湖北省武汉六中高三(国际部)数学复习课件：21有理函数的定义

Unit2:RationalFunctions

Lesson1:ReciprocalofaLinearFunctionUnit2:RationalFunctions

LesWhatisaRationalFunction?Anyfunctionoftheform:Wheref(x)andg(x)arepolynomialfunctionsBecausethedenominatorcanneverbezero,rationalfunctionshavepropertiesthatpolynomialfunctionsdonotWhatisaRationalFunction?AnWhatistheReciprocalofaLinearFunction?Wearegoingtostartbylookingatthesimplestrationalfunctions:ThisisthegeneralformforthereciprocalofalinearfunctionReciprocalmeansyouput“oneover”or,moresimply,you“flipit”WhatistheReciprocalofaLiExample1(a)UseyourTI-83ortheprogram“Graph”tographthefunction(b)Describetheendbehaviour(c)Whathappenswhenxgetscloseto½?Example1(a)UseyourTI-83orExample1:Solution(a)Example1:Solution(a)Example1:Solution(b)Asxgetslargeinboththepositiveandnegativedirections,thefunctiongetscloseto–butdoesnottouch–they-axis.Therefore,

Asx+∞,y0 Asx–∞,y0Example1:Solution(b)AsxgeExample1:SolutionStartontheleftofthegraphandmovetowardsx=½:they-valuesgetlargeandnegativeStartontherightofthegraphandmovetowardsx=½:they-valuesgetlargeandpositiveDenotedby:x½-,y-∞Denotedby:x½+,y+∞Approachx=½fromtherightApproachx=½fromtheleft(c)ThefunctionnevercrossesthisverticallineExample1:SolutionStartonthExample1:NotesAlinethatafunctiongetsclosetobutdoesnottouchiscalledanasymptoteThey-valuesgotclosetobutdidnottouchthey-axis(ahorizontalline)horizontalasymptoteistheliney=0Thereciprocalofalinearfunctionwillalwayshaveahorizontalasymptoteaty=0Thex-valuesgotclosetobutdidnottouchthelinex=½(averticalline)verticalasymptoteisthelinex=½OccursbecausethedenominatorcannotbezeroThereciprocalofalinearfunctionwillalwayshaveoneverticalasymptoteExample1:NotesAlinethataExample2(a)UseyourTI-83ortheprogram“Graph”tographthefunction(b)LabelthehorizontalandverticalasymptotesExample2(a)UseyourTI-83orExample2:SolutionThelinex=1(verticalasymptote)Theliney=0(horizontalasymptote)Example2:SolutionThelinexExample2:NotesInthisexamplethevalueofk(thenumberinfrontofx)isnegative:Asaresult,branchontheleftoftheverticalasymptoteisabovethex-axisandthebranchontherightbranchisbelowitWhenkispositivethebranchontheleftisbelowthex-axisandthebranchontherightbranchisaboveitExample2:NotesInthisexamplExample3Considerthefunction(a)Determinetheequationsoftheasymptotes(b)StatethedomainandrangeExample3ConsiderthefunctionExample3:SolutionThehorizontalasymptoteistheliney=0

See“Example1:Notes” Theverticalasymptoteoccursbecausethedenominatorcannotbezero.WeneedtofindthevalueofxthatmakesthedenominatorzeroTherefore,theverticalasymptoteisthelinex=2Example3:SolutionThehorizonExample3:Solution(b)Thedomaintellsuswhatvaluesofxthefunctioncanbeevaluatedat.Theonlyvalueofxwecan’thaveis2.Therefore, Therangetellsuswhatvaluesofythefunctioncanhave.Theonlyvalueofywewillnevergetis0.Therefore,Example3:Solution(b)ThedomExample3:NotesOurverticalasymptotewasthelinex=2andourdomainwasTheverticalasymptotegivesyouthedomainOurhorizontalasymptotewastheliney=0andourrangewasThehorizontalasymptotegivesyoutherangeExample3:NotesOurverticalaExample4Determinethex-andy-interceptsofExample4Determinethex-andExample4:SolutionThex-interceptisthevalueofxwheny=0:Thereisnovalueofxthatmakesthistrue.Thereisnox-interceptThey-interceptisthevalueofywhenx=0:They-interceptisExample4:SolutionThex-interSummaryThereciprocalofalinearfunctionhastheformTheverticalasymptoteisfoundbysettingthedenominatorequaltozeroandsolvingforxThedenominatorCANNOTbezeroThedomainisallvaluesofxexceptthisoneThehorizontalasymptoteisthex-axis(theliney=0)TherangeisallvaluesofyexceptzeroThesefunctionshavetwobranches–oneontheleftoftheverticalasymptoteandoneontherightk>0:leftbranchisbelowthex-axis,therightisaboveK<0:leftbranchisabovethex-axis,therightisbelowSummaryThereciprocalofalinPracticeProblemsP.153-154#2,3,5,7-9Note:For#7don’tbotherwithasketch.Justcalculatethey-interceptandstatethedomain,rangeandasymptotes.PracticeProblemsP.153-154#2Unit2:RationalFunctions

Lesson1:ReciprocalofaLinearFunctionUnit2:RationalFunctions

LesWhatisaRationalFunction?Anyfunctionoftheform:Wheref(x)andg(x)arepolynomialfunctionsBecausethedenominatorcanneverbezero,rationalfunctionshavepropertiesthatpolynomialfunctionsdonotWhatisaRationalFunction?AnWhatistheReciprocalofaLinearFunction?Wearegoingtostartbylookingatthesimplestrationalfunctions:ThisisthegeneralformforthereciprocalofalinearfunctionReciprocalmeansyouput“oneover”or,moresimply,you“flipit”WhatistheReciprocalofaLiExample1(a)UseyourTI-83ortheprogram“Graph”tographthefunction(b)Describetheendbehaviour(c)Whathappenswhenxgetscloseto½?Example1(a)UseyourTI-83orExample1:Solution(a)Example1:Solution(a)Example1:Solution(b)Asxgetslargeinboththepositiveandnegativedirections,thefunctiongetscloseto–butdoesnottouch–they-axis.Therefore,

Asx+∞,y0 Asx–∞,y0Example1:Solution(b)AsxgeExample1:SolutionStartontheleftofthegraphandmovetowardsx=½:they-valuesgetlargeandnegativeStartontherightofthegraphandmovetowardsx=½:they-valuesgetlargeandpositiveDenotedby:x½-,y-∞Denotedby:x½+,y+∞Approachx=½fromtherightApproachx=½fromtheleft(c)ThefunctionnevercrossesthisverticallineExample1:SolutionStartonthExample1:NotesAlinethatafunctiongetsclosetobutdoesnottouchiscalledanasymptoteThey-valuesgotclosetobutdidnottouchthey-axis(ahorizontalline)horizontalasymptoteistheliney=0Thereciprocalofalinearfunctionwillalwayshaveahorizontalasymptoteaty=0Thex-valuesgotclosetobutdidnottouchthelinex=½(averticalline)verticalasymptoteisthelinex=½OccursbecausethedenominatorcannotbezeroThereciprocalofalinearfunctionwillalwayshaveoneverticalasymptoteExample1:NotesAlinethataExample2(a)UseyourTI-83ortheprogram“Graph”tographthefunction(b)LabelthehorizontalandverticalasymptotesExample2(a)UseyourTI-83orExample2:SolutionThelinex=1(verticalasymptote)Theliney=0(horizontalasymptote)Example2:SolutionThelinexExample2:NotesInthisexamplethevalueofk(thenumberinfrontofx)isnegative:Asaresult,branchontheleftoftheverticalasymptoteisabovethex-axisandthebranchontherightbranchisbelowitWhenkispositivethebranchontheleftisbelowthex-axisandthebranchontherightbranchisaboveitExample2:NotesInthisexamplExample3Considerthefunction(a)Determinetheequationsoftheasymptotes(b)StatethedomainandrangeExample3ConsiderthefunctionExample3:SolutionThehorizontalasymptoteistheliney=0

See“Example1:Notes” Theverticalasymptoteoccursbecausethedenominatorcannotbezero.WeneedtofindthevalueofxthatmakesthedenominatorzeroTherefore,theverticalasymptoteisthelinex=2Example3:SolutionThehorizonExample3:Solution(b)Thedomaintellsuswhatvaluesofxthefunctioncanbeevaluatedat.Theonlyvalueofxwecan’thaveis2.Therefore, Therangetellsuswhatvaluesofythefunctioncanhave.Theonlyvalueofywewillnevergetis0.Therefore,Example3:Solution(b)ThedomExample3:NotesOurverticalasymptotewasthelinex=2andourdomainwasTheverticalasymptotegivesyouthedomainOurhorizontalasymptotewastheliney=0andourrangewasThehorizontalasymptotegivesyoutherangeExample3:NotesOurverticalaExample4Determinethex-andy-interceptsofExample4Determinethex-andExample4:SolutionThex-interceptisthe