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Warm-Up?Proportional RelationshipsLessonQuestionLesson GoalsDetermine whether alinear function varies.Define directproportionality.W2KCompare linearfunctions usingUse a linearfunction torepresent aproportionalrelationship.and graphs.Words to KnowFill in this table as you work through the lesson. You may also use the glossary tohelp you.a function that can be written in the form y mx b, where mand b are real numbers; consists of a set of ordered pairs alllying on the same linea statement in which two ratios are set equal to one anotherthe value of the quotient of the dependent and independentvariables in a direct variation, or the product of the dependentand independent variables in an inverse variation Edgenuity, Inc.1

Warm-UpW2KProportional RelationshipsWords to Knowan expression that compares two or more numbersa relation in which the quotient of the dependent andindependent variables remains constantReview of RatiosA ratio compares two quantities, often with different units. 30 miles 4 black socks One laptop Edgenuity, Inc.30 miles 90 miles 1hour3 hourshour2 white socksevery 2 students24 black2 black 2 white 1 white1laptop50 laptops 2 students 100 students

InstructionProportional RelationshipsSlide2Proportions A proportion sets two ratiosto each other.xy1331266 2394121Direct VariationEXAMPLEA direct variation is a relation in which the quotient of theindependent variables remainsand.In the examples, the ratios stay the same after changes, so they are proportional. Edgenuity, Inc.Here the change is in the samedirection.Here one increases and onedecreases.y6 3 x2 1y10 15 x 2 33

InstructionProportional RelationshipsSlide5Recognizing a Linear RelationshipA recipe for banana bread calls for 1.5 cups of flour for every banana used.ratio k 1.51# of Bananas# of Cups ofFlour11.523yx1.5 1234.51.5461.557.51.5 1.5The constant of variation, k, is the number that relates how the variable y isdirectly proportional to the variable Edgenuity, Inc.4.

InstructionProportional RelationshipsSlide8Preparing to Graph the DataAn animal shelter requires twice as many kennels for cats as for dogs.Which values should be represented by x?Which by y?IndependentDependent x is the # of dog kennels.# of DogKennels# of CatKennelsvariable)( y is the # of cat kennels, determined bythe # of dog kennels.(variable)   21 243648510The # of cat kennels, y, varies directly with the number of dog kennels, x. Edgenuity, Inc.5

InstructionProportional RelationshipsSlide8Graph the RelationshipEXAMPLE# of CatKennels, y12243648510Graph the relationship from the table onthe coordinate plane.y10Cat Kennels# of DogKennels, x642x(0, 0)423Dog Kennels156proportional, that means the lines willFor linear functions that arepass right through the108, which is the point (0, 0).Equation for Direct VariationyThe equation for direct variation is6y kx,where k is the constant of variation.4Ratio y : x         k   m2:x2 Edgenuity, Inc.4662y kxy 2x

InstructionProportional RelationshipsSlide12Comparing Constant VariationEXAMPLETwo animal shelters hold a fund-raiser in which guests purchase a ticket to attendto help pay for costs of housing animals. The results are shown in the equationand graph where x represents the number of guests and y represents the totaldollar amount earned.Shelter 1x ticketsy kxy 4.5xk Amount Earned ( )y 20Shelter 2y1612 3128 42844 4112x43Guestsk 4For the same number of guests attending each shelter, which shelter wouldraise more money? Shelter, because it has a larger constant ofthan shelter 2. Edgenuity, Inc.75

InstructionProportional RelationshipsSlide14Determining the RelationshipEXAMPLExyyx133 31244 23553466 3 4 257To get a direct proportion, you’d have to have the7exact ratio,so you could have the same exact constant of variation. What we have is arelationship. Edgenuity, Inc.8

InstructionProportional RelationshipsSlide14Nonproportional Relationshipsxy1324354657y GraphGraphthe relationshipfrom thethe relationshipfromtablethe onthe coordinate plane.table on the coordinate plane.y10864x (0,2)not y kx 2 Edgenuity, Inc.x2 246810does not pass through (0, 0), (0, 0). Graphs of lines that areThis line does not pass through thenot2proportions, that are nonproportional, won’t go through the origin.9

Summary?LessonQuestionProportional RelationshipsHow can linear functions represent proportional relationships?Answer2Review: Key Concepts When two variables change at theproportional.rate, they are directlyy 2x The direct variation equation is y kx, where k is thevariation.ofk Linear functions that are directly proportional are shown with a graph thatgoes through the Edgenuity, Inc.10

SummaryProportional RelationshipsUse this space to write any questions or thoughts about this lesson. Edgenuity, Inc.11

# of Dog Kennels, x # of Cat Kennels, y 1 2 2 4 3 6 4 8 5 10 Graph the relationship from the table on the coordinate plane. For linear functions that are proportional, that means the lines will pass right through the , which is the point (0, 0). x y 12 3 45 6 2 4 6 8 10 Cat Kennels Dog Kennels (0, 0) Equation for Direct Variation x y 24 6 2 4 6