chapter 1- polynomial functions - 1 unit package l3 factored form polynomial functions ... quiz...

Download Chapter 1- Polynomial Functions -   1 unit package   L3 Factored Form Polynomial Functions ... Quiz – Properties of Polynomial Functions ... Possible graphs of 5th degree polynomial functions with a negative leading coefficient: b) *:

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  • Chapter1-Polynomial

    Functions

    LessonPackage

    MHF4U

    trevorjensenStamp

  • Chapter1OutlineUnitGoal:Bytheendofthisunit,youwillbeabletoidentifyanddescribesomekeyfeaturesofpolynomialfunctions,andmakeconnectionsbetweenthenumeric,graphical,andalgebraicrepresentationsofpolynomialfunctions.

    Section Subject LearningGoals CurriculumExpectations

    L1 PowerFunctions-describekeyfeaturesofgraphsofpowerfunctions-learnintervalnotation-beabletodescribeendbehaviour

    C1.1,1.2,1.3

    L2 CharacteristicsofPolynomialFunctions

    -describecharacteristicsofequationsandgraphsofpolynomialfunctions-learnhowdegreerelatedtoturningpointsand-intercepts

    C1.1,1.2,1.3,1.4

    L3 FactoredFormPolynomialFunctions

    -connecthowfactoredformequationrelatedto-interceptsofgraphofpolynomialfunction-givengraph,determineequationinfactoredform

    C1.5,1.7,1.8

    L4 TransformationsofPolynomialFunctions

    -understandhowtheparameters, , ,andtransformpowerfunctions

    C1.6

    L5 SymmetryinPolynomialFunctions-understandthepropertiesofevenandoddpolynomialfunctions

    C1.9

    Assessments F/A/O MinistryCode P/O/C KTACNoteCompletion A P PracticeWorksheetCompletion F/A P

    QuizPropertiesofPolynomialFunctions 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%)

  • L11.1PowerFunctionsLessonMHF4UJensenThingstoRememberAboutFunctions

    Arelationisafunctionifforevery-valuethereisonly1corresponding-value.Thegraphofarelationrepresentsafunctionifitpassestheverticallinetest,thatis,ifaverticallinedrawnanywherealongthegraphintersectsthatgraphatnomorethanonepoint.

    TheDOMAINofafunctionisthecompletesetofallpossiblevaluesoftheindependentvariable()

    o Setofallpossible-valesthatwilloutputreal-values

    TheRANGEofafunctionisthecompletesetofallpossibleresultingvaluesofthedependentvariable()

    o Setofallpossible-valueswegetaftersubstitutingallpossible-values

    Forthefunction = ( 1)) + 3

    o D: }

    o R:{ | 3}

    Thedegreeofafunctionisthehighestexponentintheexpressiono = 65 3) + 4 9hasadegreeof3

    AnASYMPTOTEisalinethatacurveapproachesmoreandmorecloselybutnevertouches.

    Thefunction =

    ;hastwoasymptotes:

    VerticalAsymptote:Divisionbyzeroisundefined.Thereforetheexpressioninthedenominatorofthefunctioncannotbezero.Thereforex-3.Thisiswhytheverticallinex=-3isanasymptoteforthisfunction.HorizontalAsymptote:Fortherange,therecanneverbeasituationwheretheresultofthedivisioniszero.Thereforetheliney=0isahorizontalasymptote.Forallfunctionswherethedenominatorisahigherdegreethanthenumerator,therewillbyahorizontalasymptoteaty=0.

  • PolynomialFunctionsApolynomialfunctionhastheform

    = >> + >?@>?@ + >?)>?) + + )) + @@ + B

    Isawholenumber Isavariable thecoefficientsB, @, , >arerealnumbers thedegreeofthefunctionis,theexponentofthegreatestpowerof >,thecoefficientofthegreatestpowerof,istheleadingcoefficient B,thetermwithoutavariable,istheconstantterm ThedomainofapolynomialfunctionisthesetofrealnumbersD: } Therangeofapolynomialfunctionmaybeallrealnumbers,oritmayhavealowerboundoran

    upperbound(butnotboth) Thegraphofpolynomialfunctionsdonothavehorizontalorverticalasymptotes Thegraphsofpolynomialfunctionsofdegree0arehorizontallines.Theshapesofothergraphs

    dependsonthedegreeofthefunction.Fivetypicalshapesareshownforvariousdegrees:Apowerfunctionisthesimplesttypeofpolynomialfunctionandhastheform:

    = > isarealnumber isavariable isawholenumber

    Example1:Determinewhichfunctionsarepolynomials.Statethedegreeandtheleadingcoefficientofeachpolynomialfunction.a) = sin b) = 2Kc) = 5 5) + 6 8d) = 3N

    Thisisatrigonometricfunction,notapolynomialfunction.

    Thisisapolynomialfunctionofdegree4.Theleadingcoefficientis2

    Thisisapolynomialfunctionofdegree3.Theleadingcoefficientis1.

    Thisisnotapolynomialfunctionbutanexponentialfunction,sincethebaseisanumberandtheexponentisavariable.

  • IntervalNotationInthiscourse,youwilloftendescribethefeaturesofthegraphsofavarietyoftypesoffunctionsinrelationtoreal-numbervalues.Setsofrealnumbersmaybedescribedinavarietyofways:1)asaninequality3 < 52)interval(orbracket)notation(3, 5]3)graphicallyonanumberlineNote:

    Intervalsthatareinfiniteareexpressedusing(infinity)or(negativeinfinity) Squarebracketsindicatethattheendvalueisincludedintheinterval RoundbracketsindicatethattheendvalueisNOTincludedintheinterval Aroundbracketisalwaysusedatinfinityandnegativeinfinity

    Example2:Belowarethegraphsofcommonpowerfunctions.Usethegraphtocompletethetable.

    PowerFunction

    SpecialName Graph Domain Range

    EndBehaviouras

    EndBehaviouras

    = Linear

    (,) (,)

    Startsinquadrant3

    Endsinquadrant1

    = Quadratic

    (,) [0,)

    Startsinquadrant2

    Endsinquadrant1

    = Cubic

    (,) (,)

    Startsinquadrant3

    Endsinquadrant1

  • PowerFunction

    SpecialName Graph Domain Range

    EndBehaviouras

    EndBehaviouras

    = Quartic

    (,) [0,)

    Startsinquadrant2

    Endsinquadrant1

    = Quintic

    (,) [,)

    Startsinquadrant3

    Endsinquadrant1

    = Sextic

    (,) [0,)

    Startsinquadrant2

    Endsinquadrant1

  • KeyFeaturesofEVENDegreePowerFunctionsWhentheleadingcoefficient(a)ispositive Whentheleadingcoefficient(a)isnegative

    Endbehaviour

    as , andas ,

    Q2toQ1

    Endbehaviour

    as , andas ,

    Q3toQ4

    Domain

    (,)

    Domain (,)

    Range [0,) Range

    [0, )

    Example:

    = 2K

    Example: = 3)

    LineSymmetryAgraphhaslinesymmetryifthereisaverticalline = thatdividesthegraphintotwopartssuchthateachpartisareflectionoftheother.Note:Thegraphsofevendegreepowerfunctionshavelinesymmetryabouttheverticalline = 0(they-axis).

  • KeyFeaturesofODDDegreePowerFunctionsWhentheleadingcoefficient(a)ispositive Whentheleadingcoefficient(a)isnegative

    Endbehaviour

    as , and

    as ,

    Q3toQ1

    Endbehaviour

    as , andas ,

    Q2toQ4

    Domain

    (,)

    Domain (,)

    Range (,) Range

    (,)

    Example:

    = 3[

    Example: = 25

    PointSymmetryAgraphhaspointpointsymmetryaboutapoint(, )ifeachpartofthegraphononesideof(, )canberotated180tocoincidewithpartofthegraphontheothersideof(, ).Note:Thegraphofodddegreepowerfunctionshavepointsymmetryabouttheorigin(0,0).

  • Example3:Writeeachfunctionintheappropriaterowofthesecondcolumnofthetable.Givereasonsforyourchoices. = 2 = 5^ = 3) = _ = )

    [` = 4[ = @B = 0.5b

    EndBehaviour Functions Reasons

    Q3toQ1

    = 2

    = _

    Oddexponent

    Positiveleadingcoefficient

    Q2toQ4 =

    25

    `

    = 4[

    Oddexponent

    Negativeleadingcoefficient

    Q2toQ1 = 5^

    = @B

    Evenexponent

    Positiveleadingcoefficient

    Q3toQ4 = 3)

    = 0.5b

    Evenexponent

    Negativeleadingcoefficient

  • Example4:Foreachofthefollowingfunctionsi)Statethedomainandrangeii)Describetheendbehavioriii)Identifyanysymmetry

    a)b)c)

    =

    = .

    =

    i)Domain:(,) Range:(,) ii)As , andas , Thegraphextendsfromquadrant2to4iii)Pointsymmetryabouttheorigin(0,0)

    i)Domain:(,) Range:[0,) ii)As , andas , Thegraphextendsfromquadrant2to1iii)Linesymmetryabouttheline = 0(they-axis)

    i)Domain:(,) Range:(,) ii)As , andas , Thegraphextendsfromquadrant3to1iii)Pointsymmetryabouttheorigin(0,0)

  • L21.2CharacteristicsofPolynomialFunctionsLessonMHF4UJensenInsection1.1welookedatpowerfunctions,whicharesingle-termpolynomialfunctions.Manypolynomialfunctionsaremadeupoftwoormoreterms.Inthissectionwewilllookatthecharacteristicsofthegraphsandequationsofpolynomialfunctions.NewTerminologyLocalMin/Maxvs.AbsoluteMin/MaxLocalMinorMaxPointPointsthatareminimumormaximumpointsonsomeintervalaroundthatpoint.AbsoluteMaxorMinThegreatest/leastvalueattainedbyafunctionforALLvaluesinitsdomain.Investigation:GraphsofPolynomialFunctionsThedegreeandtheleadingcoefficientintheequationofapolynomialfunctionindicatetheendbehavioursofthegraph.Thedegreeofapolynomialfunctionprovidesinformationabouttheshape,turningpoints(localmin/max),andzeros(x-intercepts)ofthegraph.Completethefollowingtableusingtheequationandgraphsgiven:

    Inthisgraph,(-1,4)isalocalmaxand(1,-4)isalocalmin.Thesearenotabsoluteminandmaxpointsbecausethereareotherpointsonthegraphofthefunctionthataresmallerandgreater.Sometimeslocalminandmaxpointsarecalledturningpoints.

    OnthegraphofthisfunctionThereare3localmin/maxpoints.2arelocalminand1isalocalmax.Oneofthelocalminpointsisalsoanabsolutemin(itislabeled).

  • EquationandGraph DegreeEvenorOdd

    Degree?

    LeadingCoefficient EndBehaviour

    Numberofturningpoints

    Numberofx-intercepts

    = $ + 4 5

    = 3* 4+ 4$ + 5 + 5

    = + 2

    = * 2+ + $ + 2

    = 2- 12* + 18$ + 10

    = 21 + 7* 3+ 18$ + 5

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