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The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2018 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Office of Education

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MODULE 4

Polynomial Functions

ALGEBRA II

An Integrated Approach

Standard Teacher Notes

ALGEBRA II // MODULE 4

POLYNOMIAL FUNCTIONS

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

MODULE 4 - TABLE OF CONTENTS

POLYNOMIAL FUNCTIONS

4.1 Scott’s March Madness – A Develop Understanding Task Introduce polynomial functions and their rates of change (F.BF.1, F.LE.3, A.CED.2) Ready, Set, Go Homework: Polynomial Functions 4.1

4.2 You-mix Cubes – A Solidify Understanding Task Graph ! = #$ with transformations and compare to ! = #%. (F.BF.3, F.IF.4, F.IF.5, F.IF.7) Ready, Set, Go Homework: Polynomial Functions 4.2

4.3 Building Strong Roots – A Solidify Understanding Task Understand the Fundamental Theorem of Algebra and apply it to cubic functions to find roots. (A.SSE.1, A.APR.3, N.CN.9) Ready, Set, Go Homework: Polynomial Functions 4.3

4.4 Getting to the Root of the Problem – A Solidify Understanding Task Find the roots of polynomials and write polynomial equations in factored form. (A.APR.3, N.CN.8, N.CN.9) Ready, Set, Go Homework: Polynomial Functions 4.4

4.5 Is This the End? – A Solidify Understanding Task Examine the end behavior of polynomials and determine whether they are even or odd. (F.LE.3, A.SSE.1, F.IF.4, F.BF.3) Ready, Set, Go Homework: Polynomial Functions 4.5

4.6 Puzzling Over Polynomials – A Practice Understanding Task Analyze polynomials, determine roots, end behavior, and write equations (A-APR.3, N-CN.8, N-CN.9, A-CED.2) Ready, Set, Go Homework: Polynomial Functions 4.6


POLYNOMIAL FUNCTIONS– 4.1


4.1 Scott’s March Madness

A Develop Understanding Task

Eachyear,Scottparticipatesinthe“MachoMarch”promotion.The

goalof“MachoMarch”istoraisemoneyforcharitybyfinding

sponsorstodonatebasedonthenumberofpush-upscompleted

withinthemonth.Lastyear,Scottwasproudofthemoneyhe

raised,butwasalsodeterminedtoincreasethenumberofpush-

upshewouldcompletethisyear.

PartI:RevisitingthePast

BelowisthebargraphandtableScottusedlastyeartokeeptrackofthenumberofpush-upshe

completedeachday,showinghecompletedthreepush-upsondayoneandfivepush-ups(fora

combinedtotalofeightpush-ups)ondaytwo.Scott

continuedthispatternthroughoutthemonth.

1 2 3 4

1. Writetherecursiveandexplicitequationsforthenumberofpush-upsScottcompletedonany

givendaylastyear.Explainhowyourequationsconnecttothebargraphandthetableabove.

?Days A(?)

Push-ups

eachday

E(?)

Totalnumberof

pushupsinthe

month

1 3 32 5 83 7 154 9 245 11 35… … ?

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2. Writetherecursiveandexplicitequationfortheaccumulatedtotalnumberofpush-ups

Scottcompletedbyanygivendayduringthe“MachoMarch”promotionlastyear.

PartII:MarchMadness

Thisyear,Scott’splanistolookatthetotalnumberofpush-upshecompletedforthemonthlastyear

(g(n))anddothatmanypush-upseachday(m(n)).

3. Howmanypush-upswillScottcompleteondayfour?Howdidyoucomeupwiththisnumber?

Writetherecursiveequationtorepresentthetotalnumberofpush-upsScottwillcompletefor

themonthonanygivenday.

4. Howmanytotalpush-upswillScottcompleteforthemonthondayfour?

?Days A(?)

Push-upseach

daylastyear

E(?)

Totalnumber

ofpushupsin

themonth

N(?)

Push-upseach

daythisyear

T(n)

Totalpush-ups

completedfor

themonth

1 3 3 3 2 5 8 8 3 7 15 15 4 9

24 5 … … ?

2




5. Withoutfindingtheexplicitequation,makeaconjectureastothetypeoffunctionthatwould

representtheexplicitequationforthetotalnumberofpush-upsScottwouldcompleteonany

givendayforthisyear’spromotion.

6. Howdoestherateofchangeforthisexplicitequationcomparetotheratesofchangeforthe

explicitequationsinquestions1and2?

7. Testyourconjecturefromquestion5andjustifythatitwillalwaysbetrue(seeifyoucanmove

toageneralizationforallpolynomialfunctions).

3




4.1 Scott’s March Madness – Teacher Notes A Develop Understanding Task

Purpose:Thepurposeofthistaskistodevelopstudentunderstandingofhowthedegreeofa

polynomialrelatestotheoverallrateofchange.Lastyear,studentswhodidthetaskScott’sMacho

Marchdiscoveredthatquadraticfunctionscanbemodelsforthesumofalinearfunction,whichcreates

alinearrateofchange.Thisyear,studentswillseethatcubicfunctionscanbemodelsforthesumofa

quadraticfunction.Question7promptsstudentstomakeaconjectureandtomovetothe

generalizationthatthesumofapolynomialofdegreenwillproduceapolynomialofthedegreen+1.In

thistask,studentshavetheopportunitytousealgebraic,numeric,andgraphicalrepresentationsto

modelastorycontextandmakeconnections.

CoreStandardsFocus

F-BF.1Writeafunctionthatdescribesarelationshipbetweentwoquantities.*

a. Determineanexplicitexpression,arecursiveprocess,orstepsforcalculationfromacontext.

b. Combinestandardfunctiontypesusingarithmeticoperations.Forexample,buildafunctionthat

modelsthetemperatureofacoolingbodybyaddingaconstantfunctiontoadecayingexponential,

andrelatethesefunctionstothemodel.

F.LE.3Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceedsa

quantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.

A.CED.2(Createequationsthatdescribenumbersorrelationships)Createequationsintwoormore

variablestorepresentrelationshipsbetweenquantities;graphequationsoncoordinateaxeswith

labelsandscales.

RelatedStandards:F.IF.9,A.CED.1

ThistaskalsofollowsthestructuresuggestedintheModelingstandard:




StandardsforMathematicalPractice:

SMP1–Makesenseofproblemsandpersevereinsolvingthem

SMP2–Reasonabstractlyandquantitatively

SMP7–Lookforandmakeuseofstructure

SMP8–Lookforexpressregularityinrepeatedreasoning

TheTeachingCycle:

Launch:

Beginthetaskbysettingupthescenarioandclarifyingthetaskstartswiththenumberofpush-ups

Scottcompletedlastyearduringtheeventandthenmovesintothenumberofpush-upsheintendsto

completethisyear.Questions1and2fromthetaskmaybefamiliarforstudentsfromSecondaryI

(Scott’sWorkout)andSecondaryII(Scott’sMachoMarch).Youmaywishtohavestudentsworkon

questiononeasawarm-uptodiscusspriortodoingtherestofthetask.

Explore:

Havestudentsworkindependentlyforawhilebeforetheyworkwithapartner.Monitorstudent

thinkingastheywork.Sincestudentsshouldbefamiliarwithrepresentationsforlinearfunctions,do

notallowanygrouptospendtoomuchtimeonquestion1(youmayevenwishtomakethisa‘warm-

up’questioninsteadofpartofthetask).Question2isalsoreviewbutismoredifficult.Lettingstudents

workthroughthiswillbevaluabletowardmakingconnectionsandansweringthequestionsinPartII.

Forthosestudentswhohaveahardtimegettingstarted,encouragethemtouseanotherrepresentation

(visualmodel,table,orgraph)tohelpwritetheequations(bothexplicitandrecursive)astheyworkon

thetask.Question2answers:Recursive:E(1) = 3, E(?) = E(? − 1) + (2? + 1);Explicitequation

(whensimplified):E(?) = ?(? + 2).ThewholegroupdiscussionwillfocusonthequestionsfromPart




II,solookforthosewhomakeconjecturesthatmatchthepurposeofthetask.Itwillbehelpfultohave

studentswhosharetousethevisualmodeltheycreatedtoexplainhowtheyfoundsolutionsandmade

conjectures.Forexample,atableshowingthedifferencecolumnsimilartotheoneintheanswerkey

(solution3)toshowthatcubicfunctionscanbemodelsforthesumofaquadraticfunction--justas

quadraticfunctionscanbemodelsforthesumofalinearequation.

Discuss:

Startthewholegroupdiscussionbyhavingagroupgothroughtheirprocessforansweringquestions2,

3and4(besuretohavethemshowthemodeltheyused-visualdiagram,table,orgraph).Ifpossible,

haveanothergroupsharetheirvisualmodeliftheyusedadifferentrepresentation(atablehighlights

thedifferencecolumnwhereasamodeloragraphvisuallyhighlightstheratesofchangeoftheexplicit

equations).

Afterstudentssharetheirstrategiesforcompletingquestions2-4,movethediscussiontothelastthree

questionsinthetask(questions5-7).Thesequestionsfocusonthetotalnumberofpush-upscompleted

forthemonth,orinotherwords,thecubicrateofchangecreatedasaresultofaddingtheexplicit

quadraticfunctiontocurrentnumberofpush-upscompletedonanygivenday.Thegoalforthisportion

ofthediscussionisforstudentstosummarizethekeypointsthatreflectthepurposeofthistask*.

Observationstobemade:

• Thefirstdifferenceisquadratic,theseconddifferenceislinearandthethirddifferenceis

constant(acharacteristicofacubicfunction).

• Thegraph“curves”(growsatafasterrate)morethanthatofaquadratic(Thisalsohelps

forfutureworkregardingratesofchangeofpolynomialfunctionsversusthatofexponential

functions:whilecubicfunctionsgrowatafasterratethanquadratics,theyarenotasfastas

exponential,whosethirddifferenceisstillexponential).

• Cubicfunctionscanbemodelsforthesumofthetermsofaquadraticfunctioninthesame

waythatquadraticfunctionscanbemodelsforsumoftermsofalinearfunction.(Thisis

similartotherelationshipfromtheSecondaryIItaskScott’sMachoMarchwherestudents

discoveredthatquadraticfunctionscanbemodelsforthesumofalinearfunction,which

createsalinearrateofchange.)




• Recognizethereisarelationshipofthedegreeofapolynomialandtheoverallrateof

change.(Generalization:whenthenthdifferenceisconstant,thenthepolynomialisof

degreen).

• Makeaconjecturethatthesumofapolynomialofdegreenwillproduceapolynomialofthe

degreen+1.

*Ifstudentsstruggletocomeupwiththeaboveobservations,havethemfocusonthedifference

columnsandcomparethethreerecursiveequationsfromquestions1,2,and3(whoseequations

representtheformervalueplusaconstantterm,alinearexpression,andaquadraticexpression

respectively).Askstudentstoidentifysimilaritiesanddifferences.Theimportantthingtonoticeisthat

thechangeinthefunctionistheexpressionaddedtotherecursiveformula.Iftherateofchangeisa

constant,thenthefunctionislinear.Ifthechangeisalinearexpression,thenthefunctionisquadratic.

Ifthechangeisaquadraticexpression,thenthefunctioniscubic.

**Iftimepermits,itisalwaysgoodtoreviewanddiscussfeaturesoffunctions.Inthistask,thefunction

todescribethetotalnumberofpush-upsScottwouldcompletewouldbeagooddiscussionfor

domain/range,wherethesituationisincreasing/decreasing,intercepts,whetherthefunctionis

discreteorcontinuous,andthe‘endbehavior’.Whilethedomainisrestrictedtothenumberofdaysin

March,thisisagoodexampleofdistinguishingbetweentherangeofthesituationversustheend

behavioroftheactualfunction.

AlignedReady,Set,Go:PolynomialFunctions4.1

ALGEBRA 2// MODULE 4

POLYNOMIAL FUNCTIONS – 4.1

4.1


Needhelp?Visitwww.rsgsupport.org

READY Topic:Completinginequalitystatements

Foreachproblem,placetheappropriateinequalitysymbolbetweenthetwoexpressionstomakethestatementtrue.If! > #, %ℎ'(: *+- > 10, %ℎ'(: *+0 < - < 1

1. 3!____3#

4.-3____25

7.-____-3

2. # − !____! − #

5.√-____-3

8.√-____-

3.! + -____# + - 6.-3____-9 9.-____3-

SET Topic:Classifyingfunctions

Identifythetypeoffunctionforeachproblem.Explainhowyouknow.10

- +(-) 1 3 2 6 3 12 4 24 5 48

11.- +(-) 1 3 2 6 3 9 4 12 5 15

12.- +(-) 1 3 2 9 3 18 4 30 5 45

13.- +(-)1 72 93 134 215 37

14.- +(-)1 -262 -193 04 375 98

15.- +(-)1 -42 33 184 415 72

16.WhichoftheabovefunctionsareNOTpolynomials?

READY, SET, GO! Name PeriodDate

4



4.1



GO Topic:Recallinglongdivisionandthemeaningofafactor

Findthequotientwithoutusingacalculator.Ifyouhavearemainder,writetheremainderas

awholenumber.Example: remainder217.

18.

19.Is30afactorof510?Howdoyouknow?


21.

22.



25.

26.

27.Is92afactorof3405? 28.Is27afactorof3564?

21 1497

30 510 13 8359

22 14857 952 40936

92 3405 27 3564

5




4.2 You-mix Cubes A Solidify Understanding Task

InScott’sMarchMadness,thefunctionthatwasgeneratedby

thesumoftermsinaquadraticfunctionwascalledacubicfunction.Linearfunctions,

quadraticfunctions,andcubicfunctionsareallinthefamilyoffunctionscalledpolynomials,

whichincludefunctionsofhigherpowerstoo.Inthistask,wewillexploremoreaboutcubic

functionstohelpustoseesomeofthesimilaritiesanddifferencesbetweencubicfunctionsand

quadraticfunctions.

Tobegin,let’stakealookatthemostbasiccubicfunction,!(#) = #& .Itistechnicallyadegree3polynomialbecausethehighestexponentis3,butit’scalledacubicfunctionbecausethese

functionsareoftenusedtomodelvolume.Thisislikequadraticfunctionswhicharedegree2

polynomialsbutarecalledquadraticaftertheLatinwordforsquare.Scott’sMarchMadness

showedthatlinearfunctionshaveaconstantrateofchange,quadraticfunctionshavealinearrate

ofchange,andcubicfunctionshaveaquadraticrateofchange.

1. Useatabletoverifythat!(#) = #&hasaquadraticrateofchange.

2. Graph!(#) = #& .

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3. Describethefeaturesof!(#) = #&includingintercepts,intervalsofincreaseordecrease,domain,range,etc.4. Usingyourknowledgeoftransformations,grapheachofthefollowingwithoutusingtechnology.

a) !(#) = #& − 3 b) !(#) = (# + 3)&

c) !(#) = 2#& d) !(#) = −(# − 1)& + 2

5. Usetechnologytocheckyourgraphsabove.Whattransformationsdidyougetright?What

areasdoyouneedtoimproveonsothatyourcubicgraphsareperfect?

7




6. Sincequadraticfunctionsandcubicfunctionsarebothinthepolynomialfamilyoffunctions,

wewouldexpectthemtosharesomecommoncharacteristics.Listallthesimilaritiesbetween

!(#) = #&and,(#) = #- .

7. Asyoucanseefromthegraphof!(#) = #& ,therearealsosomerealdifferencesincubicfunctionsandquadraticfunctions.Eachofthefollowingstatementsdescribeoneofthose

differences.Explainwhyeachstatementistruebycompletingthesentence.

a) Therangeof!(#) = #&is(−∞,∞),buttherangeof,(#) = #-is[0, ∞)because:_______________________________________________________________________________________________________

b) For# > 1, !(#) > ,(#)because:_______________________________________________________________________________________________________________________________________________________________

c) For0 < # < 1, ,(#) > !(#)because:__________________________________________________________________________________________________________________________________________________________

8




4.2 You-mix Cubes – Teacher Notes A Solidify Understanding Task Purpose:

Thepurposeofthistaskistoexamineandextendideasaboutcubicfunctionsthatweresurfacedin

3.1Scott’sMachoMarchMadness.Studentsconsiderthebasiccubicfunction,!(#) = #& ,identifying

itscharacteristicsandgraph.Studentswillrecognizethatthegraphof!(#) = #&canbe

transformedusingthesametechniquesasquadraticfunctions.Studentswillalsomakeother

comparisonsbetweencubicandquadraticfunctions,specifically,theirendbehaviorandthevalues

ofeachfunctionwhen#isafraction.

CoreStandardsFocus:

F.BF.3Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecific

valuesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcases

andillustrateanexplanationoftheeffectsonthegraphusingtechnology.Includerecognizingeven

andoddfunctionsfromthegraphsandalgebraicexpressionsforthem.

F.IF.4.Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesof

graphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbal

descriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionis

increasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;end

behavior;andperiodicity.

F.IF.5.Relatethedomainofafunctiontoitsgraphand,whereapplicable,tothequantitative

relationshipitdescribes.Forexample,ifthefunctionh(n)givesthenumberofperson-hoursittakes

toassemblenenginesinafactory,thenthepositiveintegerswouldbeanappropriatedomainfor

thefunction.




F.IF.7:Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin

simplecasesandusingtechnologyformorecomplicatedcases.

a.Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.

c.Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,

andshowingendbehavior.

RelatedStandards:A.SSE.1


SMP2–Reasonabstractlyandquantitatively


Vocabulary:cubicfunction,polynomialfunction,degreeofapolynomial

TheTeachingCycle:

Launch:

Beginthetaskbytellingstudentsthatthecubicfunctionsintroducedintask3.1Scott’sMacho

MarchMadnessandthequadraticandlinearfunctionsthattheyhavepreviouslyworkedwithallfall

intothefamilyofpolynomialfunctions.Atechnicaldefinitionofpolynomialisgivenlater,buta

workingdefinitionisthatapolynomialfunctionisasumoftermsinthesamevariablewiththe

naturalnumberexponentsandrealcoefficients.Demonstrateforstudentshowtoidentifythe

degreeofthepolynomialbasedonthehighest-poweredterm:

Degree1Polynomial(LinearFunction):S = 3# − 1

Degree2Polynomial(QuadraticFunction):S = 3#- − 2# + 9

Degree3Polynomial(CubicFunction):S = −5#& + 3#- − 10# + 6

Degree4Polynomial(QuarticFunction):S = #W + 7#& − 4#- − 24# + 1

Explainthatinthistask,wewillbeintroducingmoreofthecharacteristicsofcubicfunctionsand

comparingthemtowhatweknowaboutquadraticfunctions.




Explore:

Monitorstudentsastheyareworkingonthetasktoseethattheyarecreatinganorganizedtableof

valuesthatcanbeusedthroughoutthetask.Inthetableforproblem#1,itiseasiesttoshowthe

quadraticrateofchangestartingfrom0andlookingatwholenumbervalues.Ifstudentsarestuck,

encouragethemtostartthere.Eventually,theywillneedtoconsiderbothpositiveandnegative

numbersforthegraph,whichcanbeeasilyaddedtothetable.Listenforstudentswhoare

expressingtheideathatthevaluesontheleftsideofthegrapharejusttheoppositeofthevalueson

therightsideofthegraph.

Asstudentsareworkingonthetransformations,supporttheminconnectingtotheirpreviouswork

withfunctionsandidentifyingthetransformationsfirst,andthenusinganchorpointsfromthe

originalgraphtofindtheirnewgraph.

Discuss:

Beginthediscussionbyhavingastudentpresentatableofvaluesthatshowsthefirst,second,and

thirddifferences.Asktheclasstoexplainhowthedifferencesrelatetoeachother,i.e.aconstant

thirddifferencemeansthattheseconddifferenceislinear,alinearseconddifferencemeansthat

thefirstdifferenceisquadratic.Remindstudentsthat,generally,functionsaredefinedbytheir

ratesofchange,andthepatternsinratesofchangethatweanticipatedin3.1Scott’sMachoMarch

Madnessareconsistentacrosspolynomials.

Movethediscussiontothegraphinproblem#2.Askastudenttosharehowtheyusedthetableto

createthegraphandhowtheycanseetherateofchangeinthegraph.Askstudentstodescribe

whythevaluestotheleftofzeroarenegativeforacubicfunctionandreinforcetheideathatwhen

negativenumbersaresquared,theproductispositiveandwhennegativenumbersarecubed,the

productisnegative.Tellstudentsthattheirtableswillhelpthemwiththeanchorpointstheyneed

totransformthegraphof!(#) = #& .Theyshouldbeabletoeasilyknowthepoints(0,0),(1,1),

(-1,-1),(2,8),and(-2,-8)andusethemforthetransformations.




Continuethediscussionbyaskingstudentstosharetheirworkonthegraphinquestion#4.Be

surethatstudentsdemonstratehowtheydeterminedthetransformationsandthenusedthe

anchorpointstolocateandgraphthenewfunction.

Finally,discusstheremainingquestions.Question7cmayproducealivelydiscussionsince

studentsoftenbelievethatcubinganumberalwaysmakesitbiggerthansquaringthenumber.

Afterdiscussinghowthisbeliefisnottrueforfractions,projectagraphof!(#) = #&and,(#) = #-

togetherandzoominoneachofthefollowingintervalstoseewhichfunctionisgreaterinthe

interval:(−∞, −1), (−1, 0), (0, 1), (1,∞).




4.2



READY Topic:Addingandsubtractingbinomials

Addorsubtractasindicated.

1. (6# + 3) + (4# + 5)

2. (# + 17) + (9# − 13) 3. (7# − 8) + (−2# + 9)

4. (4# + 9) − (# + 2)

5. (−3# − 1) − (2# + 5) 6. (8# + 3) − (−10# −

9)

7. (3# − 7) + (−3# − 7)

8. (−5# + 8) − (−5# +

7)

9. (8# + 9) − (7# + 9)

10. Usethegraphsof0(#)and1(#)tosketchthegraphsof0(#) + 1(#)and0(#)– 1(#).

0(#) + 1(#) 0(#)– 1(#).


9



4.2



SET Topic:Comparingsimplepolynomials

11. Completethetablesbelowfor5 = #7895 = #:7895 = #;

x 5 = #--−1

0

1

x 5 = #:

-1—1

0

1

x 5 = #;

-1—1

0

1

12. Whatassumptionmightyoubetemptedtomakeaboutthegraphsof

5 = #, 5 = #:7895 = #;basedonthevaluesyoufoundinthe3tablesabove?

13. Whatdoyoureallyknowaboutthegraphsof5 = #7895 = #:7895 = #;despitethe

valuesyoufoundinthe3tablesabove?

14.Completethetableswiththeadditionalvalues.

x 5 = #−1

−12Q

0

12Q

1

x 5 = #:−1

−12Q

0

12Q

1

x 5 = #;−1 −1

2Q

0

12Q

1

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4.2



15.Graph5 = #7895 = #:7895 = #;ontheinterval[−1, 1],usingthesamesetofaxes.

16.Completethetableswiththeadditionalvalues.

x 5 = #−2

−1

−12Q

0

12Q

1

2

x 5 = #:−2

−1

−12Q

0

12Q

1

2

x 5 = #;−2

−1 −1

2Q

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12Q

1

2

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4.2



17.Graph5 = #7895 = #:7895 = #;ontheinterval[-2,2],usingthesamesetofaxes.

GO Topic:UsingtheexponentrulestosimplifyexpressionsSimplify.

18.#T :Q ∙ #TVQ ∙ #

TWQ 19.7X ;Q ∙ 7

:TYQ ∙ 7

XT;Q 20.ZW

[Q ∙ Z:TWQ ∙ Z

;X\Q

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4.3 Building Strong Roots A Solidify Understanding Task

Whenworkingwithquadraticfunctions,welearned

theFundamentalTheoremofAlgebra:

An"#$degreepolynomialfunctionhas"roots.Inthistask,wewillbeexploringthisideafurtherwithotherpolynomialfunctions.

First,let’sbrushuponwhatwelearnedaboutquadratics.Theequationsandgraphsoffour

differentquadraticequationsaregivenbelow.Findtherootsforeachandidentifywhetherthe

rootsarerealorimaginary.

1.

a)%(') = '* + ' − 6

b)g(') = '* − 2' − 7

Roots: Roots:

Typeofroots: Typeofroots:

c)ℎ(') = '* − 4' + 4

d)2(') = '* − 4' + 5

Roots: Roots:

Typeofroots: Typeofroots:

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2.Didallofthequadraticfunctionshave2roots,aspredictedbytheFundamentalTheoremof

Algebra?Explain.

3.It’salwaysimportanttokeepwhatyou’vepreviouslylearnedinyourmathematicalbagoftricks

sothatyoucanpullitoutwhenyouneedit.Whatstrategiesdidyouusetofindtherootsofthe

quadraticequations?

4.Usingyourworkfromproblem1,writeeachofthequadraticequationsinfactoredform.When

youfinish,checkyouranswersbygraphing,whenpossible,andmakeanycorrectionsnecessary.

a)%(') = '* + ' − 6

b)g(') = '* − 2' − 7

Factoredform:

Factoredform:

c)ℎ(') = '* − 4' + 4

d)2(') = '* − 4' + 5

Factoredform:

Factoredform:

5.Basedonyourworkinproblem1,wouldyousaythatrootsarethesameas'-intercepts?Explain.

6.Basedonyourworkinproblem4,whatistherelationshipbetweenrootsandfactors?

14




Nowlet’stakeacloserlookatcubicfunctions.We’veworkedwithtransformationsof

%(') = '4,butwhatwe’veseensofarisjustthetipoftheiceberg.Forinstance,consider:

6(') = '4 − 3'* − 10'

7.Usethegraphtofindtherootsofthecubicfunction.Usetheequationtoverifythatyouare

correct.Showhowyouhaveverifiedeachroot.

8.Write6(')infactoredform.Verifythatthefactoredformisequivalenttothestandardform.

9.Aretheresultsyoufoundin#7consistentwiththeFundamentalTheoremofAlgebra?Explain.

15




Here’sanotherexampleofacubicfunction.

%(') = '4 + 7'* + 8' − 16

10.Usethegraphtofindtherootsofthecubicfunction.

11.Write%(')infactoredform.Verifythatthefactoredformisequivalenttothestandardform.Makeanycorrectionsneeded.

12.Aretheresultsyoufoundin#10consistentwiththeFundamentalTheoremofAlgebra?

Explain.

13.We’veseenthemostbasiccubicpolynomialfunction,ℎ(') = '4andweknowitsgraphlookslikethis:

Explainhowℎ(') = '4isconsistentwiththeFundamentalTheoremofAlgebra.

16




14.Hereisonemorecubicpolynomialfunctionforyourconsideration.Youwillnoticethatitis

giventoyouinfactoredform.Usetheequationandthegraphtofindtherootsof;(').

;(') = (' + 3)('* + 4)

15.Usetheequationtoverifyeachroot.Showyourworkbelow.

16.Aretheresultsyoufoundin#14consistentwiththeFundamentalTheoremofAlgebra?

Explain.

17.Explainhowtofindthefactoredformofapolynomial,giventheroots.

18.Explainhowtofindtherootsofapolynomial,giventhefactoredform.

17




4.3 Building Strong Roots – Teacher Notes A Solidify Understanding Task Purpose:ThepurposeofthistaskistoextendtheFundamentalTheoremofAlgebrafromquadratic

functionstocubicfunctions.Thetaskasksstudentstousegraphsandequationstofindrootsand

factorsandtoconsidertherelationshipbetweenthem.Studentswillalsoconsiderquadraticand

cubicfunctionswithmultiplerealrootsandimaginaryroots.

CoreStandardsFocus:

A.SSE.1Interpretexpressionsthatrepresentaquantityintermsofitscontext.

a.Interpretpartsofanexpression,suchasterms,factors,andcoefficients.

A.APR.3Identifyzerosofpolynomialswhensuitablefactorizationsareavailable,andusethezeros

toconstructaroughgraphofthefunctiondefinedbythepolynomial.

N.CN.9KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.

(ForSecondaryIII,limittopolynomialswithrealcoefficients).



Vocabulary:roots,factors,x-intercepts,multiplicity

TheTeachingCycle:

Launch(WholeGroup):

BeginthetaskbyintroducingtheFundamentalTheoremofAlgebra.Studentswereintroducedto

thistheoreminrelationtoquadraticfunctionsinpreviouscourses,sothefirstpartofthistaskisto

activatebackgroundknowledgesothattheirunderstandingandstrategiesforquadraticfunctions

canbeextendedtocubicfunctions.Tellstudentsthattoday’sworkwillinvolvedeterminingthe

numberofrootsforfunctionsaswellastherelationshipbetweenrootsandfactors.Studentsmay




needtoberemindedthatrootsarethevaluesof'thatmake%(') = 0.Then,askstudentstowork

onquestions1-6.

Explore(SmallGroup):

Asstudentsareworking,supporttheminusingthegraphstofindroots.Askquestionstohelp

themtomakeconnectionsbetweentheequationsandthegraphs.Listenforstudentswhoareusing

quadraticformulatofindtheexactrootswhentheyarenotobviousonthegraph.Somestudents

maytrytoestimatetheroots;nudgethemtothinkaboutwhattheycoulddotofindexactvalues.In

thecasewheretherootsarenotevidentfromthegraph,askstudentshowtheycouldusethe

equationtofindroots.Whenstudentsareworkingonfindingfactorsfromroots,youmayneedto

remindthemofhowtheysolvedequationsinfactoredform,sothattheythinkabouthowtowork

backwards.

Discuss(WholeGroup):

Beginthediscussionwiththefunctionin#1b.Askapreviously-selectedstudenttosharetheir

workinfindingtheexactvaluesoftherootsusingquadraticformula.Askstudentstofindthe

approximatevaluesandaskwherethosevaluescanbeseenonthegraph.Askanotherstudentto

showhowtheywrotethefactorsfortherootsin#1b.

Next,askastudenttosharehowtheyfoundtherootsfor#1candwroteitinfactoredform

(question2c).AsktheclassifthisfunctionhasthenumberofrootspredictedbytheFundamental

TheoremofAlgebra.Thismaygeneratesomecontroversy.Afterallowingstudentstoshare

arguments,tellthemthatthefactorizationhelpsustoseethatthisfunctionhasamultipleroot.

Explainthatitissaidthatthisisa“rootwithmultiplicityof2”.Remindstudentsthatforquadratics

withmultipleroots,thevertexisgenerallyonthe'-axis.

Haveastudentsharetheirworkfindingtherootsandfactorsforquestion2d.Thiswillserveasa

briefreminderofimaginarynumbers,whichwillbeusedinthistaskandupcomingtasks.

Discussquestions5and6.Manystudentsmayarguethatrootsand'-interceptsarethesame

thing.Ifso,askstudentsiftherootswere'-interceptsinquestion2d.Explainthatarootisdefined




asthevalueof'thatmakes%(') = 0.An'-interceptisdefinedtobethepointwhereagraph

crossesthe'-axis.Ifafunctionhasimaginaryroots,itwillnotcrossthe'-axisattheroots,sothe

rootsarenot'-intercepts.Conventionally,arootiswrittenas' = \,and'-interceptsarewritten

aspoints,e.g.(\, 0).Remindstudentsoftherelationshipbetweenrootsandfactors,thenaskthem

toworkontheremainderofthetask.

Explore(SmallGroups):

Supportstudentsastheyareworkingtoapplysomeofthetechniquesthattheylearnedwith

quadraticfunctionstocubicfunctions.Theyshouldstartwithlookingforrootsthatcanbeeasily

identifiedfromthegraphorafactor,andthenusetheremainingfactortofindtheroots.As

studentsareworkingonquestions10and11,lookforastudentthatinitiallythoughtthat(' + 4)is

asingleroot,andthendiscoveredthatitmustbeamultipleroottomaketheequationcubic.Also

lookforastudentthathasfoundtheimaginaryrootsin#14.

Discuss(WholeGroup):

Starttheseconddiscussionwithquestions#10and#11.Haveastudentselectedduringthe

Exploretimesharehis/herworkinfindingtherootsandhowtheydeterminedthat(' + 4)mustbe

amultipleroot.Asktheclasstocomparehowthecubicgraphlookswhenthereisamultipleroot

andhowthequadraticgraphlookedwiththemultipleroot.Thiswillhelpstudentstorecognize

thatwhenapolynomialtouchesthex-axiswithoutcrossingthrough,itprobablycontainsamultiple

rootofevenmultiplicityatthatpoint.

Askstudentstorespondto#13.Theremaybesomedisagreementaboutthisquestion.Again,have

studentsshareideasandcometotheconclusionthatitmusthaveamultiplerootat' = 0.Point

outthatthegraphcrossesthex-axisatthismultipleroot,whichmakesithardertoidentifyasa

multiplerootthanwhenthegraphtouchestheaxisandbouncesoff,asitdoeswithanevennumber

ofmultipleroots.

Askpreviously-selectedstudentstosharetheirworkon#14and#15thatdemonstratesone

methodforsolvingaquadratictofindimaginaryroots.Pointoutthattherearetwoimaginary




rootsandaskifthatwouldalwayshappen.Studentsmayrecognizethatusingthequadratic

formulaorsquare-rootingbothsidesoftheequationwouldalwaysleadtotwoimaginaryrootsin

similarcases.

Wrapthediscussionupwiththelasttwoquestions(17and18),demonstratinghowtousethe

rootstogetfactorsandhowtousefactoredformofapolynomialtofindroots.


ALGEBRA 2 // MODULE 4


4.3



READY Topic:Practicinglongdivisiononpolynomials

Divideusinglongdivision.(Theseproblemshavenoremainders.Ifyougetone,tryagain.)1. 2.

3.

4.

SET Topic:ApplyingtheFundamentalTheoremofAlgebra

Predictthenumberofrootsforeachofthegivenpolynomialequations.(RememberthattheFundamentalTheoremofAlgebrastates:Annthdegreepolynomialfunctionhasnroots.)5.!(#) = #& + 3# − 10 6.,(#) = #- + #& − 9# − 9 7./(#) = −2# − 48.2(#) = #3 − #- − 4#& + 4# 9.4(#) = −#& + 6# − 9 10.6(#) = #7 − 5#3 + 4#&

x + 3( ) 5x3 + 2x2 − 45x −18 x − 6( ) x3 − x2 − 44x + 84

x + 2( ) x4 + 6x3 + 7x2 − 6x − 8


x −5( ) 3x3 −15x2 +12x −60

18



4.3



Belowarethegraphsofthepolynomialsfromthepreviouspage.Checkyourpredictions.Thenusethegraphtohelpyouwritethepolynomialinfactoredform.11.!(#) = #& + 3# − 10Factoredform:

12.,(#) = #- + #& − 9# − 9Factoredform:

13./(#) = −2# − 4Factoredform:

14.2(#) = #3 − #- − 4#& + 4#Factoredform:

19



4.3



15.4(#) = −#& + 6# − 9Factoredform:

16.6(#) = #7 − 5#3 + 4#&Factoredform:

17.Thegraphsof#15and#16don’tseemtofollowtheFundamentalTheoremofAlgebra,but

thereissomethingsimilarabouteachofthegraphs.Explainwhatishappeningatthepoint(3,0)in#15andatthepoint(0,0)in#16.

GO Topic:SolvingquadraticequationsFindthezerosforeachequationusingthequadraticformula.18.4(#) = #& + 20# + 51

19.4(#) = #& + 10# + 25 20.4(#) = 3#& + 12#

21.4(#) = #& − 11

22.4(#) = #& + # − 1 23.4(#) = #& + 2# + 3

20




4.4 Getting to the Root of the Problem A Solidify Understanding Task

In4.3BuildingStrongRoots,welearnedtopredictthenumberofrootsofapolynomialusingtheFundamentalTheoremofAlgebraandtherelationshipbetweenrootsandfactors.Inthistask,wewillbeworkingonhowtofindalltherootsofapolynomialgiveninstandardform.Let’sstartbythinkingagainaboutnumbersandfactors.1.Ifyouknowthat7isafactorof147,whatwouldyoudotofindtheprimefactorizationof147?Explainyouranswerandshowyourprocesshere:2.Howisyouranswerlikeapolynomialwrittenintheform:!(#) = (# − 7)((# − 3)?Theprocessforfindingfactorsofpolynomialsisexactlyliketheprocessforfindingfactorsofnumbers.Westartbydividingbyafactorweknowandkeepdividinguntilwehaveallthefactors.Whenwegetthepolynomialbrokendowntoaquadratic,sometimeswecanfactoritbyinspection,andsometimeswecanuseourotherquadratictoolslikethequadraticformula.Let’stryit!Foreachofthefollowingfunctions,youhavebeengivenonefactor.Usethatfactortofindtheremainingfactors,therootsofthefunction,andwritethefunctioninfactoredform.3.Function:5(#) = #6 + 3#( − 4# − 12 Factor:(# + 3) Rootsoffunction:Factoredform:

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4.Function:5(#) = #6 + 6#( + 11# + 6 Factor:(# + 1) Rootsoffunction:Factoredform:5.Function:5(#) = #6 − 5#( − 3# + 15 Factor:(# − 5) Rootsoffunction:Factoredform:6.Function:5(#) = #6 + 3#( − 12# − 18 Factor:(# − 3) Rootsoffunction:Factoredform: 7.Function:5(#) = #F − 16 Factor:(# − 2) Rootsoffunction:

Factoredform:

22




8.Function:5(#) = #6 − #( + 4# − 4 Factor:(# − 2G) Rootsoffunction:Factoredform:9.Isitpossibleforapolynomialwithrealcoefficientstohaveonlyoneimaginaryroot?Explain.10.BasedontheFundamentalTheoremofAlgebraandthepolynomialsthatyouhaveseen,makeatablethatshowsallthenumberofrootsandthepossiblecombinationsofrealandimaginaryrootsforlinear,quadratic,cubic,andquarticpolynomials.

23




4.4 Getting To The Root Of The Problem – Teacher Notes A Solidify Understanding Task Purpose:

Thepurposeofthistaskisforstudentstofindrootsofpolynomialsandwritethepolynomialsin

factoredform.Thistaskbuildsonpreviousalgebraicwork,includingfactoring,polynomiallong

division,andquadraticformula.StudentsalsousetheirknowledgeoftheFundamentalTheoremof

Algebratoknowhowmanyrootsafunctionhasandtoconsiderthepossiblecombinationsofreal

andimaginaryrootsforpolynomialsofdegree1-4.Studentslearnthatimaginaryrootsoccurin

conjugatepairsandusethisknowledgetobothfindrootsandknowthepossiblecombinationsof

rootsforpolynomials.

CoreStandardsFocus:

A.APR.3:Identifyzerospolynomialswhensuitablefactorizationsareavailable,andusethezerosto

constructaroughgraphofthefunctiondefinedbythepolynomial.

N.CN.8:Extendpolynomialidentitiestothecomplexnumbers.Forexample,rewrite#( + 4\](# + 2G)(# − 2G).

N.CN.9:KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.

(InSecondaryIII,limittopolynomialswithrealcoefficients).




Vocabulary:conjugatepairs

TheTeachingCycle:




Launch:

Beginthetaskbyaskingstudentsaboutquestion1,whatistheprimefactorizationof147andhow

didyoufindit?Studentswilldescribeaprocessofdividingbytheknownfactorandthenbreaking

itdownuntilallthefactorsareprime.Thisisanalogoustotheworkthatwillbedoneinthistask.

We’llbegivenapolynomialandaknownfactor.We’lldividebytheknownfactorandthenbreak

downtheresultuntilallthefactorsarelinear.Question2illustratesthatbothpolynomialsand

numberscanbewritteninfactoredformorstandardform.Tellstudentsthattheywillneedtouse

everythingtheyknowaboutpolynomialsandallthetoolsintheirtoolkit,likequadraticformula,to

breakdownthepolynomialsandfindtheirrootsandfactors.

Explore:

Asstudentsbeginworkingonproblems#3and#4,youmayencountersomestudentswhohave

graphedthefunctionandseenthattheybothhaveintegerrootsthatareeasilyidentifiedfromthe

graph.Ifso,askthemtoverifytheroots,eitherbylongdivisionorsubstitutionintothefunction.If

thisoccurs,youmaywishtoaskoneofthesestudentstopresenttheirworktotheclass,since

connectingthegraphtotherootsisaveryusefulstrategy.

Supportstudentsastheyareworkingtousefactoring,quadraticformula,takingthesquarerootof

bothsides,etc.tosolvetheequationsandfindtheroots.Lookforstudentsthathaveselected

efficientstrategies,ratherthanjustoneconsistentstrategy,toshare.Studentsshouldbeexposedto

theideathatthequadraticformulaisapowerfultool,butnotalwaysthemostefficienttool.

Studentsarelikelytohavesometroublewith#8.Theycantrytouselongdivision,althoughitis

algebraicallymorecomplicatedthanwhattheyhaveseenpreviously.Theymaynotnoticethatthe

givenfactoristhesameasoneofthefactorsin#7,andtheymayormaynothavenoticedthat

imaginaryrootsoccurinpairs.Thisproblemisdesignedtomotivatethediscussionaboutthisidea.

Letstudentsworkontheproblem,butoncetheystartbecomingfrustrated,tellthemtomoveon

andfinishthetask.Ifyouhaveastudentthatisabletosolve#8,he/sheshouldbeaskedtoshare

duringtheclassdiscussion.




Discuss:

Beginthediscussionwithquestion3.Askastudenttosharethathasusedlongdivisiontodivideby

thegivenfactor(# + 3)andthenfactoredthequadraticthatislefttogettheremainingfactors.Ifa

studentinitiallygraphedthefunctionandthenverifiedtheroots,asdiscussedintheExplore

narrative,thenaskthemtoshare.Comparethetwoapproachesanddiscusshowagraphcan

supportthealgebraicwork.Remindstudentsthatiftheyuseagraphtofindroots,theymustverify

themalgebraically.

Next,discussquestion5.Askapreviously-selectedstudenttosharethathasdividedbythegiven

factorandthensetthequadraticthatisleftequaltozeroandsolvedbysquarerootingbothsidesof

theequation.Itisacommonerrortoforgettouseboththepositiveandnegativeroots.Ifthishas

happenedinclass,remindstudentsthatthiserrorwillcausethemtomissoneoftheroots

predictedbytheFundamentalTheoremofAlgebra.

Discuss#6.Thepresentingstudentshouldhaveusedthequadraticformulatosolvethepolynomial

thatremainedafterdividingbythegivenfactor.Thisproblemprovidestheopportunityto

reinforcehowtosimplifyexpressionsthatresultfromthequadraticformula.

Brieflydiscussquestion7.Ifpossible,haveastudentthathasfactoredtheinitialpolynomialshare

theirworktodemonstratethatthereismorethanonewaytosolvethesetypesofproblems.Be

surethatstudentsnoticethatproblem#6producedapairofimaginaryrootsusingthequadratic

formula,andproblem7producedapairofimaginaryrootsbysquarerootingbothsidesofthe

equation.Tellstudentsthesearecalledconjugatepairs.

Movetoquestion8.Ifyourclassroomclimateallowsstudentstofeelcomfortablesharingwork

thatisn’tentirelysuccessful,youmaywanttohaveastudentsharetheirattemptatlongdivision.

Askstudentsifwhattheyhavenoticedaboutcomplexrootsinpreviousproblemsmaybeusefulin

thisproblem.UsethegraphofthefunctionandtheFundamentalTheoremofAlgebratoarguethat

theremustbeanotherimaginaryroot.Thenshowstudentshowtousetheconjugateofthegiven

roottofindtheremainingroots.Verifyyourworkfortheclassbygraphingboththestandardform




andthefactoredformtoshowtheyareequivalent.Makeexplicittostudentsthatimaginaryroots

occurinconjugatepairs.

Finally,wrapupthediscussionbycompletingthetableforquestion10,showingthepossible

combinationsofrealandimaginaryrootsforeachofthefunctiontypes.




4.4



READY Topic:Orderingnumbersfromleasttogreatest

Orderthenumbersfromleasttogreatest.1.100$ √100 &'()100 100 2+,

2.2-+ −√100 &'() /181 0 (−2)+

3.2, √25 &'()8 2(5,), 5 ≠ 0 (2)-89

4.&'($3$ &'(;5-) &'(<6, &'(>4-+ &'()2$

Refertothegivengraphtoanswerthequestions.Insert>,<, 'B =ineachstatementtomakeittrue.

5.D(0)__________((0)6.D(2)__________((2)7.D(−1)__________((−1)8.D(1)__________((−1)

9.D(5)__________((5)

10.D(−2)__________((−2)


D(5)

((5)

24



4.4



SET Topic:FindingtherootsandfactorsofapolynomialUsethegivenroottofindtheremainingroots.Thenwritethefunctioninfactoredform.

Function Roots Factoredform11.D(5) = 5$ − 135) + 525 − 60 5 = 5

12.((5) = 5$ + 65) − 115 − 66 5 = −6

13.G(5) = 5$ + 175) + 925 + 150 5 = −3

14.J(5) = 5> − 65$ + 35) + 125 − 10 5 = √2

25



4.4



GO Topic:Usingthedistributivepropertytomultiplycomplexexpressions

Multiplyusingthedistributiveproperty.Simplify.Writeanswersinstandardform.15.K5 − √13LK5 + √13L 16.K5 − 3√2LK5 + 3√2L

17.(5 − 4 + 2M)(5 − 4 − 2M) 18.(5 + 5 + 3M)(5 + 5 − 3M)

19.(5 − 1 + M)(5 − 1 − M) 20.K5 + 10 − √2MLK5 + 10 + √2ML

26




4.5 Is This The End?

A Solidify Understanding Task

Inpreviousmathematicscourses,youhavecomparedand

analyzedgrowthratesofpolynomial(mostlylinearand

quadratic)andexponentialfunctions.Inthistask,weare

goingtoanalyzeratesofchangeandendbehaviorby

comparingvariousexpressions.

PartI:Seeingpatternsinendbehavior

1.Inasmanywaysaspossible,compareandcontrastlinear,quadratic,cubic,andexponentialfunctions.2.Usingthegraphprovided,writethefollowingfunctionsvertically,fromgreatesttoleastfor@ = B.Putthefunctionwiththegreatestvalueontopandthefunctionwiththesmallestvalueon

thebottom.Putfunctionswiththesamevaluesatthesamelevel.Anexample,E(F) = FG,hasbeen

placedonthegraphtogetyoustarted.

H(F) = 2J K(F) = FL + FN − 4 Q(F) = FN − 20

ℎ(F) = FT − 4FN + 1 V(F) = F + 30 X(F) = FY − 1

Z(F) = FT [(F) = \]

N^J _(F) = F`

3.WhatdeterminesthevalueofapolynomialfunctionatF = 0?Isthistrueforothertypesof

functions?

4.WritethesameexpressionsonthegraphinorderfromgreatesttoleastwhenFrepresentsa

verylargenumber(thisnumberissolarge,sowesaythatitisapproachingpositiveinfinity).Ifthe

valueofthefunctionispositive,putthefunctioninquadrant1.Ifthevalueofthefunctionis

negative,putthefunctioninquadrantIV.Anexamplehasbeenplacedforyou.

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27




5.WhatdeterminestheendbehaviorofapolynomialfunctionforverylargevaluesofF?

6.WritethesamefunctionsinorderfromgreatesttoleastwhenFrepresentsanumberthatis

approachingnegativeinfinity.Ifthevalueofthefunctionispositive,placeitinQuadrantII,ifthe

valueofthefunctionisnegative,placeitinQuadrantIII.Anexampleisshownonthegraph.

7.WhatpatternsdoyouseeinthepolynomialfunctionsforFvaluesapproachingnegativeinfinity?

Whatpatternsdoyouseeforexponentialfunctions?Usegraphingtechnologytotestthesepatterns

withafewmoreexamplesofyourchoice.

8.Howwouldtheendbehaviorofthepolynomialfunctionschangeiftheleadtermswerechanged

frompositivetonegative?

28




@ = B @ → ∞ @ → −∞

E(F) = FG

K(F) = FG m = FN

m = FL

29

SECONDARY MATH III // MODULE 3



FT

FN − 20

F + 30 FY − 1

2J

é1

2èJ

FG F`

FL + FN − 4 FT − 4FN + 1

35




4.5 Is This The End? – Teacher Notes A Solidify Understanding Task

Purpose:Thepurposeofthistaskistosolidifystudentunderstandingofhowthedegreeofthepolynomialfunctionimpactstherateofchangeandendbehavior.Bycomparingthevaluesof

expressionswith‘extreme’values,studentswillbeableto:• Understandthatthedegreeofthepolynomialisthehighestvaluedwholenumberexponent

andthatthistermdeterminestheendbehavior(regardlessoftheothertermsinthe

expression).

• Determinethatthehigherthedegreeofapolynomial,thegreaterthevalueasFapproaches

infinity.Understandthatwhilethehighestdegreepolynomialhasthegreatestvalueas

F → ∞,thatexponentialfunctionshavethegreatestrateofchange,andthereforethe

greatestvaluewhenFbecomesverylarge.

• Identifydifferencesbetweenevenandodddegreefunctions.Inthistask,studentswillknow

theendbehaviorforodddegreefunctions(asF →−∞, H(F) → −∞andthatbotheven

andodddegreepolynomialfunctionsasF → ∞, H(F) → ∞)aslongastheco-efficientis

positiveandrealizethattheoppositeistrueiftheco-efficientisnegative.

TeacherNote:Makecopiesoftheexpressionsandhavethemcutoutinadvance.Tomakethemost

ofthewholegroupdiscussion,bepreparedbyhavingseveral‘movablecopies’ofeachexpression

(largecutoutsareattachedandbeingabletomoveexpressionsonaSmartboardorusinga

documentcamerawillbehandy).Youwillwantmorethanonesetsothateveryonecanvisuallysee

thechangeinorderdependingonthevalueschosenforF.SeeDiscusssectionformoredetails.

CoreStandardsFocus:

F.LE.3Observeusinggraphsandtablesthataquantityincreasingexponentiallyeventuallyexceedsaquantityincreasinglinearly,quadratically,or(moregenerally)asapolynomialfunction.

A.SSE.1Interpretexpressionsthatrepresentaquantityintermsofitscontext.




a.Interpretpartsofanexpression,suchasterms,factors,andcoefficients.F.IF.4Forafunctionthatmodelsarelationshipbetweentwoquantities,interpretkeyfeaturesofgraphsandtablesintermsofthequantities,andsketchgraphsshowingkeyfeaturesgivenaverbaldescriptionoftherelationship.Keyfeaturesinclude:intercepts;intervalswherethefunctionisincreasing,decreasing,positive,ornegative;relativemaximumsandminimums;symmetries;end

behavior;andperiodicity.

F.BF.3Identifytheeffectonthegraphofreplacingf(x)byf(x)+k,kf(x),f(kx),andf(x+k)forspecificvaluesofk(bothpositiveandnegative);findthevalueofkgiventhegraphs.Experimentwithcasesandillustrateanexplanationoftheeffectsonthegraphusingtechnology.Includerecognizingevenandoddfunctionsfromthegraphsandalgebraicexpressionsforthem.RelatedStandards:F.IF.7StandardsforMathematicalPractice:


SMP3–Constructviableargumentsandcritiquethereasoningofothers

SMP5–Useappropriatetoolsstrategically


Vocabulary:endbehaviorTheTeachingCycle:

Launch(WholeClass):

Beginbyreadingtheintroductioninthetaskandthenexplaintostudentsthatwhiletheywillbe

comparingandorderingexpressions,theyshouldalsopayattentiontowhytheexpressionsareina

particularorder.Tellstudentsthattheyshouldbelookingforpatternsthattheycangeneralizeand

use.




Explore(SmallGroup):

Startthetaskbyhavingstudentsworkindividuallyandthenworktogetherinsmallgroupsonthe

restofthetask.Asyoumonitor,listenforstudentreasoningthroughout,payingparticularattention

whentheyarediscussingtheorderoftheexpressionsinquestions3and4(connectingtoend

behavior).Takenoteofhowdifferentgroupsaredeterminingorder.Aretheyplugginginvalues?

Aretheylookingatagraphicalrepresentation?Aretheymakingassumptionsbasedontheir

knowledgeofexponents?Asstudentsanswerquestion5,pressforthemtoexplaintheirreasoning.

Listenforanswersthatincludecomparingpolynomialstoexponentialfunctions(andthat

exponentialfunctionswithabasegreaterthanonewillalwaysoutgrowanypolynomialfunction).

Alsolistenforanswersthatdistinguishbetweenwhenthedegreeofthepolynomialisevenversus

whenitisodd(andhowthisimpactsthevaluesasF →−∞).

Discuss(WholeClass):

Whenallstudentshavehadtheopportunitytoanswerquestion6(somemayhavealready

completedallquestions),movetothewholegroupdiscussion.Beginwithastudentthatshares

theirorderingforquestion2.Askthestudenttoexplainhowtheyfiguredouttheorder.During

thisexplanation,thereshouldbesomediscussionofsubstitutingin0forxandfindingthattheonly

termleftinapolynomialistheconstantterm.Asktheclassifthiswillbethecaseforevery

polynomialfunction.Asktheclassifthisistrueofallfunctions.Theyshouldrecognizethatsome

functions,likelogarithmsareundefinedatF = 0,andsomefunctions,likeexponentials,dependon

morethanjustaconstantterm.

Next,askastudenttosharetheirorderingofthefunctionsfor#4.Useeithertheaxesprovidedin

thetaskorpostalargersetofaxesandusethelargefunctioncards,thatareprovidedfollowingthe

teachernotes,sothattheorderisveryvisualforstudents.Focusfirstontheorderofthe

polynomialfunctions.Besurethatstudentsnoticethatthehighest-poweredtermessentially

determinestheendbehaviorforlargevaluesofx.Lookspecificallyath(x)andr(x).Sincethese

twofunctionsarebothdegree5polynomials,thentheyaremoredifficulttoorder.Askforstudents

tosharedifferentideasforhowtocomparethesetwofunctions,includingusingtechnologyto

graphbothfunctionsandcomparingthemforlargevalues,substitutinglargenumbersintoboth




functionsandcomparing,andreasoningabouttheeffectoftheadditionaltermsinh(x).Itislikely

thattherewillbesomecontroversyaroundwhetherornottheexponentialfunctionisthegreatest

asF → ∞.Lookforsimilarstrategiestocomparetheexponentialwiththehighest-powered

polynomial.Then,askstudentstoconsidertheratesofchangeofthetwofunctionstothinkabout

whichoneisthegreatestforlargevaluesofx.Havestudentscalculatetheaveragerateofchangeof

bothfunctionsfrom[15,20]toseethattherapidrateofchangeoftheexponentialfunctionwill

eventuallymakeitexceedanypolynomial.

Next,askastudenttosharetheirorderingofthefunctionsforquestion#6.Again,makethevisual

availablesothatallstudentscaneasilyseethepatternthateven-poweredpolynomialsapproach

infinityasF → −∞andodd-poweredpolynomialsapproachnegativeinfinityasF → −∞.Ask

studentsifthisisconsistentwiththeirexperienceofquadraticandcubicfunctions,andwhyit

wouldbeso.Askhowthisconnectstothenumberandtypesofrootsthatwerefoundfor

polynomialfunctionsintheprevioustask.(Bepreparedtopullthechartoutaspartofthe

discussion.)Alsodiscussquestion#8andtheeffectofchangingthesignoftheleadcoefficientina

polynomial.

AskstudentstoshareasmanyoftheproblemsinPartIIastimeallows.Besurethattheyareusing

theresultsofPartIandthecorrectnotationtorecordtheirresults.

Finallyaskstudentsfortheirobservationsaboutevenfunctions.Highlightobservationsthat

includetheideathatthegraphsaresymmetricaboutthey-axis.Askstudentshowthedefinition

givenrelatestothesymmetryofthegraphsandhelpthemtoarticulatethattheoutputofthe

functionisthesameforbothpositiveandnegativevaluesofFinevenfunctions.Thisistrueofthe

parentfunctionforalleven-degreepolynomialsandiftheyhaveaverticaltranslationand/ora

verticalstretch.

Repeattheprocessbyaskingstudentsfortheirobservationsaboutoddfunctions.Itmaybeharder

torecognizethe180˚rotationabouttheorigin.Demonstratingthisideawithapieceofpattypaper

mayhelpstudentstoseeitmoreeasily.Reinforcehowthedefinitionofanoddfunctioncreatesthis




patterninthegraphofoddfunctions.Askstudentshowtotellifanodd-degreepolynomialwillbe

anoddfunction.Supportthemtothinkaboutwhichtransformationsofanodd-degreeparent

functionlikeH(F) = FLwillresultinanoddfunction.Closethediscussionbyaskingstudentsto

shareacoupleofexamplesoffunctionsthatareneitheroddnoreven.Manyoftheexamplesfrom

earlierinthetaskcanbeusedforthispurpose.





4.5


READY Topic:Recognizingspecialproducts

Multiply.1.(" + 5)(" + 5) 2.(" − 3)(" − 3) 3.(( + ))(( + ))

4.Inproblems1–3theanswersarecalledperfectsquaretrinomials.Whatabouttheseanswersmakesthembeaperfectsquaretrinomial?

5.(" + 8)(" − 8) 6.+" + √3-+" − √3- 7.(" + ))(" − ))

8.Theproductsinproblems5–7endupbeingbinomials,andtheyarecalledthedifferenceoftwosquares.Whatabouttheseanswersmakesthembethedifferenceoftwosquares?

Whydon’ttheyhaveamiddletermliketheproblemsin1–3?9.(" − 3)(". + 3" + 9) 10.(" + 10)(". − 10" + 100) 11.(( + ))((. − () + ).)

12.Theworkinproblems9–11makesthemfeelliketheanswersaregoingtohavealotofterms.Whathappensintheworkoftheproblemthatmakestheanswersbebinomials?

Theseanswersarecalledthedifferenceoftwocubes(#9)andthesumoftwocubes(#10

and#11.)Whatabouttheseanswersmakesthembethesumordifferenceoftwocubes?


30




4.5


SET Topic:Determiningvaluesofpolynomialsatzeroandat±∞.(Endbehavior)

Statethey-intercept,thedegree,andtheendbehaviorforeachofthegivenpolynomials.13.5(") = "7 + 7"9 − 9": + ". − 13" + 8y-intercept:Degree:Endbehavior:As" → −∞, 5(") → __________As" → +∞, 5(") → __________

14.?(") = 3"9 + ": + 5". − " − 15y-intercept:Degree:Endbehavior:As" → −∞, ?(") → __________As" → +∞, ?(") → __________

15.ℎ(") = −7"A + ".y-intercept:Degree:Endbehavior:As" → −∞, ℎ(") → __________As" → +∞, ℎ(") → __________

16.B(") = 5". − 18" + 4y-intercept:Degree:Endbehavior:As" → −∞, B(") → __________As" → +∞, B(") → __________

17.D(") = ": − 94". − " − 20y-intercept:Degree:Endbehavior:As" → −∞, D(") → __________As" → +∞, D(") → __________

18.F = −4" + 12y-intercept:Degree:Endbehavior:As" → −∞, F → __________As" → +∞, F → __________

31




4.5


Topic:Identifyingevenandoddfunctions19.Identifyeachfunctionaseven,odd,orneither.a)5(") = ". − 3

b)5(") = ".

c)5(") = (" + 1).

d)5(") = ":

e)5(") = ": + 2

f)5(") = (" − 2):

GO Topic:Factoringspecialproducts

Fillintheblanksonthesentencesbelow.20.TheexpressionGH + HGI+ IHiscalledaperfectsquaretrinomial.Icanrecognizeitbecause

thefirstandlasttermswillalwaysbeperfect___________________________________________.

Themiddletermwillbe2timesthe______________________________and_______________________________.

Therewillalwaysbea__________________signbeforethelastterm.

Itfactorsas(__________________)(__________________).

32




4.5


21.TheexpressionGH − IHiscalledthedifferenceof2squares.Icanrecognizeitbecauseit’sa

binomialandthefirstandlasttermsareperfect_________________________________________________.

Thesignbetweenthefirsttermandthelasttermisalwaysa______________________________.

Itfactorsas(__________________)(__________________).

22.TheexpressionGJ + IJiscalledthesumof2cubes.Icanrecognizeitbecauseit’sabinomial

andthefirstandlasttermsare_____________________________________.Theexpression(: + ):factors

intoabinomialandatrinomial.Icanrememberitasashort(______)andalong(________________).

Thesignbetweenthetermsinthebinomialisthe_____________________asthesigninthe

expression.Thefirstsigninthetrinomialisthe_________________________ofthesigninthe

binomial.That’swhyallofthemiddletermscancelwhenmultiplying.

Thelastsigninthetrinomialisalways____________.

Itfactorsas(__________________)(___________________________________).

Factorusingwhatyouknowaboutspecialproducts.23.25". + 30 + 9

24.". − 16 25.": + 27

26.49". − 36 27.": − 1

28.64". − 240 + 225

33




PartII:Usingendbehaviorpatterns

Foreachsituation:

• Determinethefunctiontype.Ifitisapolynomial,statethedegreeofthepolynomialand

whetheritisanevendegreepolynomialoranodddegreepolynomial.

• Describetheendbehaviorbasedonyourknowledgeofthefunction.Usetheformat:

AsF → −∞, H(F) → ______r[srtF → ∞H(F) → ______

1. H(F) = 3 + 2F

Functiontype:

Endbehavior:AsF → −∞, H(F) → ______

Endbehavior:AsF → ∞, H(F) → ______

2.H(F) = FY− 16

Functiontype:



3.H(F) = 3J

Functiontype:

Endbehavior:AsF →−∞, H(F) → ______


4.H(F) = FL+ 2F

N− F + 5

Functiontype:



5.H(F) = −2FL+ 2F

N− F + 5

Functiontype:



6.H(F) = EvQNF

Functiontype:



Usethegraphsbelowtodescribetheendbehaviorofeachfunctionbycompletingthe

statements.

7. 8.





34




9.Howdoestheendbehaviorforquadraticfunctionsconnectwiththenumberandtypeof

rootsforthesefunctions?Howdoestheendbehaviorforcubicfunctionsconnectwiththe

numberandtypeofrootsforcubicfunctions?

PartIII:EvenandOddFunctions

Somefunctionsthatarenotpolynomialsmaybecategorizedasevenfunctionsoroddfunctions.

Whenmathematicianssaythatafunctionisanevenfunction,theymeansomethingveryspecific.

1.Let’sseeifyoucanfigureoutwhatthedefinitionofanevenfunctioniswiththeseexamples:

Evenfunction:

H(F) = FN

Notanevenfunction:

Q(F) = 2J

Differences:

Evenfunction:

H(F) = FY− 3

Notanevenfunction:

Q(F) = F(F + 3)(F − 2)

Differences:

35




Evenfunction:

H(F) = −|F| + 4

Notanevenfunction:

Q(F) = −|F + 4|

Differences:

Evenfunction:

H(2) = 5r[sH(−2) = 5

Notanevenfunction:

Q(2) = 3r[sQ(−2) = 5

Differences:

2.Whatdoyouobserveaboutthecharacteristicsofanevenfunction?

3.Thealgebraicdefinitionofanevenfunctionis:

�(@)isanevenfunctionifandonlyif�(@) = �(−@)forallvaluesof@inthedomainof�.

Whataretheimplicationsofthedefinitionforthegraphofanevenfunction?

4.Arealleven-degreepolynomialsevenfunctions?Useexamplestoexplainyouranswer.

5.Let’strythesameapproachtofigureoutadefinitionforoddfunctions.

36




Oddfunction:

H(F) = FL

Notanoddfunction:

Q(F) = logN F

Differences:

Oddfunction:

H(F) = −FT

Notanoddfunction:

Q(F) = FL+ 3F − 7

Differences:

Oddfunction:H(F) =]

J

Notanoddfunction:

Q(F) = 2F − 3

Differences:

Oddfunction:

H(2) = 3r[sH(−2) = −3

Notanoddfunction:

Q(2) = 3r[sQ(−2) = 5

Differences:

37




6.Whatdoyouobserveaboutthecharacteristicsofanoddfunction?

7.Thealgebraicdefinitionofanoddfunctionis:

�(@)isanoddfunctionifandonlyif�(−@) = −�(@)forallvaluesof@inthedomainof�.

Explainhoweachoftheexamplesofoddfunctionsabovemeetthisdefinition.

8.Howcanyoutellifanodd-degreepolynomialisanoddfunction?

9.Areallfunctionseitheroddoreven?

38




4.6 Puzzling Over Polynomials A Practice Understanding Task

Foreachofthepolynomialpuzzlesbelow,afewpiecesofinformationhavebeengiven.Yourjobistousethosepiecesofinformationtocompletethepuzzle.Occasionally,youmayfindamissingpiecethatyoucanfillinyourself.Forinstance,althoughsomeoftherootsaregiven,youmaydecidethatthereareothersthatyoucanfillin.

1.

Function(infactoredform)Function(instandardform)Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):-2,1,and1Valueofleadingco-efficient:-2Degree:3

Graph:

CC

BY

Just

in T

aylo

r

http

s://f

lic.k

r/p/

4fU

zTo

39




2.Function(infactoredform)Function(instandardform)Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):

2 + /, 4, 0Valueofleadingco-efficient:1Degree:4

Graph:

3.

Function:)($) = 2($ − 1)($ + 3)5Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Valueofleadingco-efficient:Domain:Range:AllRealnumbers

Graph:

40




4.

Function:Endbehavior:!"$ → −∞, )($) → ∞!"$ → ∞, )($) → _____Roots(withmultiplicity):(3,0)m:1;(-1,0)m:2(0,0)m:2Valueofleadingco-efficient:-1Domain:Range:

Graph:

5.

Function:Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Valueofleadingco-efficient:1Domain:Range:Other:)(0)=16

Graph:

41




6.

Function(instandardform):)($) = $6 − 2$5 − 7$ + 2

Function(infactoredform):Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):-2Domain:Range:

Graph:

7.

Function(instandardform):)($) = $6 − 2$

Function(infactoredform):Endbehavior:!"$ → −∞, )($) → _____!"$ → ∞, )($) → _____Roots(withmultiplicity):Domain:Range:

Graph:

42




4.6 Puzzling Over Polynomials – Teacher Notes A Practice Understanding Task Purpose:Thepurposeofthistaskistietogetherwhatstudentshavelearnedaboutroots,end

behavior,operations,andgraphsofpolynomials.Eachproblemgivesafewfeaturesofapolynomial

andasksstudentstofindotherfeatures,sometimesincludingthegraphs.Thiswillrequirethemto

knowtheendbehaviorofapolynomial,basedonthedegree,andtofindalltheroots,givensomeof

theroots.Inmostcases,studentsareaskedtowritetheequationofthepolynomialortowritethe

equationinadifferentformthanisgiven.

CoreStandardsFocus:

F-IF.7:Graphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin

simplecasesandusingtechnologyformorecomplicatedcases.�

a.Graphlinearandquadraticfunctionsandshowintercepts,maxima,andminima.

c.Graphpolynomialfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and

showingendbehavior.(3.2,

A-APR.3:Identifyzerospolynomialswhensuitablefactorizationsareavailable,andusethezerosto

constructaroughgraphofthefunctiondefinedbythepolynomial.

N-CN.8:Extendpolynomialidentitiestothecomplexnumbers.Forexample,rewrite

$5 + 4!"($ + 2/)($ − 2/).

N-CN.9:KnowtheFundamentalTheoremofAlgebra;showthatitistrueforquadraticpolynomials.

(InSecondaryIII,limittopolynomialswithrealcoefficients).

A-CED:Createequationsthatdescribenumbersorrelationships

2.Createequationsintwoormorevariablestorepresentrelationshipsbetweenquantities;graph

equationsoncoordinateaxeswithlabelsandscales.






SMP6–Attendtoprecision

TheTeachingCycle:

Launch:

Launchthetaskbytellingstudentsthattheyaregivenseveralpolynomialpuzzlesthatwillhelp

themtoconnectalltheworkthattheyhavedoneinthemodulesofar.Ineachpuzzletheyaregiven

someinformationandtheyneedtousethatinformationtofindtheotherinformation.Sometimes

thepuzzlesarelittlesneakybecauseinformationthattheyneedisnotgivendirectly,butifthey

thinkaboutwhattheyknow,theycanfindit.

Explore:

Monitorstudentsastheyworktoseewhichproblemsarecausingdifficultysothattheycanbeused

intheclassdiscussion.Youmayneedtosupportstudentsinsolvingproblem2.Oneimaginary

rootisgiven,andtheywillneedtoknowthattheotherrootistheconjugate.Ifstudentsarestuck,

askquestionstohelpremindthemoftherelationshipbetweenimaginaryrootsofapolynomial.As

studentsworkonproblemslike#5and#6,allowthemtousethegivengraphtofindtheinteger

rootsandthenrequirethemtousethoserootstofindtheremainingirrationalorimaginaryroots

usinglongdivision.

Discuss:

Discussasmanyoftheproblemsastimeallows,withpreviously-selectedstudentssharingtheir

workforeachproblem.Startwiththeproblemsthatweremoredifficultsothatstudentshavea

chancetoseehowtoworkthroughsomeofthedifficulties.Afterastudentpresentstheirwork,ask

therestoftheclasstosummarizetheprocessthatthepresentingstudentusedwiththefollowing

sentenceframe:

Intheproblem,weweregiven_________________________________andsoweknew_____________________.

Thenwewereabletofind________________________________by____________________________________________.

Anexampleofusingthesentenceframewithproblem#2is:




Intheproblem,weweregiven2realrootsand1imaginaryrootandsoweknewtherewas

anotherimaginaryroot,[ − \.

Thenwewereabletofindtheequationbywritingeachrootasafactorandmultiplyingthem

together.





4.6


READY Topic:Reducingrationalnumbersandexpressions

Reducetheexpressionstolowestterms.(Assumenodenominatorequals0.)

1.!"#"$

2.&∙(∙"∙"∙"∙)!∙(∙"∙)∙)

3.*+,$

*+,$ 4.

(".&)("01)(".&)("01)

5.(!"0()(".2)("03)(!"0()

6.(&"033)(!".3*)(&"033)(!"0()

7.(4"0*)(".!)4"(".!)(&"0!)

8.!"(&".*)("03)(#"0()"(&".*)("03)(#"0()

9.Whyisitimportantthattheinstructionssaytoassumethatnodenominatorequals0?

SET Topic:Reviewingfeaturesofpolynomials

Someinformationhasbeengivenforeachpolynomial.Fillinthemissinginformation.

10.

Graph:Function:5(6) = 6!Functioninfactoredform:Endbehavior:As6 → −∞, 5(6) → _______As6 → ∞, 5(6) → ______

Roots(withmultiplicity):

Degree:

Valueofleadingco-efficient:


43




4.6


11.Graph:

Functioninstandardform:

Functioninfactoredform:=(6) = −6(6 − 2)(6 − 4)

Endbehavior:As6 → −∞,=(6) → _______As6 → ∞,=(6) → ______Roots(withmultiplicity):Degree:Valueofleadingco-efficient:

12.Graph:

Functioninstandardform:ℎ(6) = 6! − 26& − 36

Functioninfactoredform:

Endbehavior:As6 → −∞, ℎ(6) → _______As6 → ∞, ℎ(6) → ______Roots(withmultiplicity):Degree:ValueofB(C):

13.Graph:Functioninstandardform:

Functioninfactoredform:Endbehavior:As6 → −∞, 5(6) → _______As6 → ∞, 5(6) → ______


Degree:

y-intercept:

44




4.6


14.Graph:

Functioninstandardform:

Functioninfactoredform:

Endbehavior:As6 → −∞, E(6) → _______As6 → ∞, E(6) → ______


Degree:

Valueofleadingcoefficient:

15.Graph:

Functioninstandardform:F(6) = 6! + 26& + 6 + 2

Functioninfactoredform:Endbehavior:As6 → −∞, F(6) → _______As6 → ∞, F(6) → ______


6 = H

Degree:

y-intercept:

16.Finishthegraphifitis

anevenfunction.

17.Finishthegraphifit

isanoddfunction.

45




4.6


GO Topic:Writingpolynomialsgiventhezerosandtheleadingcoefficient

Writethepolynomialfunctioninstandardformgiventheleadingcoefficientandthezerosofthefunction.

18.Leadingcoefficient:2; JKKLM:2, √2,−√2

19.Leadingcoefficient:−1; JKKLM:1, 1 + √3, 1 − √3

20.Leadingcoefficient:2; JKKLM:4H, −4H

Fillintheblankstomakeatruestatement.21.If5(P) = 0,thenafactorof5(P)mustbe____________________________________.

22.Therateofchangeinalinearfunctionisalwaysa______________________________________.

23.Therateofchangeofaquadraticfunctionis_______________________________________________.

24.Therateofchangeofacubicfunctionis____________________________________________________.

25.Therateofchangeofapolynomialfunctionofdegreencanbedescribedbyafunctionofdegree

________________________.

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