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

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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%)

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.

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

(,) (,)

(,) [0,)

= Cubic

(,) (,)

• PowerFunction

SpecialName Graph Domain Range

EndBehaviouras

EndBehaviouras

= Quartic

(,) [0,)

= Quintic

(,) [,)

= Sextic

(,) [0,)

Endbehaviour

as , andas ,

Q2toQ1

Endbehaviour

as , andas ,

Q3toQ4

Domain

(,)

Domain (,)

Range [0,) Range

[0, )

Example:

= 2K

Example: = 3)

Endbehaviour

as , and

as ,

Q3toQ1

Endbehaviour

as , andas ,

Q2toQ4

Domain

(,)

Domain (,)

Range (,) Range

(,)

Example:

= 3[

Example: = 25

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

[` = 4[ = @B = 0.5b

EndBehaviour Functions Reasons

Q3toQ1

= 2

= _

Oddexponent

Q2toQ4 =

25

`

= 4[

Oddexponent

Q2toQ1 = 5^

= @B

Evenexponent

Q3toQ4 = 3)

= 0.5b

Evenexponent

• Example4:Foreachofthefollowingfunctionsi)Statethedomainandrangeii)Describetheendbehavioriii)Identifyanysymmetry

a)b)c)

=

= .

=

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

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

• EquationandGraph DegreeEvenorOdd

Degree?

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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