epression for the given value of for = for = for = for = _ 3-1 for = _ for = -8 Connect Words and Equations 8 EXAMPLE Erik s earnings equal 9 dollars per hour. e = earnings; h = hours multiplication e = 9 h Define the variables used in the situation. Identif the operation involved. Per indicates multiplication. Write the equation. Define the variables for each situation. Then write an equation. 7. Jana s age plus 5 equals her sister s age. 8. Andrew s class has 3 more students than Lauren s class. 9. The bank is 50 feet shorter than the firehouse. 10. The pencils were divided into 6 groups of. 150 Unit

87 Reading Start-Up Visualize Vocabular Use the words to complete the diagram. You can put more than one word in each section of the diagram. (, 6) m Understand Vocabular Complete the sentences using the preview words. 1. A rule that assigns eactl one output to each input is a. Reviewing Relationships = m + b b Vocabular Review Words bivariate data (datos bivariados) linear equation (ecuación lineal) nonlinear relationship (relación no lineal) ordered pair (par ordenado) proporational relationship (relación proporcional) slope (pendiente) -coordinate (coordenada ) -coordinate (coordenada ) -intercept (intersección con el eje ) Preview Words function (función) input (valor de entrada) linear function (función lineal) output (valor de salida). The value that is put into a function is the. 3. The result after appling the function machine's rule is the. Active Reading Double-Door Fold Create a double-door fold to help ou understand the concepts in this module. Label one flap Proportional Functions and the other flap Non-proportional Functions. As ou stud each lesson, write important ideas under the appropriate flap. Include an sample problems that will help ou remember the concepts when ou look back at our notes. Module 6 151

88 MDULE 6 Unpacking the Standards Understanding the standards and the vocabular terms in the standards will help ou know eactl what ou are epected to learn in this module. 8.F.1.1 Understand that a function is a rule that assigns to each input eactl one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. Ke Vocabular function (función) An input-output relationship that has eactl one output for each input. What It Means to You You will identif sets of ordered pairs that are functions. A function is a rule that assigns eactl one output to each input. UNPACKING EXAMPLE 8.F.1.1 Does the following table of inputs and outputs represent a function? Yes, it is a function because each number in the input column is assigned to onl one number in the output column. Input utput F.1. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Visit to see all Florida Common Core Standards unpacked. What It Means to You You will learn to identif and compare functions epressed as equations and tables. UNPACKING EXAMPLE 8.F.1. A spider descends a 0-foot drainpipe at a rate of.5 feet per minute. Another spider descends a drainpipe as shown in the table. Find and compare the rates of change and initial values of the linear functions in terms of the situations the model. Spider #1: f() = Spider #: Time (min) 0 1 Height (ft) For Spider #1, the rate of change is -.5, and the initial value is 0. For Spider #, the rate of change is -3, and the initial value is 3. Spider # started at 3 feet, which is 1 feet higher than Spider #1. Spider #1 is descending at.5 feet per minute, which is 0.5 feet per minute slower than Spider #. Image Credits: PhotoDisc/ Gett Images 15 Unit

89 LESSN 6.1 Identifing and Representing Functions 8.F.1.1 Understand that a function is a rule that assigns to each input eactl one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.? ESSENTIAL QUESTIN How can ou identif and represent functions? EXPLRE ACTIVITY 8.F.1.1 Understanding Relationships Carlos needs to bu some new pencils from the school suppl store at his school. Carlos asks his classmates if the know how much pencils cost. Angela sas she bought pencils for $0.50. Paige bought 3 pencils for $0.75, and Spencer bought pencils for $1.00. Carlos thinks about the rule for the price of a pencil as a machine. When he puts the number of pencils he wants to bu into the machine, the machine applies a rule and tells him the total cost of that number of pencils. Input utput A B Number of Pencils Rule Total Cost i.? ii. 3? iii.? iv. v. 1 Use the prices in the problem to fill in total cost in rows i iii of the table. Describe an patterns ou see. Use our pattern to determine the cost of 1 pencil. Lesson

90 EXPLRE ACTIVITY (cont d) C D Use the pattern ou identified to write the rule applied b the machine. Write the rule as an algebraic epression and fill in rule column row iv of the table. Carlos wants to bu 1 pencils. Use our rule to fill in row v of the table to show how much Carlos will pa for 1 pencils. Reflect 1. How did ou decide what operation to use in our rule?. What If? Carlos decides to bu erasers in a package. There are 6 penciltop erasers in packages of erasers. a. Write a rule in words for the number of packages Carlos needs to bu to get erasers. Then write the rule as an algebraic epression. b. How man packages does Carlos need to bu to get 18 erasers? Math n the Spot Identifing Functions from Mapping Diagrams A function assigns eactl one output to each input. The value that is put into a function is the input. The result is the output. A mapping diagram can be used to represent a relationship between input values and output values. A mapping diagram represents a function if each input value is paired with onl one output value. EXAMPLE 1 Determine whether each relationship is a function. A Input utput F.1.1 Since each input value is paired with onl one output value, the relationship is a function. 15 Unit

91 Determine whether each relationship is a function. B Since is paired with more than one output value (both and 5), the relationship is not a function. Reflect 3. Is it possible for a function to have more than one input value but onl one output value? Provide an illustration to support our answer. YUR TURN Determine whether each relationship is a function. Eplain Personal Math Trainer nline Assessment and Intervention Math Talk Mathematical Practices What is alwas true about a mapping diagram that represents a function? Identifing Functions from Tables Relationships between input values and output values can also be represented using tables. The values in the first column are the input values. The values in the second column are the output values. The relationship represents a function if each input value is paired with onl one output value. EXAMPLE Determine whether each relationship is a function. A Input utput Since 15 is a repeated output value, one output value is paired with two input values. If this occurs in a relationship, the relationship can still be a function. 8.F Since each input value is paired with onl one output value, the relationship is a function. Math n the Spot M Notes Lesson

92 Determine whether each relationship is a function. B Input utput Since 1 is a repeated input value, one input value is paired with two output values. Look back at the rule for functions. Is this relationship a function? Since the input value 1 is paired with more than one output value (both 10 and ), the relationship is not a function. Reflect 6. What is alwas true about the numbers in the first column of a table that represents a function? Wh must this be true? YUR TURN Determine whether each relationship is a function. Eplain 7. Input utput 8. Input utput Personal Math Trainer nline Assessment and Intervention 156 Unit

93 Identifing Functions from Graphs Graphs can be used to displa relationships between two sets of numbers. Each point on a graph represents an ordered pair. The first coordinate in each ordered pair is the input value. The second coordinate is the output value. The graph represents a function if each input value is paired with onl one output value. Math n the Spot EXAMPLE 3 8.F.1.1 The graph shows the relationship between the number of hours students spent studing for an eam and the eam grades. Is the relationship represented b the graph a function? The input values are the number of hours spent studing b each student. The output values are the eam grades. The points represent the following ordered pairs: Eam grade Hours Studied and Eam Grade Hours studied (1, 70) (, 70) (, 85) (3, 75) (5, 80) (6, 8) (7, 88) (9, 90) (9, 95) (1, 98) Notice that is paired with both 70 and 85, and 9 is paired with both 90 and 95. Therefore, since these input values are paired with more than one output value, the relationship is not a function. Reflect 9. Man real-world relationships are functions. For eample, the amount of mone made at a car wash is a function of the number of cars washed. Give another eample of a real-world function. YUR TURN 10. The graph shows the relationship between the heights and weights of the members of a basketball team. Is the relationship represented b the graph a function? Eplain. Weight (lb) Heights and Weights of Team Members Height (in.) Personal Math Trainer nline Assessment and Intervention Lesson

94 Guided Practice Complete each table. In the row with as the input, write a rule as an algebraic epression for the output. Then complete the last row of the table using the rule. (Eplore Activit) 1. Input utput. Input utput 3. Input utput Tickets Cost ($) Minutes Pages Muffins Cost ($) Determine whether each relationship is a function. (Eamples 1 and ) Input utput ? 6. The graph shows the relationship between the weights of 5 packages and the shipping charge for each package. Is the relationship represented b the graph a function? Eplain. ESSENTIAL QUESTIN CHECK-IN 7. What are four different was of representing functions? How can ou tell if a relationship is a function? Weights and Shipping Costs Shipping cost ($) Weight (lb) 158 Unit

95 Name Class Date 6.1 Independent Practice 8.F.1.1 Personal Math Trainer nline Assessment and Intervention Determine whether each relationship represented b the ordered pairs is a function. Eplain. 8. (, ), (3, 1), (5, 7), (8, 0), (9, 1) 9. (0, ), (5, 1), (, 8), (6, 3), (5, 9) 10. Draw Conclusions Joaquin receives $0.0 per pound for 1 to 99 pounds of aluminum cans he reccles. He receives $0.50 per pound if he reccles more than 100 pounds. Is the amount of mone Joaquin receives a function of the weight of the cans he reccles? Eplain our reasoning. 11. A biologist tracked the growth of a strain of bacteria, as shown in the graph. a. Eplain wh the relationship represented b the graph is a function. Number of Bacteria Bacteria B b. What If? Suppose there was the same number of bacteria for two consecutive hours. Would the graph still represent a function? Eplain. 1. Multiple Representations Give an eample of a function in everda life, and represent it as a graph, a table, and a set of ordered pairs. Describe how ou know it is a function. 6 Time (h) Lesson

96 The graph shows the relationship between the weights of si wedges of cheese and the price of each wedge. 13. Is the relationship represented b the graph a function? Justif our reasoning. Use the words input and output in our eplanation, and connect them to the contet represented b the graph. Price ($) Cost of Cheese 1. Analze Relationships Suppose the weights and prices of additional wedges of cheese were plotted on the graph. Might that change our answer to question 13? Eplain our reasoning Weight (lb) FCUS N HIGHER RDER THINKING Work Area 15. Justif Reasoning A mapping diagram represents a relationship that contains three different input values and four different output values. Is the relationship a function? Eplain our reasoning. 16. Communicate Mathematical Ideas An onion farmer is hiring workers to help harvest the onions. He knows that the number of das it will take to harvest the onions is a function of the number of workers he hires. Eplain the use of the word function in this contet. Image Credits: Brand X Pictures/ Gett Images 160 Unit

97 ? LESSN 6. Describing Functions ESSENTIAL QUESTIN 8.F.1.3 Interpret the equation = m + b as defining a linear function, whose graph is a straight line; give eamples of functions that are not linear. Also 8.F.1.1 What are some characteristics that ou can use to describe functions? EXPLRE ACTIVITY 8.F.1.1 Investigating a Constant Rate of Change The U.S. Department of Agriculture defines heav rain as rain that falls at a rate of 1.5 centimeters per hour. A The table shows the total amount of rain that falls in various amounts of time during a heav rain. Complete the table. Time (h) Total Amount of Rain (cm) B C Plot the ordered pairs from the table on the coordinate plane at the right. How much rain falls in 3.5 hours? Heav Rainfall D E F Plot the point corresponding to 3.5 hours of heav rain. What do ou notice about all of the points ou plotted? Is the total amount of rain that falls a function of the number of hours that rain has been falling? Wh or wh not? Reflect 1. Suppose ou continued to plot points for times between those in the table, such as 1. hours or.5 hours. What can ou sa about the locations of these points? Total Amount of Rain (cm) Time (h) Lesson

98 Math n the Spot Graphing Linear Functions The relationship ou investigated in the previous activit can be represented b the equation = 1.5, where is the time and is the total amount of rain. The graph of the relationship is a line, so the equation is a linear equation. Since there is eactl one value of for each value of, the relationship is a function. It is a linear function because its graph is a nonvertical line. EXAMPLE 1 8.F.1.3 Math Talk Mathematical Practices Carrie said that for a function to be a linear function, the relationship it represents must be proportional. Do ou agree or disagree? Eplain. The temperature at dawn was 8 F and increased steadil F ever hour. The equation = + 8 gives the temperature after hours. State whether the relationship between the time and the temperature is proportional or nonproportional. Then graph the function. STEP 1 STEP Compare the equation with the general linear equation = m + b. = + 8 is in the form = m + b, with m = and b = 8. Therefore, the equation is a linear equation. Since b 0, the relationship is nonproportional. Choose several values for the input. Substitute these values for in the equation to find the output. + 8 (, ) 0 (0) (0, 8) () (, 1) () (, 16) 6 (6) (6, 0) STEP 3 YUR TURN Graph the ordered pairs. Then draw a line through the points to represent the solutions of the function.. State whether the relationship between and in = 0.5 is proportional or nonproportional. Then graph the function. -10 Temperature ( F) Temperatures Time (h) Personal Math Trainer nline Assessment and Intervention Unit

99 Determining Whether a Function is Linear The linear equation in Eample 1 has the form = m + b, where m and b are real numbers. Ever equation in the form = m + b is a linear equation. The linear equations represent linear functions. Equations that cannot be written in this form are not linear equations, and therefore are not linear functions. Math n the Spot EXAMPLE 8.F.1.3 A square tile has a side length of inches. The equation = gives the area of the tile in square inches. Determine whether the relationship between and is linear and, if so, if it is proportional. STEP 1 STEP STEP 3 STEP YUR TURN Choose several values for the input. Substitute these values for in the equation to find the output. Graph the ordered pairs. Identif the shape of the graph. The points suggest a curve, not a line. Draw a curve through the points to represent the solutions of the function. Describe the relationship between and. The graph is not a line so the relationship is not linear. nl a linear relationship can be proportional, so the relationship is not proportional. 3. A soda machine makes _ 3 gallon of soda ever minute. The total amount that the machine makes in minutes is given b the equation = _ 3. Determine whether the relationship between and is linear and, if so, if it is proportional. Time (min), Amount (gal), (, ) (1, 1) (, ) (3, 9) 16 (, 16) Amount (gal) Making Soda Time (min) Animated Math Math Talk Mathematical Practices How can ou use the numbers in the table to decide whether or not the relationship between and is linear? Personal Math Trainer nline Assessment and Intervention Lesson

100 Guided Practice Plot the ordered pairs from the table. Then graph the function represented b the ordered pairs and tell whether the function is linear or nonlinear. (Eamples 1 and ) 1. = 5 -. = - Input, utput, Input, utput, Eplain whether each equation is a linear equation. (Eample ) 3. = 1. = 1? ESSENTIAL QUESTIN CHECK-IN 5. Eplain how ou can use a table of values, an equation, and a graph to determine whether a function represents a proportional relationship. 16 Unit

101 Name Class Date 6. Independent Practice 8.F.1.1, 8.F.1.3 Personal Math Trainer nline Assessment and Intervention 6. State whether the relationship between and in = 5 is proportional or nonproportional. Then graph the function The Fortaleza telescope in Brazil is a radio telescope. Its shape can be approimated with the equation = Is the relationship between and linear? Is it proportional? Eplain Kile spent $0 on rides and snacks at the state fair. If is the amount she spent on rides, and is the amount she spent on snacks, the total amount she spent can be represented b the equation + = 0. Is the relationship between and linear? Is it proportional? Eplain. 9. Represent Real-World Problems The drill team is buing new uniforms. The table shows, the total cost in dollars, and, the number of uniforms purchased. Number of uniforms, Total cost ($), a. Use the data to draw a graph. Is the relationship between and linear? Eplain. b. Use our graph to predict the cost of purchasing 1 uniforms. 10. Marta, a whale calf in an aquarium, is fed a special milk formula. Her handler uses a graph to track the number of gallons of formula the calf drinks in hours. Is the relationship between and linear? Is it proportional? Eplain. Total cost ($) Total amount of formula (gal) Drill Team Uniforms Number of uniforms Marta s Feedings Time (h) Lesson

102 11. Critique Reasoning A student claims that the equation = 7 is not a linear equation because it does not have the form = m + b. Do ou agree or disagree? Wh? 1. Make a Prediction Let represent the number of hours ou read a book and represent the total number of pages ou have read. You have alread read 70 pages and can read 30 pages per hour. Write an equation relating hours and pages ou read. Then predict the total number of pages ou will have read after another 3 hours. FCUS N HIGHER RDER THINKING Work Area 13. Draw Conclusions Rebecca draws a graph of a real-world relationship that turns out to be a set of unconnected points. Can the relationship be linear? Can it be proportional? Eplain our reasoning. 1. Communicate Mathematical Ideas Write a real-world problem involving a proportional relationship. Eplain how ou know the relationship is proportional. 15. Justif Reasoning Show that the equation + 3 = 3( + 1) is linear and that it represents a proportional relationship between and. Image Credits: Photodisc/ Gett Images 166 Unit

103 ? LESSN 6.3 Comparing Functions ESSENTIAL QUESTIN How can ou use tables, graphs, and equations to compare functions? 8.F.1. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Also 8.EE..5, 8.F.. Comparing a Table and an Equation To compare a function written as an equation and another function represented b a table, find the equation for the function in the table. EXAMPLE 1 8.F.1., 8.F.. Math n the Spot Josh and Maggie bu MP3 files from different music services. The monthl cost, dollars, for songs is linear. The cost of Josh s service is = The cost of Maggie s service is shown below. Monthl Cost of MP3s at Maggie s Music Service Image Credits: Houghton Mifflin Harcourt A B Songs, Cost ($), Write an equation to represent the monthl cost of Maggie s service. STEP 1 STEP STEP 3 Choose an two ordered pairs from the table to find the slope. m = = = = 0.99 The points (5,.95) and (10, 9.90) were used. Find the -intercept. Use the slope and an point. = m + b Slope-intercept form..95 = b 0 = b Substitute the slope and -intercept. = or = 0.99 Substitute for, m, and. Substitute 0.99 for m and 0 for. Which service is cheaper when 30 songs are downloaded? Math Talk Mathematical Practices Describe each service s cost in words using the meanings of the slopes and -intercepts. Josh s service: Maggie s service: = = = 5 = 9.7 Josh s service is cheaper. Lesson

104 YUR TURN Personal Math Trainer nline Assessment and Intervention 1. Quentin is choosing between buing books at the bookstore or buing online versions of the books for his tablet. The cost, dollars, of ordering books online for books is = The cost of buing the books at the bookstore is shown in the table. Which method of buing books is more epensive if Quentin wants to bu 6 books? Cost of Books at the Bookstore Books, Cost ($), EXPLRE ACTIVITY 1 Comparing a Table and a Graph The table and graph show how man words Morgan and Brian tped correctl on a tping test. For both students, the relationship between words tped correctl and time is linear. A Morgan s Tping Test Time (min) Words Find Morgan s unit rate. 8.F.1., 8.EE..5 Words Brian s Tping Test Time (min) B C Find Brian s unit rate. Which student tpes more correct words per minute? Reflect. Katie tpes 17 correct words per minute. Eplain how a graph of Katie s test results would compare to Morgan s and Brian s. 168 Unit

105 EXPLRE ACTIVITY Comparing a Graph and a Description Jamal wants to bu a new game sstem that costs $00. He does not have enough mone to bu it toda, so he compares laawa plans at different stores. The plan at Store A is shown on the graph. Store B requires an initial pament of $60 and weekl paments of $0 until the balance is paid in full. 8.F.1., 8.F.. Balance owed ($) Number of weeks A Write an equation in slope-intercept form for Store A s laawa plan. Let represent number of weeks and represent balance owed. B Write an equation in slope-intercept form for Store B s laawa plan. Let represent number of weeks and represent balance owed. C D Sketch a graph of the plan at Store B on the same grid as Store A. How can ou use the graphs to tell which plan requires the greater down pament? How can ou use the equations? E F How can ou use the graphs to tell which plan requires the greater weekl pament? How can ou use the equations? Which plan allows Jamal to pa for the game sstem faster? Eplain. Lesson

106 Guided Practice Doctors have two methods of calculating maimum heart rate. With the first method, maimum heart rate,, in beats per minute is = 0 -, where is the person s age. Maimum heart rate with the second method is shown in the table. (Eample 1) Age, Heart rate (bpm), Which method gives the greater maimum heart rate for a 70-ear-old?. Are heart rate and age proportional or nonproportional for each method? Aisha runs a tutoring business. With Plan 1, students ma choose to pa $15 per hour. With Plan, the ma follow the plan shown on the graph. (Eplore Activit 1 and ) 3. Describe the plan shown on the graph.. Sketch a graph showing the $15 per hour option. Cost ($) Time (h)? 5. What does the intersection of the two graphs mean? 6. Which plan is cheaper for 10 hours of tutoring? 7. Are cost and time proportional or nonproportional for each plan? ESSENTIAL QUESTIN CHECK-IN 8. When using tables, graphs, and equations to compare functions, wh do ou find the equations for tables and graphs? Image Credits: PhotoAlto/Gett Images 170 Unit

107 Name Class Date 6.3 Independent Practice 8.EE..5, 8.F.1., 8.F.. Personal Math Trainer nline Assessment and Intervention The table and graph show the miles driven and gas used for two scooters. Scooter A Distance (mi), Gas used (gal), Gas used (gal) Scooter B 9. Which scooter uses fewer gallons of gas when 1350 miles are driven? Distance (mi) 10. Are gas used and miles proportional or nonproportional for each scooter? A cell phone compan offers two teting plans to its customers. The monthl cost, dollars, of one plan is = , where is the number of tets. The cost of the other plan is shown in the table. Number of tets, Cost ($), Which plan is cheaper for under 00 tets? 1. The graph of the first plan does not pass through the origin. What does this indicate? 13. Brianna wants to bu a digital camera for a photograph class. ne store offers the camera for $50 down and a pament plan of $0 per month. The pament plan for a second store is described b = , where is the total cost in dollars and is the number of months. Which camera is cheaper when the camera is paid off in 1 months? Eplain. Lesson

108 1. The French club and soccer team are washing cars to earn mone. The amount earned, dollars, for washing cars is a linear function. Which group makes the most mone per car? Eplain. Soccer Team French Club Number of cars, Amount earned ($), Amount earned ($) Number of cars FCUS N HIGHER RDER THINKING Work Area 15. Draw Conclusions Gm A charges $60 a month plus $5 per visit. The monthl cost at Gm B is represented b = 5 + 0, where is the number of visits per month. What conclusion can ou draw about the monthl costs of the gms? 16. Justif Reasoning Wh will the value of for the function = alwas be greater than that for the function = + when > 1? 17. Analze Relationships The equations of two functions are = and = Which function is changing more quickl? Eplain. 17 Unit

109 ? LESSN 6. Analzing Graphs ESSENTIAL QUESTIN 8.F..5 Describe qualitativel the functional relationship between two quantities b analzing a graph.... Sketch a graph that ehibits the qualitative features of a function that has been described verball. How can ou describe a relationship given a graph and sketch a graph given a description? EXPLRE ACTIVITY 1 Interpreting Graphs 8.F..5 A roller coaster park is open from Ma to ctober each ear. The graph shows the number of park visitors over its season. 3 Visitors 1 5 Time (months) Image Credits: Purestock/ Gett Images A B C Segment 1 shows that attendance during the opening weeks of the park s season staed constant. Describe what Segment shows. Based on the time frame, give a possible eplanation for the change in attendance represented b Segment. Which segments of the graph show decreasing attendance? Give a possible eplanation. Reflect 1. Eplain how the slope of each segment of the graph is related to whether attendance increases or decreases. Lesson

110 EXPLRE ACTIVITY 8.F..5 Matching Graphs to Situations Grace, Jet, and Mike are studing 100 words for a spelling bee. Grace started b learning how to spell man words each da, but then learned fewer and fewer words each da. Jet learned how to spell the same number of words each da. Mike started b learning how to spell onl a few words each da, but then learned a greater number of words each da. F-U-N-C-T-I--N A B C Words Learned Time (das) Words Learned Time (das) Words Learned Time (das) A B Describe the progress represented b Graph A. Describe the progress represented b Graph B. Math Talk Mathematical Practices Tell whether each graph is linear or nonlinear and proportional or nonproportional. C D Describe the progress represented b Graph C. Determine which graph represents each student s stud progress and write the students names under the appropriate graphs. Reflect. What would it mean if one of the graphs slanted downward? 17 Unit

111 EXPLRE ACTIVITY 3 Sketching a Graph for a Situation Mrs. Sutton provides free math tutoring to her students ever da after school. No one comes to tutoring sessions during the first week of school. ver the net two weeks, use of the tutoring service graduall increases. A 8.F..5 Sketch a graph showing the number of students who use the tutoring service over the first three weeks of school. Students B Time (weeks) Mrs. Sutton s students are told that the will have a math test at the end of the fifth week of school. How do ou think this will affect the number of students who come to tutoring? C Considering our answer to B, sketch a graph showing the number of students who might use the tutoring service over the first si weeks of school. Students Time (weeks) Reflect 3. If Mrs. Sutton offers bonus credit to students who come to tutoring, how might this affect the number of students?. How would our answer to Question 3 affect the graph? Lesson

112 Guided Practice In a lab environment, colonies of bacteria follow a predictable pattern of growth. The graph shows this growth over time. (Eplore Activit 1) 1. What is happening to the population during Phase?. What is happening to the population during Phase? Number of Microbes Phase 1 Bacterial Growth Curve Phase Phase 3 Phase The graphs give the speeds of three people who are riding snowmobiles. Tell which graph corresponds to each situation. (Eplore Activit ) Time Graph 1 Graph Graph 3 Speed (mi/h) Speed (mi/h) Speed (mi/h) Time Time Time 3. Chip begins his ride slowl but then stops to talk with some friends. After a few minutes, he continues his ride, graduall increasing his speed.. Linda steadil increases her speed through most of her ride. Then she slows down as she nears some trees. 5. Paulo stood at the top of a diving board. He walked to the end of the board, and then dove forward into the water. He plunged down below the surface, then swam straight forward while underwater. Finall, he swam forward and upward to the surface of the water. Draw a graph to represent Paulo s elevation at different distances from the edge of the pool. (Eplore Activit 3) Distance Above or Below Water 0 Distance from Edge of Pool 176 Unit

113 Name Class Date 6. Independent Practice 8.F..5 Personal Math Trainer nline Assessment and Intervention Tell which graph corresponds to each situation below. Graph 1 Graph Graph 3 Distance from home Distance from home Distance from home Time Time Time 6. Arnold started from home and walked to a friend s house. He staed with his friend for a while and then walked to another friend s house farther from home. 7. Francisco started from home and walked to the store. After shopping, he walked back home. Image Credits: Adobe Image Librar/Gett Images 8. Celia walks to the librar at a stead pace without stopping. Regina rented a motor scooter. The graph shows how far awa she is from the rental site after each half hour of riding. 9. Represent Real-World Problems Use the graph to describe Regina s trip. You can start the description like this: Regina left the rental shop and rode for an hour 10. Analze Relationships Determine during which half hour Regina covered the greatest distance. Distance (mi) Distance from Rental Site 3 5 Time (h) 6 Lesson

114 The data in the table shows the speed of a ride at an amusement park at different times one afternoon. Time 3:0 3:1 3: 3:3 3: 3:5 Speed (mi/h) Sketch a graph that shows the speed of the ride over time. 1. Between which times is the ride s speed increasing the fastest? 13. Between which times is the ride s speed decreasing the fastest? Speed (mi/h) :0 3:13: Time 3:3 3:3:5 FCUS N HIGHER RDER THINKING Work Area A woodland area on an island contains a population of foes. The graph describes the changes in the population over time. 1. Justif Reasoning What is happening to the fo population before time t? Eplain our reasoning. Population Fo Population t Time 15. What If? Suppose at time t, a conservation organization moves a large group of foes to the island. Sketch a graph to show how this action might affect the population on the island after time t. 16. Make a Prediction At some point after time t, a forest fire destros part of the woodland area on the island. Describe how our graph from problem 15 might change. Population Fo Population t Time Image Credits: Eric Isselée/ Shutterstock 178 Unit

115 MDULE QUIZ Read 6.1 Identifing and Representing Functions Determine whether each relationship is a function. Personal Math Trainer nline Assessment and Intervention 1.. Input, utput, 3. (, 5), (7, ), ( 3, ), 0 (, 9), (1, 1) Describing Functions Determine whether each situation is linear or nonlinear, and proportional or nonproportional.. Joanna is paid $1 per hour. 5. Alberto started out bench pressing 50 pounds. He then added 5 pounds ever week. 6.3 Comparing Functions 6. Which function is changing more quickl? Eplain. 6. Analzing Graphs Function Describe a graph that shows Sam running at a constant rate. ESSENTIAL QUESTIN 8. How can ou use functions to solve real-world problems? Function Input, utput, Module 6 179

116 MDULE 6 MIXED REVIEW PARCC Assessment Readiness Personal Math Trainer nline Assessment and Intervention Selected Response 1. Which table shows a proportional function? A B C D What is the slope and -intercept of the function shown in the table? A m = -; b = - B m = -; b = C m = ; b = D m = ; b = 3. The table below shows some input and output values of a function. Input utput What is the missing output value? A 0 B 1 C D 3. Tom walked to school at a stead pace, met his sister, and the walked home at a stead pace. Describe this graph. A V-shaped B upside down V-shaped C Straight line sloping up D Straight line sloping down Mini-Task 5. Linear functions can be used to find the price of a building based on its floor area. Below are two of these functions. = ,000 Floor Area (ft ) ,000 Price ($1,000s) a. Find and compare the slopes. b. Find and compare the -intercepts. c. Describe each function as proportional or nonproportional. 180 Unit

117 ? UNIT Stud Guide Review MDULE 13 ESSENTIAL QUESTIN Proportional Relationships How can ou use proportional relationships to solve real-world problems? EXAMPLE 1 Write an equation that represents the proportional relationship shown in the graph. Ke Vocabular constant of proportionalit (constante de proporcionalidad) proportional relationship (relación proporcional) slope (pendiente) Profit ($) Bracelets sold Use the points on the graph to make a table. Bracelets sold Profit ($) Let represent the number of bracelets sold. Let represent the profit. The equation is = 3. EXAMPLE Find the slope of the line. slope = rise run = 3 - = - 3 _ Run Rise Unit 181

118 EXERCISES 1. The table represents a proportional relationship. Write an equation that describes the relationship. Then graph the relationship represented b the data. (Lessons 3.1, 3.3, 3.) 6 Distance Time () Distance () Time Find the slope and the unit rate represented on each graph. (Lesson 3.). Words Feet Minutes Seconds? 18 MDULE ESSENTIAL QUESTIN Nonproportional Relationships How can ou use nonproportional relationships to solve real-world problems? EXAMPLE 1 Jai is saving to bu his mother a birthda gift. Each week, he saves $5. He started with $5. The equation = gives the total Jai has saved,, after weeks. Draw a graph of the equation. Then describe the relationship. Use the equation to make a table. Then, graph the ordered pairs from the table, and draw a line through the points. (weeks) (savings in dollars) The relationship is linear but nonproportional. Unit Savings ($) Time (wk) Ke Vocabular linear equation (ecuación lineal) slope-intercept form of an equation (forma de pendiente-intersección) -intercept (intersección con el eje )

119 EXAMPLE Graph = The slope is -1, or - 1_. The -intercept is Rise intercept Run EXERCISES Complete each table. Eplain whether the relationship between and is proportional or nonproportional and whether it is linear. (Lesson.1) 1. = 10. = Find the slope and -intercept for the linear relationship shown in the table. Graph the line. Is the relationship proportional or nonproportional? (Lessons.,.) slope - -intercept - The relationship is.. Tom s Tais charges a fied rate of $ per ride plus $0.50 per mile. Carla s Cabs does not charge a fied rate but charges $1.00 per mile. (Lessons.3) a. Write an equation that represents the cost of Tom s Tais. b. Write an equation that represents the cost of Carla s cabs. c. Steve calculated that for the distance he needs to travel, Tom s Tais will charge the same amount as Carla s Cabs. Graph both equations. How far is Steve going to travel and how much will he pa? Total cost ($) Distance (mi) Unit 183

120 ? MDULE 5 Writing Linear Equations ESSENTIAL QUESTIN How can ou use linear equations to solve real-world problems? Ke Vocabular bivariate data (datos bivariados) nonlinear relationship (relación no lineal) EXAMPLE 1 Jose is renting a backhoe for a construction job. The rental charge for a month is based on the number of das in the month and a set charge per month. In September, which has 30 das, Jose paid $700. In August, which has 31 das, he paid $715. Write an equation in slope-intercept form that represents this situation. ( 1, 1 ), (, ) (30, 700), (31, 715) m = = = 15 = m + b 715 = 15(31) + b 50 = b = Write the information given as ordered pairs. Find the slope. Slope-intercept form Substitute for, m, and to find b. Solve for b. Write the equation. EXAMPLE Determine if the graph shown represents a linear or nonlinear relationship. 10 Points Rate of Change (0, 0) and (6, 5) m = = 5 _ 6 (6, 5) and (9, 7) m = = _ 3 (0, 0) and (9, 7) m = = 7_ 9 The rates of change are not constant. The graph represents a nonlinear relationship. 18 Unit

121 EXERCISES 1. Ms. Thompson is grading math tests. She is giving everone that took the test a 10-point bonus. Each correct answer is worth 5 points. Write an equation in slope-intercept form that represents the scores on the tests. (Lesson 5.1) The table shows a pa scale based on ears of eperience. (Lessons 5.1, 5.) Eperience (ears), Hourl pa ($), Find the slope for this relationship. 3. Find the -intercept.. Write an equation in slope-intercept form that represents this relationship. 5. Graph the equation, and use it to predict the hourl pa of someone with 10 ears of eperience. 50 Hourl pa ($) Does each of the following graphs represent a linear relationship? Wh or wh not? (Lesson 5.3) Eperience (r) Unit 185

122 ? MDULE 6 Functions ESSENTIAL QUESTIN How can ou use functions to solve real-world problems? EXAMPLE 1 Determine whether each relationship is a function. A Input utput Ke Vocabular function (función) input (valor de entrada) linear function (función lineal) output (valor de salida) The relationship is not a function, because an input,, is paired with different outputs, and 0. B Since each input value is paired with onl one output value, the relationship is a function. EXAMPLE Sall and Louis are on a long-distance bike ride. Sall bikes at a stead rate of 18 miles per hour. The distance that Sall covers in hours is given b the equation = 18. Louis s speed can be found b using the numbers in the table. Who will travel farther in hours and b how much? Louis s Biking Speed Time (h), Distance (mi), Sall s ride: Louis s ride: = 18 = 0 = 18() = 0() = 7 = 80 Each distance in the table is 0 times each number of hours. Louis s speed is 0 miles per hour, and his distance covered is represented b = 0. Sall will ride 7 miles in hours. Louis will ride 80 miles in hours. Louis will go 8 miles farther. 186 Unit

123 EXERCISES Determine whether each relationship is a function. (Lesson 6.1) Input utput Tell whether the function is linear or nonlinear. (Lesson 6.) 3. = = Elaine has a choice of two health club memberships. The first membership option is to pa $500 now and then pa $150 per month. The second option is shown in the table. Elaine plans to go to the club for 1 months. Which option is cheaper? Eplain. (Lesson 6.3) Months, 1 3 Total paid ($), Jenn rode her bike around her neighborhood. Use the graph to describe Jenn s bike ride. (Lesson 6.) Speed (mi/h) Time (h) Unit 187

124 Unit Performance Tasks 1. CAREERS IN MATH Cost Estimator To make MP3 plaers, a cost estimator determined it costs a compan $1500 per week for overhead and $5 for each MP3 plaer made. a. Define a variable to represent the number of plaers made. Then write an equation to represent the compan s total cost c. b. ne week, the compan spends $560 making MP3 plaers. How man plaers were made that week? Show our work. c. If the compan sells MP3 plaers for $10, how much profit would it make if it sold 80 plaers in one week? Eplain how ou found our answer.. A train from Portland, regon, to Los Angeles, California, travels at an average speed of 60 miles per hour and covers a distance of 963 miles. Susanna is taking the train from Portland to Los Angeles to see her aunt. She needs to arrive at her aunt s house b 8 p.m. It takes 30 minutes to get from the train station to her aunt s house. a. B what time does the train need to leave Portland for Susanna to arrive b 8 p.m.? Eplain how ou got our answer. As part of our eplanation, write a function that ou used in our work. b. Susanna does not want to leave Portland later than 10 p.m. or earlier than 6 a.m. Does the train in part a meet her requirements? If not, give a new departure time that would allow her to still get to her aunt s house on time, and find the arrival time of that train. 188 Unit

125 UNIT MIXED REVIEW PARCC Assessment Readiness Personal Math Trainer nline Assessment and Intervention Selected Response 1. Rickie earns $7 an hour babsitting. Which table represents this proportional relationship? 5. The graph of which equation is shown below? 5 A Hours 6 8 Earnings ($) 8 56 B Hours Earnings ($) 8 35 C Hours 3 Earnings ($) D Hours 3 Earnings ($) 1 1. Which of the relationships below is a function? A (6, 3), (5, ), (6, 8), (0, 7) B (8, ), (1, 7), ( 1, ), (1, 9) C (, 3), (3, 0), ( 1, 3), (, 7) D (7, 1), (0, 0), (6, ), (0, ) 3. Which set best describes the numbers used on the scale for a standard thermometer? A whole numbers B rational numbers C real numbers D integers. Which term refers to slope? A rate of change B equation C -intercept D coordinate A = + 3 B = 0.75 C = + 3 D = Which equation represents a nonproportional relationship? A = 5 B = 5 C = D = 1_ 5 7. Which number is written in standard notation? A B C D Unit 189

126 8. Which term does not correctl describe the relationship shown in the table? B linear C proportional D nonproportional 9. As part of a science eperiment, Greta measured the amount of water flowing from Container A to Container B. Container B had half a gallon of water in it to start the eperiment. Greta found that the water was flowing at a rate of two gallons per hour. Which equation represents the amount of water in Container B? A = B = 0.5 C = D = Carl and Jeannine both work at appliance stores. Carl earns a weekl salar of $600 plus $0 for each appliance he sells. The equation p = 50n represents the amount of mone Jeannine earns in a week, p ($), as a function of the number of appliances she sells, n. Which of the following statements is true? A Carl has a greater salar and a greater rate per appliance sold. B Jeannine has a greater salar and a greater rate per appliance sold. C Carl will earn more than Jeannine if the each sell 10 appliances in a given week. D Both Carl and Jeannine earn the same amount if the each sell 5 appliances in a given week. Mini-Task 11. The table below represents a linear relationship A function a. Find the slope for this relationship. b. Find the -intercept. Eplain how ou found it. c. Write an equation in slope-intercept form that represents this relationship. Hot Tip! Estimate our answer before solving the problem. Use our estimate to check the reasonableness of our answer. 1. Jac has a choice of cell phone plans. Plan A is to pa $60 for the phone and then pa $70 per month for service. Plan B is to get the phone for free and pa $8 per month for service. a. Write an equation to represent the total cost, c, of Plan A for m months. b. Write an equation to represent the total cost, c, of Plan B for m months. c. If Jac plans to keep the phone for months, which plan is cheaper? Eplain. 190 Unit