Post on 16-Jun-2020
Chapter1-Polynomial
Functions
LessonPackage
MHF4U
Chapter1OutlineUnitGoal:Bytheendofthisunit,youwillbeabletoidentifyanddescribesomekeyfeaturesofpolynomialfunctions,andmakeconnectionsbetweenthenumeric,graphical,andalgebraicrepresentationsofpolynomialfunctions.
Section Subject LearningGoals CurriculumExpectations
L1 PowerFunctions-describekeyfeaturesofgraphsofpowerfunctions-learnintervalnotation-beabletodescribeendbehaviour
C1.1,1.2,1.3
L2 CharacteristicsofPolynomialFunctions
-describecharacteristicsofequationsandgraphsofpolynomialfunctions-learnhowdegreerelatedtoturningpointsand𝑥-intercepts
C1.1,1.2,1.3,1.4
L3 FactoredFormPolynomialFunctions
-connecthowfactoredformequationrelatedto𝑥-interceptsofgraphofpolynomialfunction-givengraph,determineequationinfactoredform
C1.5,1.7,1.8
L4 TransformationsofPolynomialFunctions
-understandhowtheparameters𝑎, 𝑘, 𝑑,and𝑐transformpowerfunctions
C1.6
L5 SymmetryinPolynomialFunctions
-understandthepropertiesofevenandoddpolynomialfunctions C1.9
Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P
Quiz–PropertiesofPolynomialFunctions F P
PreTestReview F/A P Test-Functions O C1.1,1.2,1.3,1.4,1.5,1.6,1.7,
1.8,1.9 P K(21%),T(34%),A(10%),C(34%)
L1–1.1–PowerFunctionsLessonMHF4UJensenThingstoRememberAboutFunctions
• Arelationisafunctionifforevery𝑥-valuethereisonly1corresponding𝑦-value.Thegraphofarelationrepresentsafunctionifitpassesthe________________________________________,thatis,ifaverticallinedrawnanywherealongthegraphintersectsthatgraphatnomorethanonepoint.
• The________________ofafunctionisthecompletesetofallpossiblevaluesoftheindependentvariable(𝑥)
o Setofallpossible𝑥-valesthatwilloutputreal𝑦-values
• The________________ofafunctionisthecompletesetofallpossibleresultingvaluesofthedependentvariable(𝑦)
o Setofallpossible𝑦-valueswegetaftersubstitutingallpossible𝑥-values
• Forthefunction𝑓 𝑥 = (𝑥 − 1)) + 3
• Thedegreeofafunctionisthehighestexponentintheexpressiono 𝑓 𝑥 = 6𝑥- − 3𝑥) + 4𝑥 − 9hasadegreeof_____.
• An__________________________isalinethatacurveapproachesmoreandmorecloselybutnevertouches.
Thefunction𝒚 = 𝟏
𝒙3𝟑hastwoasymptotes:
VerticalAsymptote:Divisionbyzeroisundefined.Thereforetheexpressioninthedenominatorofthefunctioncannotbezero.Thereforex≠-3.Thisiswhytheverticallinex=-3isanasymptoteforthisfunction.HorizontalAsymptote:Fortherange,therecanneverbeasituationwheretheresultofthedivisioniszero.Thereforetheliney=0isahorizontalasymptote.Forallfunctionswherethedenominatorisahigherdegreethanthenumerator,therewillbyahorizontalasymptoteaty=0.
PolynomialFunctionsApolynomialfunctionhastheform
𝑓 𝑥 = 𝑎6𝑥6 + 𝑎678𝑥678 + 𝑎67)𝑥67) + ⋯+ 𝑎)𝑥) + 𝑎8𝑥8 + 𝑎:
• 𝑛Isawholenumber• 𝑥Isavariable• the_________________________𝑎:, 𝑎8, … , 𝑎6arerealnumbers• the___________________ofthefunctionis𝑛,theexponentofthegreatestpowerof𝑥• 𝑎6,thecoefficientofthegreatestpowerof𝑥,isthe________________________________• 𝑎:,thetermwithoutavariable,isthe___________________________• Thedomainofapolynomialfunctionisthesetofrealnumbers__________________• Therangeofapolynomialfunctionmaybeallrealnumbers,oritmayhavealowerboundoran
upperbound(butnotboth)• Thegraphofpolynomialfunctionsdonothavehorizontalorverticalasymptotes• Thegraphsofpolynomialfunctionsofdegree0are__________________________.Theshapesofother
graphsdependsonthedegreeofthefunction.Fivetypicalshapesareshownforvariousdegrees:A___________________________isthesimplesttypeofpolynomialfunctionandhastheform:
𝑓 𝑥 = 𝑎𝑥6• 𝑎isarealnumber• 𝑥isavariable• 𝑛isawholenumber
Example1:Determinewhichfunctionsarepolynomials.Statethedegreeandtheleadingcoefficientofeachpolynomialfunction.a)𝑔 𝑥 = sin 𝑥b)𝑓 𝑥 = 2𝑥Cc)𝑦 = 𝑥- − 5𝑥) + 6𝑥 − 8d)𝑔 𝑥 = 3F
IntervalNotationInthiscourse,youwilloftendescribethefeaturesofthegraphsofavarietyoftypesoffunctionsinrelationtoreal-numbervalues.Setsofrealnumbersmaybedescribedinavarietyofways:1)asaninequality−3 < 𝑥 ≤ 52)interval(orbracket)notation(−3, 5]3)graphicallyonanumberlineNote:
• Intervalsthatareinfiniteareexpressedusing_____________________or_____________________________• ________________________indicatethattheendvalueisincludedintheinterval• ________________________indicatethattheendvalueisNOTincludedintheinterval• A________________bracketisalwaysusedatinfinityandnegativeinfinity
Example2:Belowarethegraphsofcommonpowerfunctions.Usethegraphtocompletethetable.
PowerFunction
SpecialName Graph Domain Range
EndBehaviouras𝒙 →−∞
EndBehaviouras
𝒙 → ∞
𝒚 = 𝒙 Linear
𝒚 = 𝒙𝟐 Quadratic
𝒚 = 𝒙𝟑 Cubic
PowerFunction
SpecialName Graph Domain Range
EndBehaviouras𝒙 →−∞
EndBehaviouras
𝒙 → ∞
𝒚 = 𝒙𝟒 Quartic
𝒚 = 𝒙𝟓 Quintic
𝒚 = 𝒙𝟔 Sextic
KeyFeaturesofEVENDegreePowerFunctionsWhentheleadingcoefficient(a)ispositive Whentheleadingcoefficient(a)isnegative
Endbehaviour
Endbehaviour
Domain
Domain
Range Range
Example:
𝑓 𝑥 = 2𝑥C
Example:𝑓 𝑥 = −3𝑥)
LineSymmetryAgraphhaslinesymmetryifthereisaverticalline𝑥 = 𝑎thatdividesthegraphintotwopartssuchthateachpartisareflectionoftheother.Note:
KeyFeaturesofODDDegreePowerFunctionsWhentheleadingcoefficient(a)ispositive Whentheleadingcoefficient(a)isnegative
Endbehaviour
Endbehaviour
Domain
Domain
Range Range
Example:
𝑓 𝑥 = 3𝑥Q
Example:𝑓 𝑥 = −2𝑥-
PointSymmetryAgraphhaspointpointsymmetryaboutapoint(𝑎, 𝑏)ifeachpartofthegraphononesideof(𝑎, 𝑏)canberotated180°tocoincidewithpartofthegraphontheothersideof(𝑎, 𝑏).Note:
Example3:Writeeachfunctionintheappropriaterowofthesecondcolumnofthetable.Givereasonsforyourchoices.𝑦 = 2𝑥 𝑦 = 5𝑥U 𝑦 = −3𝑥) 𝑦 = 𝑥V𝑦 = − )
Q𝑥W 𝑦 = −4𝑥Q 𝑦 = 𝑥8: 𝑦 = −0.5𝑥Y
EndBehaviour Functions Reasons
Q3toQ1
Q2toQ4
Q2toQ1
Q3toQ4
Example4:Foreachofthefollowingfunctionsi)Statethedomainandrangeii)Describetheendbehavioriii)Identifyanysymmetry
a)b)c)
𝒚 = −𝒙
𝒚 = 𝟎. 𝟓𝒙𝟐
𝒚 = 𝟒𝒙𝟑
i)Domain: Range:ii)As_____________________andas______________________Thegraphextendsfromquadrant_____to_____iii)
i)Domain: Range:ii)As_____________________andas______________________Thegraphextendsfromquadrant_____to_____iii)
i)Domain: Range:ii)As_____________________andas______________________Thegraphextendsfromquadrant_____to_____iii)
L2–1.2–CharacteristicsofPolynomialFunctionsLessonMHF4UJensenInsection1.1welookedatpowerfunctions,whicharesingle-termpolynomialfunctions.Manypolynomialfunctionsaremadeupoftwoormoreterms.Inthissectionwewilllookatthecharacteristicsofthegraphsandequationsofpolynomialfunctions.NewTerminology–LocalMin/Maxvs.AbsoluteMin/MaxLocalMinorMaxPoint–Pointsthatareminimumormaximumpointsonsomeintervalaroundthatpoint.AbsoluteMaxorMin–Thegreatest/leastvalueattainedbyafunctionforALLvaluesinitsdomain.Investigation:GraphsofPolynomialFunctionsThedegreeandtheleadingcoefficientintheequationofapolynomialfunctionindicatetheendbehavioursofthegraph.Thedegreeofapolynomialfunctionprovidesinformationabouttheshape,turningpoints(localmin/max),andzeros(x-intercepts)ofthegraph.Completethefollowingtableusingtheequationandgraphsgiven:
Inthisgraph,(-1,4)isa___________________and(1,-4)isa___________________.Thesearenotabsoluteminandmaxpointsbecausethereareotherpointsonthegraphofthefunctionthataresmallerandgreater.Sometimeslocalminandmaxpointsarecalled______________________________.
Onthegraphofthisfunction…Thereare____localmin/maxpoints.____arelocalminand____isalocalmax.Oneofthelocalminpointsisalsoanabsolutemin(itislabeled).
EquationandGraph DegreeEvenorOdd
Degree?
LeadingCoefficient EndBehaviour
Numberofturningpoints
Numberofx-intercepts
𝑓 𝑥 = 𝑥$ + 4𝑥 − 5
𝑓 𝑥 = 3𝑥* − 4𝑥+ − 4𝑥$ + 5𝑥 + 5
𝑓 𝑥 = 𝑥+ − 2𝑥
𝑓 𝑥 = −𝑥* − 2𝑥+ + 𝑥$ + 2𝑥
𝑓 𝑥 = 2𝑥- − 12𝑥* + 18𝑥$ + 𝑥 − 10
𝑓 𝑥 = 2𝑥1 + 7𝑥* − 3𝑥+ − 18𝑥$ + 5
SummaryofFindings:
• Apolynomialfunctionofdegree𝑛hasatmost_____________localmax/minpoints(turningpoints)• Apolynomialfunctionofdegree𝑛mayhaveupto_____distinctzeros(x-intercepts)• Ifapolynomialfunctionis_________degree,itmusthaveatleastonex-intercept,andanevennumberof
turningpoints• Ifapolynomialfunctionis________degree,itmayhavenox-intercepts,andanoddnumberofturningpoints• Anodddegreepolynomialfunctionextendsfrom…
o _____quadrantto_____quadrantifithasapositiveleadingcoefficiento _____quadrantto_____quadrantifithasanegativeleadingcoefficient
• Anevendegreepolynomialfunctionextendsfrom…o _____quadrantto_____quadrantifithasapositiveleadingcoefficiento _____quadrantto_____quadrantifithasanegativeleadingcoefficient
EquationandGraph DegreeEvenorOdd
Degree?
LeadingCoefficient EndBehaviour
Numberofturningpoints
Numberofx-intercepts
𝑓 𝑥 = 5𝑥1 + 5𝑥* − 2𝑥+ + 4𝑥$ − 3𝑥
𝑓 𝑥 = −2𝑥+ + 4𝑥$ − 3𝑥 − 1
𝑓 𝑥 = 𝑥* + 2𝑥+ − 3𝑥 − 1
Note:OdddegreepolynomialshaveOPPOSITEendbehaviours
Note:EvendegreepolynomialshaveTHESAMEendbehaviour.
Example1:Describetheendbehavioursofeachfunction,thepossiblenumberofturningpoints,andthepossiblenumberofzeros.Usethesecharacteristicstosketchpossiblegraphsofthefunctiona)𝑓 𝑥 = −3𝑥1 + 4𝑥+ − 8𝑥$ + 7𝑥 − 5Possiblegraphsof5thdegreepolynomialfunctionswithanegativeleadingcoefficient:b)𝑔 𝑥 = 2𝑥* + 𝑥$ + 2Possiblegraphsof4thdegreepolynomialfunctionswithapositiveleadingcoefficient:Example2:Filloutthefollowingchart
Degree Possible#of𝒙-intercepts Possible#ofturningpoints1 2 3 4 5
Note:Odddegreefunctionsmusthaveanevennumberofturningpoints.
Note:Evendegreefunctionsmusthaveanoddnumberofturningpoints.
Example3:Determinethekeyfeaturesofthegraphofeachpolynomialfunction.Usethesefeaturestomatcheachfunctionwithitsgraph.Statethenumberof𝑥-intercepts,thenumberoflocalmax/minpoints,andthenumberofabsolutemax/minpointsforthegraphofeachfunction.Howarethesefeaturesrelatedtothedegreeofeachfunction?a)𝑓 𝑥 = 2𝑥+ − 4𝑥$ + 𝑥 + 1b)𝑔 𝑥 = −𝑥* + 10𝑥$ + 5𝑥 − 4c)ℎ 𝑥 = −2𝑥1 + 5𝑥+ − 𝑥d)𝑝 𝑥 = 𝑥- − 16𝑥$ + 3a)b)c)d)FiniteDifferencesForapolynomialfunctionofdegree𝑛,where𝑛isapositiveinteger,the𝑛9:differences…
• areequal• havethesamesignastheleadingcoefficient• areequalto𝑎 ∙ 𝑛!,where𝑎istheleadingcoefficient
Note:𝑛!isreadas𝑛factorial.𝑛! = 𝑛×(𝑛 − 1)×(𝑛 − 2)×…×2×15! = 5×4×3×2×1 = 120
Example4:Thetableofvaluesrepresentsapolynomialfunction.Usefinitedifferencestodetermine
a) thedegreeofthepolynomialfunctionb) thesignoftheleadingcoefficientc) thevalueoftheleadingcoefficient
a)b)c) Example5:Forthefunction2𝑥* − 4𝑥$ + 𝑥 + 1whatisthevalueoftheconstantfinitedifferences?𝐹𝑖𝑛𝑖𝑡𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 =
𝒙 𝒚 Firstdifferences Seconddifferences Thirddifferences
-3 -36 -2 -12 -1 -2 0 0 1 0 2 4 3 18 4 48
L3–1.3–FactoredFormPolynomialFunctionsLessonMHF4UJensenInthissection,youwillinvestigatetherelationshipbetweenthefactoredformofapolynomialfunctionandthe𝑥-interceptsofthecorrespondinggraph,andyouwillexaminetheeffectofrepeatedfactoronthegraphofapolynomialfunction.
FactoredFormInvestigation
Ifwewanttographthepolynomialfunction𝑓 𝑥 = 𝑥$ + 3𝑥' + 𝑥( − 3𝑥 − 2accurately,itwouldbemostusefultolookatthefactoredformversionofthefunction:𝑓 𝑥 = 𝑥 + 1 ( 𝑥 + 2 𝑥 − 1
Let’sstartbylookingatthegraphofthefunctionandmakingconnectionstothefactoredformequation.
Graphof𝑓(𝑥): Fromthegraph,answerthefollowingquestions…
a)Whatisthedegreeofthefunction? b)Whatisthesignoftheleadingcoefficient? c)Whatarethe𝑥-intercepts? d)Whatisthe𝑦-intercept?e)The𝑥-interceptsdividethegraphintointofourintervals.Writetheintervalsinthefirstrowofthetable.Inthesecondrow,chooseatestpointwithintheinterval.Inthethirdrow,indicatewhetherthefunctionispositive(abovethe𝑥-axis)ornegative(belowthe𝑦-axis).
Interval
TestPoint
Signof𝒇(𝒙)
f)Whathappenstothesignoftheof𝑓(𝑥)neareach𝑥-intercept?
Conclusionsfrominvestigation:The𝑥-interceptsofthegraphofthefunctioncorrespondtotheroots(zeros)ofthecorrespondingequation.Forexample,thefunction𝑓 𝑥 = (𝑥 − 2)(𝑥 + 1)has𝑥-interceptsat___and___.Thesearetherootsoftheequation 𝑥 − 2 𝑥 + 1 = 0.Ifapolynomialfunctionhasafactor(𝑥 − 𝑎)thatisrepeated𝑛times,then𝑥 = 𝑎isazeroof_________𝑛.Order–theexponenttowhicheachfactorinanalgebraicexpressionisraised.Forexample,thefunction𝑓 𝑥 = 𝑥 − 3 ((𝑥 − 1)hasazerooforder______at𝑥 = 3andazerooforder______at𝑥 = 1.Thegraphofapolynomialfunctionchangessignatzerosof_________orderbutdoesnotchangesignatzerosof_________order.Shapesbasedonorderofzero:
𝑓 𝑥 = 0.01(𝑥 − 1) 𝑥 + 2 ((𝑥 − 4)'
ORDER2(-2,0)isan𝑥-interceptoforder2.Therefore,itdoesn’tchangesign.“Bouncesoff”𝑥-axis.Parabolicshape.
ORDER1(1,0)isan𝑥-interceptoforder1.Therefore,itchangessign.“Goesstraightthrough”𝑥-axis.LinearShape
ORDER3(4,0)isan𝑥-interceptoforder3.Therefore,itchangessign.“S-shape”through𝑥-axis.Cubicshape.
Example1:AnalyzingGraphsofPolynomialFunctionsForeachgraph,
i) theleastpossibledegreeandthesignoftheleadingcoefficientii) the𝑥-interceptsandthefactorsofthefunctioniii) theintervalswherethefunctionispositive/negative
a) i) ii) iii)b) i) ii) iii)
Example2:AnalyzeFactoredFormEquationstoSketchGraphs
Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-interceptTheexponenton𝑥whenallfactorsof𝑥aremultipliedtogether
OR
Addtheexponentsonthefactorsthatincludean𝑥.
Theproductofallthe𝑥coefficients
Usedegreeandsignofleadingcoefficienttodeterminethis
Seteachfactorequaltozeroandsolvefor𝑥
Set𝑥 = 0andsolvefor𝑦
Sketchagraphofeachpolynomialfunction:a)𝑓 𝑥 = (𝑥 − 1)(𝑥 + 2)(𝑥 + 3)
Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept
b)𝑔 𝑥 = −2 𝑥 − 1 ((𝑥 + 2)
Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept
c)ℎ 𝑥 = − 2𝑥 + 1 '(𝑥 − 3)
Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept
d)𝑗 𝑥 = 𝑥$ − 4𝑥' + 3𝑥(
Degree LeadingCoefficient EndBehaviour 𝒙-intercepts 𝒚-intercept
Example3:RepresentingtheGraphofaPolynomialFunctionwithitsEquationa)Writetheequationofthefunctionshownbelow:
Note:mustputintofactoredformtofind𝑥-intercepts
Steps:1)Writetheequationofthefamilyofpolynomialsusingfactorscreatedfrom𝑥-intercepts2)Substitutethecoordinatesofanotherpoint(𝑥, 𝑦)intotheequation.3)Solvefor𝑎4)Writetheequationinfactoredform
b)Findtheequationofapolynomialfunctionthatisdegree4withzeros−1(order3)and1,andwitha𝑦-interceptof−2.
L4–1.4–TransformationsLessonMHF4UJensenInthissection,youwillinvestigatetherolesoftheparameters𝑎, 𝑘, 𝑑,and𝑐inpolynomialfunctionsoftheform𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄.Youwillapplytransformationstothegraphsofbasicpowerfunctionstosketchthegraphofitstransformedfunction.Part1:TransformationsInvestigationInthisinvestigation,youwillbelookingattransformationsofthepowerfunction𝑦 = 𝑥4.Completethefollowingtableusinggraphingtechnologytohelp.Thegraphof𝑦 = 𝑥4isgivenoneachsetofaxes;sketchthegraphofthetransformedfunctiononthesamesetofaxes.Thencommentonhowthevalueoftheparameter𝑎, 𝑘, 𝑑,or𝑐transformstheparentfunction.Effectsof𝑐on𝑦 = 𝑥4 + 𝑐TransformedFunction
Valueof𝒄 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒
𝑦 = 𝑥4 + 1
𝑦 = 𝑥4 − 2
Effectsof𝑑on𝑦 = (𝑥 − 𝑑)4TransformedFunction
Valueof𝒅 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒
𝑦 = (𝑥 − 2)4
𝑦 = (𝑥 + 3)4
Effectsof𝑎on𝑦 = 𝑎𝑥4TransformedFunction
Valueof𝒂 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒
𝑦 = 2𝑥4
𝑦 =12𝑥
4
𝑦 = −2𝑥4
Effectsof𝑘on𝑦 = (𝑘𝑥)4TransformedFunction
Valueof𝒌 Transformationsto𝒚 = 𝒙𝟒 Graphoftransformedfunctioncomparedto𝒚 = 𝒙𝟒
𝑦 = (2𝑥)4
𝑦 =13𝑥
4
𝑦 = (−2𝑥)4
Summaryofeffectsof𝑎, 𝑘, 𝑑,and𝑐inpolynomialfunctionsoftheform𝑓(𝑥) = 𝑎 𝑘 𝑥 − 𝑑 ; + 𝑐
Valueof𝒄in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄
𝑐 > 0
𝑐 < 0
Valueof𝒅in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄
𝑑 > 0
𝑑 < 0
Valueof𝒂in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄
𝑎 > 1or𝑎 < −1
−1 < 𝑎 < 1
𝑎 < 0
Valueof𝒌in𝒇(𝒙) = 𝒂 𝒌 𝒙 − 𝒅 𝒏 + 𝒄
𝑘 > 1or𝑘 < −1
−1 < 𝑘 < 1
𝑘 < 0
Note:𝑎and𝑐cause_________________transformationsandthereforeeffectthe𝑦-coordinatesofthefunction.𝑘and𝑑cause_________________transformationsandthereforeeffectthe𝑥-coordinatesofthefunction.Whenapplyingtransformationstoaparentfunction,makesuretoapplythetransformationsrepresentedby𝑎and𝑘BEFOREthetransformationsrepresentedby𝑑and𝑐.
Part2:DescribingTransformationsfromanEquationExample1:Describethetransformationsthatmustbeappliedtothegraphofeachpowerfunction,𝑓(𝑥),toobtainthetransformedfunction,𝑔(𝑥).Then,writethecorrespondingequationofthetransformedfunction.Then,statethedomainandrangeofthetransformedfunction.a)𝑓 𝑥 = 𝑥4,𝑔 𝑥 = 2𝑓 @
A𝑥 − 5
b)𝑓 𝑥 = 𝑥C,𝑔 𝑥 = @
4𝑓 −2 𝑥 − 3 + 4
Part3:ApplyingTransformationstoSketchaGraphExample2:Thegraphof𝑓(𝑥) = 𝑥Aistransformedtoobtainthegraphof𝑔(𝑥) = 3 −2 𝑥 + 1 A + 5.a)Statetheparametersanddescribethecorrespondingtransformationsb)Makeatableofvaluesfortheparentfunctionandthenusethetransformationsdescribedinparta)tomakeatableofvaluesforthetransformedfunction.
c)Graphtheparentfunctionandthetransformedfunctiononthesamegrid.
𝑓 𝑥 = 𝑥A𝒙 𝒚
Note:Whenchoosingkeypointsfortheparentfunction,alwayschoose𝑥-valuesbetween-2and2andcalculatethecorrespondingvaluesof𝑦.
Example3:Thegraphof𝑓(𝑥) = 𝑥4istransformedtoobtainthegraphof𝑔(𝑥) = − @A𝑥 + 2
4− 1.
a)Statetheparametersanddescribethecorrespondingtransformationsb)Makeatableofvaluesfortheparentfunctionandthenusethetransformationsdescribedinparta)tomakeatableofvaluesforthetransformedfunction.
c)Graphtheparentfunctionandthetransformedfunctiononthesamegrid.
𝑓 𝑥 = 𝑥4𝒙 𝒚
Note:𝑘valuemustbefactoredoutintotheform[𝑘(𝑥 + 𝑑)]
Part4:DetermininganEquationGiventheGraphofaTransformedFunctionExample4:Transformationsareappliedtoeachpowerfunctiontoobtaintheresultinggraph.Determineanequationforthetransformedfunction.Thenstatethedomainandrangeofthetransformedfunction.a)b)
f(1)
f(-1)
f(1)f(-1)
𝑓(𝑥) = 2𝑥' + 3𝑥* − 2Notice:𝑓(1) =𝑓(−1) =∴
L5–1.3–SymmetryinPolynomialFunctionsMHF4UJensenInthissection,youwilllearnaboutthepropertiesofevenandoddpolynomialfunctions.SymmetryinPolynomialFunctions__________________–thereisaverticallineoverwhichthepolynomialremainsunchangedwhenreflected.___________________________________–thereisapointaboutwhichthepolynomialremainsunchangedwhenrotated180°Section1:PropertiesofEvenandOddFunctionsApolynomialfunctionofevenorodddegreeisNOTnecessarilyandevenoroddfunction.Thefollowingarepropertiesofallevenandoddfunctions:
EvenFunctions OddFunctionsAnevendegreepolynomialfunctionisanEVENFUNCTIONif:
• Linesymmetryoverthe__________• Theexponentofeachtermis_______• Mayhaveaconstantterm
AnodddegreepolynomialfunctionisanODDFUNCTIONif:
• Pointsymmetryaboutthe__________• Theexponentofeachtermis_____• Noconstantterm
Rule:
Rule:
Example:
Example:
𝑓(𝑥) = 2𝑥1 + 3𝑥Notice:𝑓(1) =𝑓(−1) =∴
Example1:Identifyeachfunctionasanevenfunction,oddfunction,orneither.Explainhowyoucantell.a)𝑦 = 𝑥1 − 4𝑥b)𝑦 = 𝑥1 − 4𝑥 + 2c)𝑦 = 𝑥' − 4𝑥* + 2
d)𝑦 = 3𝑥' + 𝑥1 − 4𝑥* + 2e)𝑦 = −3𝑥* − 6𝑥Example2:Chooseallthatapplyforeachfunctiona) b)
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
c)𝑃 𝑥 = 5𝑥1 + 3𝑥* + 2 d)𝑃 𝑥 = 𝑥8 + 𝑥* − 11
e) f)g)𝑃 𝑥 = 5𝑥9 − 4𝑥1 + 8𝑥Example3:Withoutgraphing,determineifeachpolynomialfunctionhaslinesymmetryaboutthey-axis,pointsymmetryabouttheorigin,orneither.Verifyyourresponsealgebraically.a)𝑓 𝑥 = 2𝑥' − 5𝑥* + 4
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
i)nosymmetry
ii)pointsymmetry
iii)linesymmetry
iv)oddfunction
v)evenfunction
Note:
(-1, 3)
b)𝑓 𝑥 = −3𝑥9 + 9𝑥1 + 2𝑥c)𝑥8 − 4𝑥1 + 6𝑥* − 4Section2:ConnectingfromthroughouttheunitExample4:Usethegivengraphtostate:a)𝑥-interceptsb)numberofturningpointsc)leastpossibledegreeb)anysymmetrypresentc)theintervalswhere𝑓 𝑥 < 0
d)Findtheequationinfactoredform