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NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations1
Study Guide: Slope and Linear Equations
Rates and Unit Rates Arateisaproportionalrelationshipbetweentwoquantities.Unitrateisaratewherethesecondquantityis1.Example:Paulinecanmow35squarefeetoflawnis2.5minutes.(thisisarate).Shecanmow14squarefeetperminute(thisisaunitrate).Wesetupproportionstofindunitratesandwritelinearequationsin2variables.Wesolvefory.Example:
𝑦𝑥 =
352.5
Setupproportion
2.5𝑦 = 35𝑥
Multiplythenumeratorsbythedenominatoroftheotherfraction.
𝑦 = 14𝑥 Solvefory.RatesaredirectlyproportionaliftheirgraphliesonastraightlineANDpassesthroughtheorigin.
Slope Slopeisameasureofsteepnessorslantofaline.Itisalsothemeasureoftherateofthechangeofaline.Slopecanbeexpressedasaunitrate:Howmanyverticalunitsdoesalinemoveforeveryonehorizontalunit?Theslopeformulastatesthat:
𝒎 =𝒑𝟐 − 𝒓𝟐𝒑𝟏 − 𝒓𝟏
= 𝒓𝒊𝒔𝒆𝒓𝒖𝒏 =
𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒚− 𝒗𝒂𝒍𝒖𝒆𝒔𝒅𝒊𝒇𝒇𝒆𝒓𝒆𝒏𝒄𝒆 𝒊𝒏 𝒙− 𝒗𝒂𝒍𝒖𝒆𝒔
WherethecoordinatesofpointPonalinearerepresentedby𝑃 = (𝑝!,𝑝!)andthecoordinatesofpointRonthelinearerepresentedby𝑅 = (𝑟!, 𝑟!).
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations2
Slope Triangles Wecanusetheconceptofslopetrianglestoprovetheslopeofalineisthesamebetweenanytwopointsontheline:
• ShowAAsimilarityoftwotriangles.• Knowthatcorrespondingsidesofsimilartrianglesare
proportional…andthismeansthattheslopeofalineisthesamefortheentiredistanceoftheline.
Ifyoucreateseveraldifferentrighttrianglesusingthepointsabove,youcanseethatseethatallofthetrianglesaresimilartrianglesusingtheconceptofAAsimilarity.Example:∆𝑃𝐿𝐾 ~∆𝑅𝑀𝐾becausetheyshareacommonangleANDtheyeachhaverightangles…hence,theymeettheAAcriterionforsimilarity.Thismeanstheproportionsofcorrespondingsidesofeachtriangleareequal.Example:Theproportionsofthecorrespondingsidesofthetrianglesmentionedaboveareequal…
𝑅𝑀𝑀𝐾
=𝑃𝐿𝐿𝐾
126 =
105
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NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations3
Slope (cont’d)
PositiveSlopeLeft-to-rightinclining
lines
NegativeslopeLeft-to-rightdeclining
lines
ZeroslopeHorizontalLines
Thegreatertheabsolutevalueoftheslope,thesteepertheline!!Horizontallineshavezeroslope,andarewrittenas𝑦 = 𝑐.Verticallineshaveanundefinedslopeandarewrittenas𝑥 = 𝑐.
Equations of Lines StandardFormofanequation𝒂𝒙 + 𝒃𝒚 = 𝒄where𝑎, 𝑏, and 𝑐areallintegersand𝑎 > 0Wecangraphalineknowingthestandardformofalinearequationbyidentifyingthex-interceptandy-intercept.SlopeInterceptformofanequation𝒚 = 𝒎𝒙 + 𝒃wheremistheslopeandbisthey-intercept.Wecangraphalineknowingtheslope-interceptformofalinearequationbyidentifyingthey-interceptandusingtheslopetographanotherpoint.Minimumrequirementsforfindingtheequationofaline:1.Slopeandonepointontheline(oftentheonepointwillbethey-intercept)2.Twopointsontheline.(thiscanbethex-interceptandy-intercept)Convertfromslope-interceptformtostandardform:
• Gatheraxandbytermsonleftsideoftheequation.• Makesurethata,b,andcareallintegersandaispositive.
Convertfromstandardformtoslope-interceptform:
• Solvefory.
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations4
Problem set Completetheproblemsbelowbeforetakingourtest.Ifyouhaveanyquestions,besuretoaskeitheratareviewsession.Aphonecompanyhastwoplansfordatathatareasfollows:PlanAincludesanupfrontservicechargeof$45permonth,andcharges$2.50foreachGBofdataused.PlanBincludesnoservicecharge,andcharges$7.50foreachGBofdata.1.Writeequationsthatrepresenteachcellphoneplan,whereyisthecostofthedata,andxistheamountofdata(inGB).Identifytheslopeandexplainwhatitmeans.PlanA:𝑦 = 2.5𝑥 + 45PlanB:𝑦 = 7.5𝑥2.Graphtheequationsonthecoordinateplane.Seecoordinateplane.3.WhenisitbetterforyoutogowithplanA?WhendoesitmakesensetogowithplanB?Ifyou’replanningtousemorethan9GBofdata,I’dgowithplanA,butlessthan9GBofdata,gowithplanB.thebreakevencostis$67.50.SophiaandColbyweregiventhesetwopoints(2,−5)and(−3, 5),andtoldtowritetheequationoftheline.Sophiathoughttheequationwas𝑦 = −2𝑥 − 1.Colbythoughtshewaswrongandthattheequationwas𝑦 = − !
!𝑥 − 4.Who’sright?Why?
Whatdidtheotherpersondowrong?Sophiaiscorrect.Shecomputedtheslopeas!!!"#$ !" !
!!!"#$ !" !andfoundthat!!(!!)
!!!!= −2.
Shethenfoundthatthey-interceptwas−1bysubstitutinginapointandtheslopeinthe𝑦 = 𝑚𝑥 + 𝑏formula.It’slikelythatColbyconfused!"#$
!!"andinsteadcalculated
theslopeas!"#!"#$
incorrectly.
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations5
AgallonofmilkatWholeFoodscost$6.ThegraphbelowcomparesthecostofmilkatWholeFoodsandSafeway.Estimatethecostofahalf-gallonofmilkatSafeway.Justifyyourreasoning.TheslopeofthelinerepresentingthecostofmilkatSafewayislesssteepthanthatofWholeFoodsmilk,indicatingit’slessexpensive.ItlooksliketheSFYlineisabouthalfassteepastheWFMline,whichwouldleadmetobelievethepriceofahalfgallonofmilkis$1.50…thisisbecauseaGALLONofmilkatWholeFoodsis$6,andifthisistobelinear,thehalfgallonwouldbe$3,andhalfof$3is$1.50.gotit??Also,ifyoucanfindmilkat$1.50forahalfgallon,I’dstockup—that’sagoodpricethesedays.12candybarsatTargetcost$15.00.ThegraphbelowcomparesthecostofcandybarsatTargetandCostco.Estimatethecostof1candybaratCostco.Justifyyourreasoning.ThecandybarsatTargetcostMOREthanthecandybarsatCostco.IknowbecausethelinedepictingthepricesatTGTaresteeperthantheofthelinedepictingpricesatCOST,andtheslopeisgreater.TheunitcostofcandybarsatTGTis$1.25($15/12).TheCOSTlinelookstobeabout2/3thesteepnessoftheTGTline,soI’destimatetheCOSTcandybarstobeabout$0.83each…butifyou’rethinking$0.75,youcansayyou’regoodtoo.$1.00istoohigh,and$0.63istoolow.Inadditiontothesefourproblems,thefollowingexerciseshavebeenassignedinKhanAcademy.Pleasechecktheboxtoindicatethatyouhaveeithercompleted5inarowor8outof10.
¨ Graphfromslope-interceptequation¨ Slope-interceptequationfromgraph¨ ConvertingBetweenSlope-InterceptandStandardForm
Cost($)
Milk(gallons)
WholeFoods
Safeway
Costco
Target
Candybars
Cost($)
NAME:___________________________ Math________,Period____________
Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations6
Usethegraphbelowtocompletethequestions.Onpage2,weidentifiedtwosimilartriangles(∆𝑃𝐿𝐾 𝑎𝑛𝑑 ∆𝑅𝑀𝐾).Identify3othersimilartriangles,andshowthattheyaresimilarbylookingattheproportionsofthecorrespondingsides.3triangles:∆𝑆𝑇𝑃,∆𝑆𝑁𝐾,∆𝑅𝑄𝑃Proportionsofcorrespondingsides(seepage2foranexample):
!"!"= !"
!"= !"
!"!!= !"
!= !
!= 𝑠𝑙𝑜𝑝𝑒 𝑜𝑓 2
Whatistheproportionofthecorrespondingsidesalsocalled?(ifyouconsiderthecorrespondingverticalsidesinthenumeratorandthecorrespondinghorizontalsidesinthedenominator).Thisistheslope.Oncewecanprovesimilaritybetweenapairofslopetriangles(andyoushouldbeabletoprovethisonatest),wecanstatepositivelythattheproportionofthecorrespondingsidesareequal…also,thatproportionmeasuringthechangeinverticaldistanceoverthechangeinhorizontal
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Mr.Rogove Date:__________
StudyGuide:SlopeandLinearEquations7
distanceiscalledslope,andthisisproofthattheslopeofalineisthesamebetweenanytwopointsontheline.YOUWILLNEEDTOKNOWTHIS!!