Nonproportional Relationships MODULE 17? ESSENTIAL QUESTION How can ou use nonproportional relationships to solve real-world problems? LESSON 17.1 Representing Linear Nonproportional Relationships 8.F.3 LESSON 17. Determining Slope and -intercept 8.EE.6, 8.F. LESSON 17.3 Graphing Linear Nonproportional Relationships using Slope and -intercept 8.F.3, 8.F. LESSON 17. Proportional and Nonproportional Situations Houghton Mifflin Harcourt Publishing Compan Image Credits: viapp/shutterstock m.hrw.com 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. 8.F., 8.F.3, 8.F. m.hrw.com m.hrw.com Math On the Spot Animated Math 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. 53
Are YOU Read? Complete these eercises to review skills ou will need for this module. Integer Operations m.hrw.com EXAMPLE 7 ( ) = 7 + 7 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. 1. 3 ( 5). 5 3. 6 10. 5 ( 3) 5. 8 ( 8) 6. 9 5 7. 3 9 8. 0 ( 6) 9. 1 - (-9) 10. -6 - (-) 11. -7-10 1. 5-1 Graph Ordered Pairs (First Quadrant) EXAMPLE 8 6 O A 6 8 To graph a point at (6, ), start at the origin. Move 6 units right. Then move units up. Graph point A(6, ). Graph each point on the coordinate grid. 13. B (0, 5) 1. C (8, 0) 15. D (5, 7) 8 6 Houghton Mifflin Harcourt Publishing Compan 16. E (, 3) O 6 8 5 Unit 8
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 -coordinate of the point where a graph of a line intersects the -ais 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 Houghton Mifflin Harcourt Publishing Compan 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 17 55
GETTING READY FOR Nonproportional Relationships 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.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. 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. 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. EXAMPLE 8.F.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. - - 3 - - 1 8.F.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. 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. What It Means to You You will distinguish linear relationships from nonlinear relationships b looking at graphs. EXAMPLE 8.F.3 Which relationship is linear and which is nonlinear? P = s A = s P - - - O s - - - A O s Houghton Mifflin Harcourt Publishing Compan Visit m.hrw.com to see all CA Common Core Standards eplained. P = s is linear because its graph is a line. A = s is not linear because its graph is not a line. m.hrw.com 56 Unit 8
? LESSON 17.1 ESSENTIAL QUESTION Representing Linear Nonproportional Relationships 8.F.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. 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 On the Spot m.hrw.com EXAMPLE 1 Prep for 8.F.3 The equation = 3 + gives the total charge for one person,, renting a pair of shoes and 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. (number of games) 1 3 (total cost in dollars) 5 8 11 1 Houghton Mifflin Harcourt Publishing Compan YOUR TURN Substitute 1 for : = 3(1) + = 5. 1. 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) m.hrw.com Lesson 17.1 57
EXPLORE ACTIVITY 8.F.3 Eamining Linear Relationships Recall Math On that the Spot a proportional relationship is a relationship between two quantities in which m.hrw.com the ratio of one quantit to the other quantit is constant. The graph of a proportional relationship is a line through the origin. When ratios between quantities are not constant, a relationship ma be linear but not proportional and the graph does not pass through the origin. 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) 0 6 8 (total cost in dollars) 0 STEP STEP 3 STEP Plot ordered pairs from the information in the table. Describe the shape of the graph. Find the rate of change between each point and the net. Is the rate constant? Calculate rates of change for the values in the table. Eplain wh the relationship between number of tickets and total cost is not proportional. Cost ($) Theme Park Costs 0 3 16 8 O 6 8 10 Number of tickets Reflect. Analze Relationships Would it be possible to add more points to the graph from = 0 to = 10? Would it make sense to connect the points with a line? Eplain. Houghton Mifflin Harcourt Publishing Compan m.hrw.com 58 Unit 8
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 On the Spot m.hrw.com EXAMPLE 8.F.3 The diameter of a Douglas fir tree is currentl 10 inches when measured at chest height. Over the net 50 ears, the diameter is epected to increase b an average growth rate of _ 5 inch per ear. The equation = _ 5 + 10 gives, the diameter of the tree in inches, after ears. Draw a graph of the equation. Describe the relationship. STEP 1 STEP Make a table. Choose several values for that make sense in contet. To make calculations easier, choose multiples of 10. (ears) 0 10 0 30 50 (diameter in inches) 10 1 18 30 Plot ordered pairs from the information in the table. Then draw a line connecting the points to represent all the possible solutions. Math Talk Mathematical Practices Wh can a line be drawn connecting the points in this eample, but not in the preceding Eplore Activit? Houghton Mifflin Harcourt Publishing Compan Image Credits: Don Mason/Corbis STEP 3 Diameter (in.) 0 3 16 YOUR TURN The relationship is linear but nonproportional. The graph is a line but it does not go through the origin. 8 O Fir Tree Growth 10 0 30 0 50 Time (r) 3. Make a table and graph the solutions of the equation = + 1. -1 0 1 - - O - - m.hrw.com Lesson 17.1 59
Guided Practice Make a table of values for each equation. (Eample 1) 1. = + 5 - -1 0 1. = 3 _ 8-5 -8 0 8 Eplain wh each relationship is not proportional. (Eplore Activit) 3. 0 6 8 3 7 11 15 19. First calculate _ for the values in the table. - - O - - Complete the table for the equation. Then use the table to graph the equation. (Eample ) 5. = - 1 - -1 0 1 - - O -? ESSENTIAL QUESTION CHECK-IN 6. How can ou choose values for when making a table of values representing a real world situation? - Houghton Mifflin Harcourt Publishing Compan 530 Unit 8
Name Class Date 17.1 Independent Practice 8.F.3 m.hrw.com State whether the graph of each linear relationship is a solid line or a set of unconnected points. Eplain our reasoning. 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. 56 Houghton Mifflin Harcourt Publishing Compan d. Does it make sense to connect the points on the graph with a solid line? Eplain. 8 0 3 16 8 O 6 8 10 1 1 Lesson 17.1 531
10. Analze Relationships A proportional relationship is a linear relationship because the rate of change is constant (and equal to the constant of proportionalit). What is required of a proportional 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. FOCUS ON HIGHER ORDER 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. 1 3 5 90 150 10 70 330 13. Make a Conjecture Two parallel lines are graphed on a coordinate plane. How man of the lines could represent proportional relationships? Eplain. Houghton Mifflin Harcourt Publishing Compan 53 Unit 8
? LESSON 17. ESSENTIAL QUESTION Determining Slope and -intercept 8.EE.6 Use similar triangles to eplain wh the slope m is the same between an two distinct points on a non-vertical line in the coordinate plane; 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? EXPLORE 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? - - O - (0, ) 6 8 Houghton Mifflin Harcourt Publishing Compan STEP 3 STEP 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 17. 533
Math On the Spot m.hrw.com 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 = 630-80 0-10 = 150 10 = 15 change in income change in phones sold = 780-630 30-0 = 150 10 = 15 change in income change in phones sold = 930-780 0-30 = 150 10 = 15 Number of Phones Sold Weekl Income ($) 10 $80 0 $630 30 $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 $15 commission for each phone sold. Find the initial value when the number of phones sold is 0. -10-10 Number of phones sold 0 10 0 Weekl income ($) 330 80 630 Work backward from = 10 to = 0 to find the initial value. YOUR TURN -150-150 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. Houghton Mifflin Harcourt Publishing Compan m.hrw.com 1. 6 8. 3 5 1 3 8 15 9 53 Unit 8
EXPLORE 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. Math On the Spot m.hrw.com 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 = - - 0 STEP 3 Solve the equation for. m = - b m = - b Simplif the denominator. Multipl both sides of the equation b. m = - b m + = - b + Add to both sides of the equation. m + = Houghton Mifflin Harcourt Publishing Compan = m + Write the equation with on the left side. = m + b is the Slope-Intercept form of an equation. Reflect 3. Critical Thinking Write the equation of a line with slope m that passes through the origin. Eplain our reasoning. m.hrw.com Lesson 17. 535
Guided Practice Find the slope and -intercept of the line in each graph. (Eplore Activit 1) 1.. 0 - - O (0, 1) - - 10 (3, 0) O - - (, -3) -10-0 (0, -15) slope m = -intercept b = slope m = -intercept b = 3.. 1 6 - - O - - O - -6 - -1 slope m = -intercept b = slope m = -intercept b = Find the slope and -intercept of the line represented b each table. (Eample 1) 5. 0 6 8 6. 0 5 10 15 0 1 7 13 19 5 10 10 100 80 60? slope m = -intercept b = ESSENTIAL QUESTION CHECK-IN 7. How can ou determine the slope and the -intercept of a line from a graph? slope m = -intercept b = Houghton Mifflin Harcourt Publishing Compan 536 Unit 8
Name Class Date 17. Independent Practice 8.EE.6, 8.F. m.hrw.com 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. Rooms cleaned 1 3 Cost ($) 15 175 5 75 9. 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 $53 10. Multi-Step Ramond s parents will pa for him to take sailboard lessons during the summer. He can take half-hour group lessons or half-hour private lessons. The relationship between cost and number of lessons is linear. Lessons 1 3 Group ($) 55 85 115 15 Private ($) 75 15 175 5 a. Find the rate of change and the initial value for the group lessons. Houghton Mifflin Harcourt Publishing Compan 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 17. 537
Vocabular Eplain wh each relationship is not linear. 11. 1 3.5 6.5 8.5 11.5 1. 3 5 7 9 10 16 110 9 13. Communicate Mathematical Ideas Describe the procedure ou performed to derive the slope-intercept form of a linear equation. FOCUS ON HIGHER ORDER 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 750 600 50 300 150 O 10 0 30 0 50 60 70 80 90 Cars parked Houghton Mifflin Harcourt Publishing Compan 538 Unit 8
? LESSON 17.3 ESSENTIAL QUESTION Graphing Linear Nonproportional Relationships Using Slope and -intercept 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 or from two (, ) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. Also 8.F.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 On the Spot m.hrw.com EXAMPLE 1 8.F.3 A Graph = _ 3-1. STEP 1 STEP STEP 3 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. +3 + (3, 1) - - O (0, -1) - - Animated Math m.hrw.com Math Talk Mathematical Practices Is a line with a positive slope alwas steeper than a line with a negative slope? Eplain. Houghton Mifflin Harcourt Publishing Compan B Graph = - 5 _ + 3. STEP 1 STEP 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). 1 - (-, 8) +5 (0, 3) -8 - -5 8 + (, -) - STEP 3 Draw a line through the points. Note that the line passes through all three points: (-, 8), (0, 3), and (, -). Lesson 17.3 539
Reflect 1. Draw Conclusions How can ou use the slope of a line to predict the wa the line will be slanted? Eplain. YOUR TURN m.hrw.com Graph each equation.. = 1_ + 1 3. = -3 + - - O - - O - - 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 On the Spot m.hrw.com EXAMPLE 8.F. Ken has a weekl goal of burning 00 calories b taking brisk walks. The equation = -300 + 00 represents the number of calories Ken has left to burn after hours of walking which burns 300 calories per hour. A Graph the equation = -300 + 00. TIME HOURS 0:30 1:30 CALORIES 150 STEP 1 STEP STEP 3 STEP Write the slope as a fraction. m = -300 1 = -600 = -900 3 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. Using the slope as -900 helps in 3 drawing a more accurate graph. Calories remaining 3000 00 1800 100 600 6 8 Time (h) Houghton Mifflin Harcourt Publishing Compan 50 Unit 8
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 3000 00 1800 100 600 6 8 Time (h) Ken will reach his goal after 8 hours of brisk walking. YOUR 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 = -00 + 00. 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 3000 00 1800 100 600 6 8 10 1 1 Time (h) 6. Will Ken have to eercise more or less to meet his goal? Eplain. Houghton Mifflin Harcourt Publishing Compan 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? m.hrw.com Lesson 17.3 51
Guided Practice Graph each equation using the slope and the -intercept. (Eample 1) 1. = 1_ - 3 slope = -intercept =. = -3 + slope = -intercept = - - O - - O - - - - 3. A friend gives ou two baseball cards for our birthda. Afterward, ou begin collecting them. You bu cards each week. The equation = + describes the number of cards,, ou have after weeks. (Eample ) 0 16 a. Find and interpret the slope and the -intercept of the line that represents this situation. Graph = +. Include ais labels. 1 8 6 8 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.? ESSENTIAL QUESTION CHECK-IN. Wh might someone choose to use the -intercept and the slope to graph a line? Houghton Mifflin Harcourt Publishing Compan 5 Unit 8
Name Class Date 17.3 Independent Practice 8.F.3, 8.F. m.hrw.com 5. Science A spring stretches in relation to the weight hanging from it according to the equation = 0.75 + 0.5 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. 1 1 3 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 Houghton Mifflin Harcourt Publishing Compan 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 = -1 7. slope = -1, -intercept = 8 8. slope = 0., -intercept = 0.3 9. slope = 1.5, -intercept = -3 10. slope = - 1_, -intercept = 11. slope = _, -intercept = -5 3 1. A music school charges a registration fee in addition to a fee per lesson. Music lessons last 0.5 hour. The equation = 0 + 30 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 17.3 53
13. A public pool charges a membership fee and a fee for each visit. The equation = 3 + 50 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 = 3 + 50. Include ais labels. Then interpret the slope and -intercept. 50 00 150 100 50 50 150 50 c. How man visits to the pool can a member get for $00? FOCUS ON HIGHER ORDER THINKING 1. Eplain the Error A student sas that the slope of the line for the equation = 0-15 is 0 and the -intercept is 15. Find and correct the error. Work Area 15. Critical Thinking Suppose ou know the slope of a linear relationship and a point that its graph passes through. Can ou graph the line even if the point provided does not represent the -intercept? Eplain. 16. Make a Conjecture Graph the lines = 3, = 3-3, and = 3 + 3. What do ou notice about the lines? Make a conjecture based on our observation. - - O Houghton Mifflin Harcourt Publishing Compan - - 5 Unit 8
? LESSON 17. ESSENTIAL QUESTION Proportional and Nonproportional Situations How can ou distinguish between proportional and nonproportional situations? Distinguish Between Proportional and Nonproportional Situations Using a Graph If a relationship is nonlinear, it is nonproportional. If it is linear, it ma be either proportional or nonproportional. When the graph of the linear relationship contains the origin, the relationship is proportional. 8.F. Compare properties of two functions each represented in a different wa (algebraicall, graphicall, numericall in tables, or b verbal descriptions). Also 8.F.3, 8.F. Math On the Spot m.hrw.com EXAMPLE 1 The graph shows the sales ta charged based on the amount spent at a video game store in a particular cit. Does the graph show a linear relationship? Is the relationship proportional or nonproportional? The graph shows a linear proportional relationship because it is a line that contains the origin. Sales ta ($) 6.0.8 3.6. 1. O 8.F.3 0 0 60 80 100 Amount spent ($) Houghton Mifflin Harcourt Publishing Compan YOUR TURN Determine if each of the following graphs represents a proportional or nonproportional relationship. 10 8 6 1.. 0 16 1 8 Math Talk Mathematical Practices What do the slope and the -intercept of the graph represent in this situation? O 6 8 10 O 8 16 3 0 m.hrw.com Lesson 17. 55
Math On the Spot m.hrw.com 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? Animated Math m.hrw.com = a - 1 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 - 1.. What If? Suppose another equation represents Keith s age in months given his age in ears a. Is this relationship proportional? Eplain. YOUR TURN Determine if each of the following equations represents a proportional or nonproportional relationship. Houghton Mifflin Harcourt Publishing Compan 5. d = 65t 6. p = 0.1s + 000 m.hrw.com 7. n = 50-3p 8. 36 = 1d 56 Unit 8
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 On the Spot m.hrw.com A linear relationship represented b a table is a proportional relationship when the quotient of each pair of numbers is constant. Otherwise, the linear relationship is nonproportional. EXAMPLE 3 8.F. Houghton Mifflin Harcourt Publishing Compan 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 1,690 55 3,315 505 6,565 1,690 130 = 169 13 = 13 3,315 55 = 1 17 = 13 6,565 505 = 1,313 101 = 13 The ratio of pesos received to dollars traded is constant at 13 Meican pesos per U.S. dollar. This is a proportional relationship. YOUR TURN Determine if the linear relationship represented b each table is a proportional or nonproportional relationship. 9. 30 8 90 1 150 Simplif the ratios to compare the pesos received to the dollars traded. 10. 5 1 0 8 65 13 Animated Math m.hrw.com Math Talk Mathematical Practices How could ou confirm that the values in the table have a linear relationship? m.hrw.com Lesson 17. 57
Math On the Spot m.hrw.com 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. Math Talk Mathematical Practices How might graphing the equation for Arena A help ou to compare the situations? M Notes A B 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, $5, is greater than Arena B s, but there is no additional fee. Total cost ($) 500 00 300 00 100 O Arena B 0.5 1.0 1.5.0 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. 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 0 1 3 0 55 90 15 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. Houghton Mifflin Harcourt Publishing Compan 58 Unit 8
YOUR 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. m.hrw.com Guided Practice Determine if each relationship is a proportional or nonproportional situation. Eplain our reasoning. (Eample 1, Eample, Eample ) 1. 50. 30 0 30 18 0 1 10 O 6 8 10 6 O 6 1 18 30 Look at the origin. Houghton Mifflin Harcourt Publishing Compan 3. q = p + 1_ Compare the equation with = m + b.. v = 1 10 u Lesson 17. 59
The tables represent linear relationships. Determine if each relationship is a proportional or nonproportional situation. (Eample 3, Eample ) 5. 6. 3 1 9 36 6 8 1 8 58 10 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 1 0,000,000 16 5,000,000 0? ESSENTIAL QUESTION CHECK-IN 8. How are using graphs, equations, and tables similar when distinguishing between proportional and nonproportional situations? Houghton Mifflin Harcourt Publishing Compan 550 Unit 8
Name Class Date 17. Independent Practice 8.F., 8.F.3, 8.F. m.hrw.com 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) 0 16 1 8 O Cooler Weight 8 1 16 0 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 in feet above the first floor on the escalator, h, is given b h = 0.75t, where t is the time in seconds. Houghton Mifflin Harcourt Publishing Compan 1. Analze Relationships Compare and contrast the two graphs. Graph A Graph B 6 6 = 1_ 3 = _ - O 6 - O 6 - - - - Lesson 17. 551
13. Represent Real-World Problems Describe a real-world situation where the relationship is linear and nonproportional. FOCUS ON HIGHER ORDER 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 + 73.15 Table C Degrees Celsius kelvins 8 81.15 15 88.15 36 309.15 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? Houghton Mifflin Harcourt Publishing Compan 55 Unit 8
MODULE QUIZ Read 17.1 Representing Linear Nonproportional Relationships 1. Complete the table using the equation = 3 +. -1 0 1 3 m.hrw.com 17. Determining Slope and -intercept 5. Find the slope and -intercept of the line in the graph. -5 O 5 17.3 Graphing Linear Nonproportional Relationships 3. Graph the equation = - 3 using slope and -intercept. 5-5 O 5 17. Proportional and Nonproportional Situations. Does the table represent a proportional or a nonproportional linear relationship? 1 3 5 8 1 16 0 5. Does the graph in Eercise represent a proportional or a nonproportional linear relationship? Houghton Mifflin Harcourt Publishing Compan 6. Does the graph in Eercise 3 represent a proportional or a nonproportional relationship? ESSENTIAL QUESTION 7. How can ou identif a linear nonproportional relationship from a table, a graph, and an equation? Module 17 553
MODULE 17 MIXED REVIEW Assessment Readiness m.hrw.com 1. Consider each table. Does the table represent a proportional relationship? Select Yes or No for tables A C. A. - - 0-10 - 8 1 B. 8 1 16 0 3 6 9 1 15 C. -3-1 1 3 5-1 - 1 0 Yes Yes Yes No No No. The graph shows the relationship between the hours Leann works and the mone she earns. Choose True or False for each statement. A. Slope = 8. True False B. Unit rate = $8/h. True False C. Equation is = _ 8 True False Earnings ($) 16 8 O 6 Time (h) 3. There is an initial fee to join a gm, plus a monthl charge. The table represents the linear relationship between the total cost of joining the gm and the number of months a person is a member. What is the -intercept of the line, and what does it represent in this situation? Time (months), Total Cost ($), 1 15 155 3 185. The equation = 35-5 gives as the cost of tickets to a theme park with a $5 coupon. After the park raises ticket prices b $3, what is the new equation? How are the graphs of the two equations different? Houghton Mifflin Harcourt Publishing Compan 55 Unit 8