4.4 Are You Rational? g z - Utah Education Networkthe case when the numerator of the fraction is...

$: 4.4 Are You Rational? g z - Utah Education Networkthe case when the numerator of the fraction is greater than the denominator. So, let’s take a closer look at the rational function$
SECONDARY MATH III // MODULE 4

RATIONAL EXPRESSIONS & FUNCTIONS -4.4

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

4.4 Are You Rational?

A Solidify Understanding Task

BackinModule3whenwewereworkingwithpolynomials,

itwasusefultodrawconnectionsbetweenpolynomialsand

integers.Inthistask,wewilluseconnectionsbetween

rationalnumbersandrationalfunctionstohelpustothink

aboutoperationsonrationalfunctions.

1.Inyourownwords,definerationalnumber.

Circlethenumbersbelowthatarerationalandrefineyourdefinition,ifneeded.

3 − 5 <=<>

=142.7√52=3C=DEF<9

H>

2.Theformaldefinitionofarationalfunctionisasfollows:

AfunctionL(N)iscalledarationalfunctionifandonlyifitcanbewrittenintheformL(N) = Q(N)R(N)

whereQSTURarepolynomialsinNandRisnotthezeropolynomial.

Interpretthisdefinitioninyourownwordsandthenwritethreeexamplesofrationalfunctions.

3.Howarerationalnumbersandrationalfunctionssimilar?Different?

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Nowwearegoingtousewhatweknowaboutrationalnumberstoperformoperationsonrational

expressions.Thefirstthingweoftenneedtodoistosimplifyor“reduce”arationalnumberor

expression.Thenumbersandexpressionsarenotreallybeingreducedbecausethevalueisn’t

actuallychanging.Forinstance,2/4canbesimplifiedto½,butasthediagramshows,thesearejust

twodifferentwaysofexpressingthesameamount.

Let’stryusingwhatweknowaboutsimplifyingrationalnumberstosimplifyrationalexpressions.

Fillinanymissingpartsinthefractionsbelow.

Given: 24

30 4.

j< − j − 6j< − 4

5.j< + 8j + 15j< + 9j + 18

Lookforcommonfactors:

2 ∙ 2 ∙ 2 ∙ 32 ∙ 3 ∙ 5

(j + 2)(j − 2)

Dividenumeratoranddenominatorbythesamefactor(s):

o ∙ 2 ∙ 2 ∙ po ∙ p ∙ 5

Writethesimplifiedform:

45

j − 3j − 2

j + 5j + 6

6.Whydoesdividingthenumeratoranddenominatorbythesamefactorkeepthevalueoftheexpressionthesame?

7.Ifyouweregiventheexpressionr

rsCt,woulditbeacceptabletoreduceitlikethis:

NNo − 1

=1

j − 1

Explainyouranswer.

22




In4.3RationalThinking,welearnedtopredictverticalandhorizontalasymptotes,andtofindinterceptsforgraphingrationalfunctions.

8.Givenv(j) = rsCrCwrsCx

,predicttheverticalandhorizontalasymptotesandfindtheintercepts.9.Usetechnologytoviewthegraph.Wereyourpredictionscorrect?Whatoccursonthegraphatj = −2?Rationalnumberscanbewrittenaseitherproperfractionsorimproperfractions.10.Describethedifferencebetweenproperfractionsandimproperfractionsandwritetwoexamplesofeach.Arationalexpressionissimilar,exceptthatinsteadofcomparingthenumericvalueofthenumeratoranddenominator,thecomparisonisbasedonthedegreeofeachpolynomial.Therefore,arationalexpressionisproperifthedegreeofthenumeratorislessthanthedegreeofthedenominator,andimproperotherwise.Inotherwords,improperrationalexpressionscanbewrittenas{(r)

|(r),where}(j)}~�Ä(j)arepolynomialsandthedegreeof}(j)isgreaterthanorequaltothe

degreeofÄ(j).11.Labeleachrationalexpressionasproperorimproper.

(rÅt)

(rC<)(rÅ<) rÇC=rsÅÉrCt

rsCxrÅx (rÅ=)(rÅ<)

rÑCx rÅ=

rÅÉr

ÇCÉrÅ<rCt>

Aswemayremember,improperfractionscanberewritteninanequivalentformwecallamixed

number.Ifthenumeratorisgreaterthanthedenominatorthenwedividethenumeratorbythe

denominatorandwritetheremainderasaproperfraction.Inmathtermswewouldsay:

23




If} > Ä, thenthefraction {|canberewritenas {

|= Ü + á

|,whereqrepresentsthequotientandr

representstheremainder.

12.Rewriteeachimproperfractionasanequivalentmixednumber.

a)=HÉ= b)tÉ>

t<=

Rationalexpressionsworktheverysameway.Iftheexpressionisimproper,thenumeratorcanbe

dividedbythedenominatorandtheremainderiswrittenasafraction.Inmathematicalterms,we

wouldsay:{(r)|(r)

= Ü(j) + á(r)|(r)

whereÜ(j)representsthequotientandâ(j)representstheremainder.

Tryityourself!Labeleachrationalexpressionasproperorimproper.Ifitisimproper,thendivide

thenumeratorbythedenominatorandwriteitinanequivalentform.

13.rsÅÉrÅHrÅ<

14. CÉrÅt>rÇÅwrsÅ=rCt

15.rsÅ<rÅÉrÅ=

16.=rÅãrCt

24




In4.3RationalThinking,whenwelookedatthegraphsofrationalfunctions,wedidnotconsider

thecasewhenthenumeratorofthefractionisgreaterthanthedenominator.So,let’stakeacloser

lookattherationalfunctionfrom#13.

16.Letv(j) = j2+5j+7j+2 .Wheredoyouexpecttheverticalasymptoteandtheinterceptsto

be?

17.Usetechnologytographthefunction.Relatethegraphofthefunctiontotheequivalent

expressionthatyouwrote.Whatdoyounotice?

18.Let’strythesamethingwith#15.Letv(j) = j2+2j+5j+3 .Findtheverticalasymptote,the

intercepts,andthenrelatethegraphtotheequivalentexpressionforv(j).

19.Usingthetwoexamplesabove,writeaprocessforpredictingthegraphsofrationalfunctions

whenthedegreeofthenumeratorisgreaterthanthedegreeofthedenominator.

25




4.4 Are You Rational? – Teacher Notes A Solidify Understanding Task Purpose:

Inthistask,rationalfunctionsareformallydefinedandconnectedtorationalnumbers.Students

willusetheseconnectionstoreducerationalfunctionsandtoidentifyrationalfunctionsthatare

improperandwritetheminanequivalentform.Studentswillgraphrationalfunctionsthatcanbe

reduced,seeingthatthegraphisequivalenttothereducedfunctionexceptthatthereisaholein

thegraph,producedbythefactorthatisreduced.Theywillalsographimproperrationalfunctions

andidentifytheslantasymptote.

CoreStandardsFocus:

F.IF.7dGraphfunctionsexpressedsymbolicallyandshowkeyfeaturesofthegraph,byhandin

simplecasesandusingtechnologyformorecomplicatedcases.*

d.Graphrationalfunctions,identifyingzeroswhensuitablefactorizationsareavailable,and

showingendbehavior.

A.APR.6Rewritesimplerationalexpressionsindifferentforms;write{(r)|(r)

intheformsuchthat

{(r)|(r)

= Ü(j) + á(r)|(r)

where}(j), Ä(j), Ü(j)}~�â(j)arepolynomialswithdegreeofâ(j)lessthan

thedegreeofÄ(j),usinginspection,longdivision,orforthemorecomplicatedexamples,a

computeralgebrasystem.

A.APR.7Understandthatrationalexpressionsformasystemanalogoustotherationalnumbers,

closedunderaddition,subtraction,multiplicationanddivisionbyanonzerorationalexpression;

add,subtract,multiplyanddividerationalexpressions.

A.SSE.3Chooseandproduceanequivalentformofanexpressiontorevealandexplainproperties

ofthequantityrepresentedbytheexpression.




StandardsforMathematicalPractice:

SMP2–Reasonabstractlyandquantitatively

SMP8–Lookforandexpressregularityinrepeatedreasoning

Vocabulary:Slantasymptote

TheTeachingCycle:

Launch--Part1(WholeGroup):

Beginthetaskbyaskingstudentsquestion#1,“Whatisarationalnumber?”Askthemtolookatthe

numbersgiveninquestion#1andtodecidewhicharerational.Discussresponsesanddefine

rationalnumbersas“anynumberthatcanbeexpressedasthequotientorratio,p/qoftwo

integers,anumeratorpandanon-zerodenominatorq.”Then,discusstheformaldefinitionof

rationalfunctionsthatisgiveninquestion#2andconnectittotheinformationdefinitiongivenin

task4.3,thatrationalfunctionsarearatioofpolynomials.Shareafewoftheexamplesthat

studentshavewrittenandtellstudentsthatrationalfunctionsbehavemuchlikerationalnumbers.

Tellstudentsthatinthistask,theywillbeworkingwithrationalfunctionsthatcanbereducedor

areimproper.Youmaywishtogoovertheprocessforreducingrationalnumbersbeforeasking

studentstostartworkingwithrationalfunctions.Then,tellstudentsthattheyshouldrelyontheir

experienceswithrationalnumberstohelpthemthinkaboutrationalfunctions.Askstudentsto

workproblemsupto#9beforehavingaclassdiscussionandre-launchingtheremainderofthe

task.

Explore(Individually,FollowedbySmallGroup):

Monitorstudentsastheyworkonquestions#4and#5toseethattheyaremakingsenseofthe

necessarystepsinworkingwithrationalexpressions.Thescaffoldingisgiveninthetask,but

studentswillneedtobeabletofactorthefunctionsandreducethem.Theanswersaregiveninthe

tasksothatstudentscanfocusontheprocessthatwillgetthemtheanswer.Makesurethatthey

areworkingontheprocessandcorrectingerrorsiftheirworkisnotleadingthemtotheright

answers.Helpstudentstofocusontheideathatreducingisjustfindinganequivalentformby

dividingthesamefactoroutofthenumeratoranddenominator,whichisjustdividingbyone.




Selectstudentstosharethathavearticulatedthisideainsomeway,sothatotherstudentscanhear

severalversionsoftheidea.

Asstudentsprogress,supporttheminusingtechnologyforquestion#9toexploretheareaofthe

graphnear-2wherethereisahole.Asthecurveistracedonsomegraphingtechnology,like

Desmos,thevalueat-2isshowntobeundefined.Onothertechnology,likesomegraphing

calculators,thereisnovalueshownat-2.Eitherway,studentswillneedsupportforinterpreting

thegraphduringthediscussion.Listenfortheirideasthatwillbeusefultoshare.

Discuss—Part1(WholeGroup):

Beginthediscussionbyaskingastudentsharehis/herworkonquestion#4.Focusonhowthe

numeratorwasfactoredandthecommonfactorsarereduced.Similarly,askastudenttoshare

question#5.Askpreviously-selectedstudentstoshareideasforquestion#6.Emphasizethat

commonfactorsmustbedividedfromthenumeratoranddenominatorsothevalueofthefraction

isunchanged.Then,askstudentsaboutquestion#7.Sincethisproblemisbaseduponacommon

misconception,theremaybesomecontroversy.Letstudentsmakeargumentsaboutwhetheror

notitiscorrect.Afterhearingarguments,bringtheclasstoconsensusthatitisnotcorrectbecause

jisnotafactorinthedenominator,so“reducing”itwillnotresultinanequivalentexpression.

Discussquestions#8and#9.Askstudentsfortheirpredictionsandthenprojectagraphofthe

function.Askstudentswhythereisnotanasymptoteatj = −2.Helpthemtoseethattheoriginal

functionisnotdefinedat-2,sothereisnovaluethere,andalltheothervaluesarethesameasthe

functionthatremainsafterthefactor(j + 2)isreduced.

Launch--Part2(WholeGroup):

Re-launchthesecondpartofthetaskbydiscussingquestions#10and#11together.Discuss

question#12,emphasizingtheprocessforwritingamixednumeralfromanimproperfraction.Tell

studentsthatthisisthesameprocessforimproperrationalfunctionsandthenaskthemtoworkon

theremainingquestionsinthetask.




Explore(Individually,FollowedbySmallGroup):

Studentsmayneedalittlenudgetodecidetouselongdivisionofpolynomials.Remindthemthat

fractionsarejustanotherwaytowritedivision,sothehintofwhattodoisrightintheexpression

thattheyaregiven.

Asstudentsareworkingonquestion17,watchforastudentthatnoticesthatthefunctionis

approachingthelineå = j + 3.Thismaytakealittlepromptingwithquestionslike,“Whatisthe

endbehaviorofthefunction?Whatlinedoesitappeartobeapproaching?Howisthatlinerelated

totheequivalentfunctionthatyoufound?”

Discuss—Part2(WholeGroup):

Beginthediscussionwithquestion#13.Askastudenttosharehis/herworkinwritingan

equivalentexpression.Then,projectagraphofthefunctionandaskthepreviouslyselected

studenttodescribetherelationshipbetweentheequivalentexpressionandthegraph,specifically

howtoidentifytheendbehaviorofthefunction.Tellstudentsthatthistypeofendbehavioris

calledaslant(oroblique)asymptote.Repeatthesameprocesswithquestion#15.Discuss

question#19,bringingtheclasstoconsensusonhowtographarationalfunctionwherethedegree

ofthenumeratorisgreaterthanthedegreeofthedenominatorthatincludes:

• Dividingthenumeratorbythedenominatortofindanequivalentexpression.• Findingtheverticalasymptotebyfindingtherootsofthedenominator.• Usingtheequivalentexpressiontoidentifytheslantasymptote.• Findtheinterceptsusingthesameprocessasotherrationalfunctions.

Iftimeallows,askastudenttosharehis/herworkinwritinganequivalentexpressionforquestion

#16.Thenasktheclasswhattheywouldexpectofthegraphoftheoriginalfunction.Theyshould

noticethatthedegreeofthenumeratoristhesameasthedegreeofthedenominator,sotheycan

predicttheverticalandhorizontalasymptotes.Connecttheirpredictedasymptotestothe

equivalentform.Theyshouldbeabletoseethatthisfunctionturnsouttobeatransformationof

å = 1/j.

AlignedReady,Set,Go:RationalExpressionsandFunctions4.4


RATIONAL EXPRESSIONS & FUNCTIONS – 4.4


4.4

Needhelp?Visitwww.rsgsupport.org

READY Topic:Connectingfeaturesofpolynomialsandrationalfunctions

Findtherootsanddomainforeachfunction.

1.!(#) = (# + 5)(# − 2)(# − 7)

2.+(#) = #, + 7# + 6

3..(#) =/

(012)(03,)(034)

4.ℎ(#) =/

(0614017)

5.Makeaconjecturethatcomparesthedomainofapolynomialwiththedomainofthereciprocalofthepolynomial.(Notethatthereciprocalofapolynomialisarationalfunction.)

6.Dotherootsofthepolynomialtellyouanythingaboutthegraphofthereciprocalofthepolynomial?Explain.

7.Findthey-interceptfor#1and#2.Whatisthey-interceptfor#3and#4?

SET Topic:Distinguishingbetweenproperandimproperrationalfunctions.

Determineifeachofthefollowingisaproperoranimproperrationalfunction.8. 9. 10.

!(#) =#> + 3#, + 7

7#, − 2# + 1

!(#) = #> − 5#, − 4!(#) =

3#, − 2# + 7

#2 − 5

READY, SET, GO! Name PeriodDate

26




4.4


11.!(#) =0A1B061,0

/C014 12.!(#) =

2063B01B

40D3,01>

13.Whichoftheabovefunctionshavethefollowingendbehavior?

EF# → ∞, !(#) → 0EJKEF# → −∞, !(#) → 014.Completethestatement:

ALLproperrationalfunctionshaveendbehaviorthat___________________________________________

Determineifeachrationalexpressionisproperorimproper.Ifimproper,uselongdivisiontorewritetherationalexpressionssuchthatL(M)

N(M)= O(M) +

P(M)

N(M)whereO(M)representsthe

quotientandP(M)representstheremainder.

15.,0A340617

03/ 16.

(01/)

(03,)(01,)

17.0A3>061203/

063B01B 18.

0A3201,

03/C

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4.4


GO Topic:Findingthedomainofrationalfunctionsthatcanbereduced

Statethedomainofthefollowingrationalfunctions.

19.Q =(03,)

(03,)(012) 20.Q =

(017)

(03B)(017) 21.Q =

(034)(01/C)

(01/C)(03>)(034)

a)Eachofthepreviousfunctionshasonlyoneverticalasymptote.Writetheequationoftheverticalasymptotefor#19,#20,and#21below.

19a)V.A. 20a)V.A. 21a)V.A.

b)Thegraphsof#19,#20,and#21arebelow.Foreachgraph,sketchintheverticalasymptote.Putanopencircleonthegraphanywhereitisundefined.

19b)

20b)

21b)

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