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NameClassDate6.1 Independent PracticePersonalMath Trainer8.F.1my.hrw.comOnline Practiceand HelpDetermine whether each relationship represented by the orderedpairs is a function. Explain.8. (2, 2), (3, 1), (5, 7), (8, 0), (9, 1)9. (0, 4), (5, 1), (2, 8), (6, 3), (5, 9)10. Draw Conclusions Joaquin receives 0.40 per pound for 1 to 99 poundsof aluminum cans he recycles. He receives 0.50 per pound if he recyclesmore than 100 pounds. Is the amount of money Joaquin receives afunction of the weight of the cans he recycles? Explain your reasoning.11. A biologist tracked the growth of a strain of bacteria, as shown inthe graph.Bacteria B800Number ofBacteriaa. Explain why the relationship represented by the graph is afunction.y600400200xO24 Houghton Mifflin Harcourt Publishing CompanyTime (h)6b. What If? Suppose there was the same number of bacteria for twoconsecutive hours. Would the graph still represent a function?Explain.y12. Multiple Representations Give an example of a function in everydaylife, and represent it as a graph, a table, and a set of ordered pairs.Describe how you know it is a function.xyxOLesson 6.1161

13. Is the relationship represented by the graph a function? Justify yourreasoning. Use the words “input” and “output” in your explanation, andconnect them to the context represented by the graph.Cost of Cheesey1210Price ( )The graph shows the relationship between the weights of six wedgesof cheese and the price of each wedge.864214. Analyze Relationships Suppose the weights and prices of additionalwedges of cheese were plotted on the graph. Might that change youranswer to question 13? Explain your reasoning.FOCUS ON HIGHER ORDER THINKINGOx0.4 0.8 1.2 1.6 2.0Weight (lb)Work Area16. Communicate Mathematical Ideas An onion farmer is hiring workersto help harvest the onions. He knows that the number of days it will taketo harvest the onions is a function of the number of workers he hires.Explain the use of the word “function” in this context.162Unit 2 Houghton Mifflin Harcourt Publishing Company Image Credits: Brand X Pictures/Getty Images15. Justify Reasoning A mapping diagram represents a relationship thatcontains three different input values and four different output values.Is the relationship a function? Explain your reasoning.

LESSON8.F.36.2 Describing Functions?Interpret the equation y mx b as defining a linearfunction, whose graph is astraight line; give examples offunctions that are not linear.Also 8.F.1.1ESSENTIAL QUESTIONHow can you tell if a function is linear are not?8.F.1EXPLORE ACTIVITYInvestigating a ConstantRate of ChangeThe U.S. Department of Agriculture defines heavy rain as rain thatfalls at a rate of 1.5 centimeters per hour.A The table shows the total amount of rain that falls in variousamounts of time during a heavy rain. Complete the table.Time (h)01Total Amountof Rain (cm)01.52345B Plot the ordered pairs from the table on the coordinate planeat the right.D Plot the point corresponding to 3.5 hours of heavy rain.E What do you notice about all of the points you plotted?Total Amountof Rain (cm)C How much rain falls in 3.5 hours?Heavy Rainfally108642 Houghton Mifflin Harcourt Publishing CompanyOx1234Time (h)5F Is the total amount of rain that falls a function of the numberof hours that rain has been falling? Why or why not?Reflect1.Suppose you continued to plot points for times between those inthe table, such as 1.2 hours or 4.5 hours. What can you say about thelocations of these points?Lesson 6.2163

Graphing Linear FunctionsThe relationship you investigated in the previous activity can be represented bythe equation y 1.5x, where x is the time and y is the total amount of rain. Thegraph of the relationship is a line, so the equation is a linear equation. Sincethere is exactly one value of y for each value of x, the relationship is a function.It is a linear function because its graph is a nonvertical line.Math On the Spotmy.hrw.comEXAMPLE 18.F.3The temperature at dawn was 8 F and increased steadily 2 F every hour.The equation y 2x 8 gives the temperature y after x hours. State whetherthe relationship between the time and the temperature is proportional ornonproportional. Then graph the function.STEP 1Carrie said that for a functionto be a linear function, therelationship it representsmust be proportional. DoSTEP 2you agree or disagree?Explain.Compare the equation with the general linear equation y mx b.y 2x 8 is in the form y mx b, with m 2 and b 8.Therefore, the equation is a linear equation. Since b 0, therelationship is nonproportional.Choose several values forthe input x. Substitutethese values for x in theequation to find theoutput y.0246STEP 32x 8xy(x, y)8(0, 8)12(2, 12)16(4, 16)20(6, 20)2(0) 82(2) 82(4) 82(6) 8Graph the ordered pairs. Then draw aline through the points to representthe solutions of the function.yTemperatures20161284xO4yYOUR TURN102. State whether the relationshipbetween x and y in y 0.5x isproportional or nonproportional.Then graph the function.x-10OPersonalMath TrainerOnline Practiceand Helpmy.hrw.com164Unit 28 12 16 20Time (h)-1010 Houghton Mifflin Harcourt Publishing CompanyMathematical PracticesTemperature ( F)Math Talk

Determining Whether aFunction is LinearThe linear equation in Example 1 has the form y mx b,where m and b are real numbers. Every equation in theform y mx b is a linear equation. The linear equationsrepresent linear functions. Equations that cannot be writtenin this form are not linear equations, and therefore are not linear functions.EXAMPL 2EXAMPLEMath On the Spotmy.hrw.com8.F.3A square tile has a side length of x inches. The equation y x2 gives thearea of the tile in square inches. Determine whether the relationshipbetween x and y is linear and, if so, if it is proportional.STEP 1STEP 2STEP 3STEP 4Choose several values for theinput x. Substitute these valuesfor x in the equation to find theoutput y.xx2y(x, y)1121(1, 1)2224(2, 4)Graph the ordered pairs.3329(3, 9)4416(4, 16)2Identify the shape of the graph. The pointssuggest a curve, not a line. Draw a curvethrough the points to represent thesolutions of the function.Describe the relationship between x and y.The graph is not a line so the relationshipis not linear.my.hrw.comMath TalkMathematical PracticesHow can you use thenumbers in the table todecide whether or not therelationship betweenx and y is linear?y20161284xO1Only a linear relationship can beproportional, so the relationship is not proportional.2345YOUR TURN3. A yogurt machine makes 32 gallon of yogurtevery minute. The total amount y that themachine makes in x minutes is given bythe equation y 32 x . Determine whether therelationship between x and y is linear and, ifso, if it is proportional.Time (min), xAmount (gal), y0394yMaking Yogurt10Amount (gal) Houghton Mifflin Harcourt Publishing CompanyAnimatedMath8642Ox2468 10Time (min)PersonalMath TrainerOnline Practiceand Helpmy.hrw.comLesson 6.2165

Guided PracticePlot the ordered pairs from the table. Then graph the function representedby the ordered pairs and tell whether the function is linear or nonlinear.(Examples 1 and 2)1. y 5 - 2x2. y 2 - x2-1Input, x135Output, yInput, x-2-12y866442x2x-41Output, yy-60-2O24-6-46-2O246-2-2-4-4-6-6-8Explain whether each equation is a linear equation. (Example 2)?4. y 1 – xESSENTIAL QUESTION CHECK-IN5. Explain how you can use a table of values, an equation, and a graph todetermine whether a function represents a proportional relationship.166Unit 2 Houghton Mifflin Harcourt Publishing Company3. y x2 – 1

NameClassDate6.2 Independent PracticePersonalMath Trainer8.F.1, 8.F.3my.hrw.com6. State whether the relationship between x and y in y 4x – 5 isproportional or nonproportional. Then graph the function.7. The Fortaleza telescope in Brazil is a radio telescope. Its shapecan be approximated with the equation y 0.013x2. Is therelationship between x and y linear? Is it proportional? Explain.Online Practiceand Helpy10xO-1010-108. Kiley spent 20 on rides and snacks at the state fair. If x is theamount she spent on rides, and y is the amount she spent on snacks, thetotal amount she spent can be represented by the equation x y 20.Is the relationship between x and y linear? Is it proportional? Explain.1359Total cost ( ), y60180300540a. Use the data to draw a graph. Is the relationship between xand y linear? Explain.y600Total cost ( )Number of uniforms, xDrill Team Uniforms480360240120xO2468 10Number of uniformsb. Use your graph to predict the cost of purchasing 12 uniforms.Marta’s Feedingsy10. Marta, a whale calf in an aquarium, is fed a special milk formula.Her handler uses a graph to track the number of gallons offormula left in the 20-gallon container after each feeding every4 hours. Is the relationship between x and y linear? Explain.Total amount offormula (gal) Houghton Mifflin Harcourt Publishing Company9. Represent Real-World Problems The drill team is buying newuniforms. The table shows y, the total cost in dollars, and x, thenumber of uniforms purchased.20161284Ox48 12 16 20Time (h)Lesson 6.2167

11. Critique Reasoning A student claims that the equation y 7 isnot a linear equation because it does not have the form y mx b.Do you agree or disagree? Why?12. Make a Prediction Let x represent the number of hours you reada book and y represent the total number of pages you have read.You have already read 70 pages and can read 30 pages per hour. Writean equation relating x hours and y pages you read. Then predict the totalnumber of pages you will have read after another 3 hours.FOCUS ON HIGHER ORDER THINKINGWork Area14. Communicate Mathematical Ideas Give two different examples ofgeometry formulas that represent linear functions and two examplesthat do not.15. Justify Reasoning Show that the equation y 3 3(2x 1) is linearand that it represents a proportional relationship between x and y.168Unit 2 Houghton Mifflin Harcourt Publishing Company Image Credits: Photodisc/Getty Images13. Draw Conclusions Rebecca draws a graph of a real-world relationshipthat turns out to be a set of unconnected points. Can the relationship belinear? Can it be proportional? Explain your reasoning.

LESSON6.3 Comparing Functions8.F.2Compare properties of twofunctions each represented ina different way (algebraically,graphically, numericallyin tables, or by verbaldescriptions). Also 8.EE.5,8.F.4?ESSENTIAL QUESTIONHow can you use tables, graphs, and equations tocompare functions?Comparing a Table and an EquationTo compare a function written as an equation and another functionrepresented by a table, find the equation for the function in the table.Math On the SpotEXAMPL 1EXAMPLE8.F.2, 8.F.4my.hrw.comJosh and Maggie buy MP3 files from different music services.The monthly cost, y dollars, for x songs is linear. The cost of Josh’sservice is y 0.50x 10. The cost of Maggie’s service is shown below. Houghton Mifflin Harcourt Publishing Company Image Credits: Houghton Mifflin HarcourtMonthly Cost of MP3s at Maggie’s Music ServiceSongs, x510152025Cost ( ), y4.959.9014.8519.8024.75A Write an equation to represent the monthly cost of Maggie’sservice.STEP 1Choose any two ordered pairs from the table to find the slope.The points (5, 4.95) andy2 - y19.90- 4.954.95m 0.99x2 - x1(10, 9.90) were used.510 - 5STEP 2Find the y-intercept. Use the slope and any point.y mx b4.95 0.99 · 5 bSlope-intercept form.Substitute for y, m, and x.0 bSTEP 3Substitute the slope and y-intercept.y 0.99x 0 or y 0.99xMath TalkMathematical PracticesDescribe each service’scost in words using themeanings of the slopesand y-intercepts.Substitute 0.99 for m and 0 for y.B Which service is cheaper when 30 songs are downloaded?Josh’s service:Maggie’s service:y 0.50 30 10y 0.99 30y 25y 29.7Josh’s service is cheaper.Lesson 6.3169

YOUR TURNPersonalMath TrainerOnline Practiceand Helpmy.hrw.com1. Quentin is choosing between buying books at the bookstore or buyingonline versions of the books for his tablet. The cost, y dollars, of orderingbooks online for x books is y 6.95x 1.50. The cost of buying the booksat the bookstore is shown in the table. Which method of buying books ismore expensive if Quentin wants to buy 6 books?Cost of Books at the BookstoreBooks, x12345Cost ( ), y7.5015.0022.5030.0037.508.F.2, 8.EE.5EXPLORE ACTIVITY 1Comparing a Table and a Graphy180WordsThe table and graph show how many words Morgan and Briantyped correctly on a typing test. For both students, the relationshipbetween words typed correctly and time is linear.Brian’s Typing Test120Morgan’s Typing TestTime (min)246810Words30609012015060xO2468 10Time (min)A Find Morgan’s unit rate.C Which student types more correct words per minute?Reflect2.170Katie types 17 correct words per minute. Explain how a graph of Katie’stest results would compare to Morgan’s and Brian’s.Unit 2 Houghton Mifflin Harcourt Publishing CompanyB Find Brian’s unit rate.

8.F.2, 8.F.4Comparing a Graph anda DescriptionJamal wants to buy a new game system that costs 200.He does not have enough money to buy it today, so hecompares layaway plans at different stores.The plan at Store A is shown on the graph.Layaway PlanyBalance owed ( )EXPLORE ACTIVITY 21601208040xStore B requires an initial payment of 60 and weeklypayments of 20 until the balance is paid in full.O2468Number of weeks10A Write an equation in slope-intercept form for Store A’s layaway plan.Let x represent number of weeks and y represent balance owed.B Write an equation in slope-intercept form for Store B’s layaway plan.Let x represent number of weeks and y represent balance owed.C Sketch a graph of the plan at Store B on the same grid as Store A. Houghton Mifflin Harcourt Publishing CompanyD How can you use the graphs to tell which plan requires the greaterdown payment? How can you use the equations?E How can you use the graphs to tell which plan requires the greaterweekly payment? How can you use the equations?F Which plan allows Jamal to pay for the game system faster? Explain.Lesson 6.3171

Guided PracticeDoctors have two methods of calculating maximum heart rate.With the first method, maximum heart rate, y, in beats per minuteis y 220 - x, where x is the person’s age. Maximum heart ratewith the second method is shown in the table. (Example 1)Age, x2030405060Heart rate (bpm), y1941871801731661. Which method gives the greater maximum heart rate for a 70-year-old?2. Are heart rate and age proportional or nonproportional for each method?y80Cost ( )Aisha runs a tutoring business. With Plan 1, students may choose topay 15 per hour. With Plan 2, they may follow the plan shown onthe graph. (Explore Activity 1 and 2)Tutoring Fees3. Describe the plan shown on the graph.604020Ox2468 10Time (h)4. Sketch a graph showing the 15 per hour option.6. Which plan is cheaper for 10 hours of tutoring?7. Are cost and time proportional or nonproportional for each plan?ESSENTIAL QUESTION CHECK-IN8. When using tables, graphs, and equations to compare functions, why doyou find the equations for tables and graphs?172Unit 2 Houghton Mifflin Harcourt Publishing Company Image Credits: PhotoAlto/Getty Images5. What does the intersection of the two graphs mean?

NameClassDate6.3 Independent PracticePersonalMath Trainer8.EE.5, 8.F.2, 8.F.4my.hrw.comOnline Practiceand HelpThe table and graph show the miles driven and gas used for two scooters.Scooter ADistance (mi), xGas used (gal), y1502300445066008750Gas used (gal)y54321O10Scooter Bx90270450Distance (mi)9. Which scooter uses fewer gallons of gas when 1350 miles are driven?10. Are gas used and miles proportional or nonproportional for eachscooter?A cell phone company offers two texting plans to its customers. Themonthly cost, y dollars, of one plan is y 0.10x 5, where x is the numberof texts. The cost of the other plan is shown in the table.Number of texts, x100200300400500Cost ( ), y2025303540 Houghton Mifflin Harcourt Publishing Company11. Which plan is cheaper for under 200 texts?12. The graph of the first plan does not pass through the origin. What doesthis indicate?13. Brianna wants to buy a digital camera for a photography class. One storeoffers the camera for 50 down and a payment plan of 20 per month. Thepayment plan for a second store is described by y 15x 80, where y isthe total cost in dollars and x is the number of months. Which camera ischeaper when the camera is paid off in 12 months? Explain.Lesson 6.3173

Number of cars, xAmount earned ( ), y2104206308401050Amount earned ( )14. The French club and soccer team are washing cars to earn money. Theamount earned, y dollars, for washing x cars is a linear function. Whichgroup makes the most money per car? Explain.Soccer TeamyFrench Club724824xO2468 10Number of carsFOCUS ON HIGHER ORDER THINKINGWork Area15. Draw Conclusions Gym A charges 60 a month plus 5 per visit. Themonthly cost at Gym B is represented by y 5x 40, where x is thenumber of visits per month. What conclusion can you draw about themonthly costs of the gyms?17. Analyze Relationships The equations of two functions arey -21x 9 and y -24x 8. Which function is changing morequickly? Explain.174Unit 2 Houghton Mifflin Harcourt Publishing Company16. Justify Reasoning Why will the value of y for the function y 5x 1always be greater than that for the function y 4x 2 when x 1?

LESSON6.4 Analyzing Graphs?ESSENTIAL QUESTION8.F.5Describe qualitatively thefunctional relationshipbetween two quantitiesby analyzing a graph.Sketch a graph that exhibitsthe qualitative features ofa function that has beendescribed verbally.How can you describe a relationship given a graph and sketch agraph given a description?EXPLORE ACTIVITY 18.F.5Interpreting GraphsA roller coaster park is open from May to October each year.The graph shows the n

Personal Math Trainer Interactively explore key concepts to see how math works. Animated Math Go digital with your write-in student edition, accessible on any device. my.hrw.com Scan with your smart phone to jump directly to the online edition, video tutor, and more. Math On the Spot MODU