4.4 Are You Rational? g z - Utah Education Networkthe case when the numerator of the fraction is...
Transcript of 4.4 Are You Rational? g z - Utah Education Networkthe case when the numerator of the fraction is...
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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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
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.
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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
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:
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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
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
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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
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.
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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? – 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.
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
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.
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
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.
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
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
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
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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
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SECONDARY MATH III // MODULE 4
RATIONAL EXPRESSIONS & FUNCTIONS – 4.4
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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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RATIONAL EXPRESSIONS & FUNCTIONS – 4.4
Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org
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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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