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Section2:PolynomialandRationalFunctionsThefollowingmapsthevideosinthissectiontotheTexasEssentialKnowledgeandSkillsforMathematicsTAC§111.42(c).2.01QuadraticFunctions
• Precalculus(1)(A)• Precalculus(1)(B)• Precalculus(1)(C)• Precalculus(1)(G)• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(J)• Precalculus(2)(N)
2.02ComplexNumbers
• Precalculus(2)(I)• Precalculus(2)(N)
2.03PolynomialandPowerFunctions
• Precalculus(1)(G)• Precalculus(2)(F)• Precalculus(2)(G)• Precalculus(2)(I)• Precalculus(2)(J)• Precalculus(2)(N)• Precalculus(5)(J)
2.04LongDivision
• Precalculus(5)(J)2.05RationalFunctions
• Precalculus(1)(G)• Precalculus(2)(F)• Precalculus(2)(I)• Precalculus(2)(J)• Precalculus(2)(K)• Precalculus(2)(L)• Precalculus(2)(M)• Precalculus(2)(N)
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2.06Inequalities
• Precalculus(5)(K)• Precalculus(5)(L)
Note:Unlessstatedotherwise,anydataisfictitiousandusedsolelyforthepurposeofinstruction.
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2.01QuadraticFunctions
Quadraticfunction–Apolynomialofdegreetwowhosegraphisaparabola
• Quadraticform–𝑦 = 𝑎𝑥& + 𝑏𝑥 + 𝑐
Vertexis ℎ, 𝑘 ,whereℎ = − .&/and𝑘 = 𝑓(ℎ)
• Vertexform–𝑦 = 𝑎 𝑥 − ℎ & + 𝑘
Vertexis ℎ, 𝑘
• Axisofsymmetry–The𝑥-coordinateofthevertex
• Leadingcoefficient–Representedby𝑎
o If𝑎 > 0,thenthegraphopens_______,andthevertexisa________. as𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →
o If𝑎 < 0,thenthegraphopens_______,andthevertexisa________. as𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →
1. Findthevertex,domain,range,axisofsymmetry,andvertexformof𝑓 𝑥 = 𝑥& − 4𝑥 + 3.
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2. Findtheequationofaparabolawithavertexof(5, −8)thatpassesthroughthepoint(2, 1).
Whenaskedtofindthemaximumorminimumvalueofafunctioninawordproblemthatinvolvesquadraticfunctions,youarefindingthevertexofthatequation.
3. SupposethepriceandcostofbuyingTexasLonghornsfootballjerseysinbulkcanbeexpressedas𝑝 𝑥 = 400 − ?
&𝑥and𝐶 𝑥 = 20𝑥 + 300respectively.Findthenumberof
jerseysthatneedtobesoldtomaximizeprofit.
4. Therearetwopositivenumbers;thesumofthefirstnumberandtwicethesecondnumberis40.Determinethetwonumbersthatmaximizetheirproduct.
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2.02ComplexNumbers
Imaginaryunit–Thenumberrepresentedby𝑖suchthat𝑖 = −1
Complexnumbers–Thecombinationofrealandimaginarynumbers
Intheform𝑎 + 𝑏𝑖,𝑎istherealpartand𝑏𝑖istheimaginarypart.
Noticethatpowersoftheimaginaryunitrepeateveryfourterms.
𝑖? = 𝑖& = 𝑖B = 𝑖C =
𝑖D = 𝑖E = 𝑖F = 𝑖G =
Whentheimaginaryunithasalargeexponent,youcandividethelasttwodigitsby4.Theremainderwillbeequivalenttothenewpowerof𝑖.
1. Findtheequivalentexpressionforeachofthefollowing:i. 𝑖?BF
ii. 𝑖HCD
iii. 𝑖&?II
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Complexconjugatesdifferinthesignbetweentherealandimaginarynumber.
Theconjugateof𝑎 + 𝑏𝑖is𝑎 − 𝑏𝑖.Multiplyingthemtogethergivesus
2. Findthecomplexconjugateforeachofthefollowing:i. 5 − 4𝑖
ii. 7 + 3𝑖
iii. 3𝑖
iv. −7
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2.03PolynomialandPowerFunctions
Polynomialfunctionsareintheform𝑓 𝑥 = 𝑎K𝑥K + 𝑎KH?𝑥KH? + ⋯+ 𝑎&𝑥& + 𝑎?𝑥 + 𝑎I,where𝑛isawholenumberand𝑎sarerealnumbers.
CharacteristicsofPolynomials
• Turn–Thecoordinatewhereapolynomialhasalocalmaximumorminimum
Apolynomialofdegree𝑛hasatmost𝑛 − 1turnsinitsgraph.
Examples:
second-degreepolynomial: third-degreepolynomial:
fourth-degreepolynomial: fifth-degreepolynomial:
• Endbehavior–Wherethe“arrows”ofthegraphofthepolynomialarepointingas𝑥approachesinfinityandnegativeinfinity
o Foreven-degreepolynomials:§ Iftheleadingcoefficientispositive,theendbehavioristhatbothsidesof
thegraphareup.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →§ Iftheleadingcoefficientisnegative,theendbehavioristhatbothsides
ofthegrapharedown.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →
o Forodd-degreepolynomials:§ Iftheleadingcoefficientispositive,theendbehaviorisdowntotheleft
anduptotheright.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →§ Iftheleadingcoefficientisnegative,theendbehaviorisdowntotheright
anduptotheleft.Meaningas𝑥 → ∞,𝑓(𝑥) → andas𝑥 → −∞,𝑓(𝑥) →
• Degreeofapolynomial
o Ifallliketermsofapolynomialarecombined,thedegreeisthehighestpower.o Ifthepolynomialiscompletelyfactored,thenthedegreeisfoundbyaddingup
themultiplicitiesofthezeroes.§ Zeroes–Thesolution(s)totheequation;alsoknownasthe𝑥-intercepts§ Multiplicities–Thepowersonthefactorsofeachzero
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1. Listthezeroesandmultiplicitiesofeachpolynomial.Findthedegreeofeachpolynomialanditsendbehavior.
i. 𝑓 𝑥 = 2𝑥 + 3 𝑥 − 2 B
ii. 𝑔 𝑥 = −2𝑥B 𝑥 − 5 2𝑥 − 1 D
HowtoGraphaPolynomialFunction
1. Findthezeroes,multiplicitiesofthezeroes,andendbehavior.
2. Graphthezeroes,andthengraphtheendbehavior.
3. Usemultiplicitiestodetermineifthegraphtouchesorcrossesthezero.
o Ifthemultiplicityiseven,ittouchesthezero.o Ifthemultiplicityisodd,itcrossesthezero.
AquickacronymtousewhengraphingisZEM:graphbasedontheZeroes,Endbehavior,andMultiplicities.
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2. Graphthepolynomialfunction𝑓 𝑥 = 2 𝑥 − 1 𝑥 + 3 &.
3. Graphthepolynomialfunction𝑓 𝑥 = 𝑥C − 4𝑥&.
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2.04LongDivision
Longdivisionissimilartodividingintegers,butinthecaseofpolynomialdivision,youwilluse0asaplaceholderforanypowermissingbeforethehighestpowerofthepolynomial.
1. Uselongdivisiontowriteanequivalentexpressionfor 9𝑥B − 3𝑥& + 5 ÷ 3𝑥& + 1 .
Syntheticdivisionisashortcutoflongdivisionthatcanbeusedwhenthedivisorisabinomialintheform𝑥 − 𝑘.Onlythecoefficientsareusedforsyntheticdivision.
2. Usesyntheticdivisiontowriteanequivalentexpressionfor 2𝑥B − 4𝑥 + 1 ÷ 𝑥 + 2 .
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RemainderTheorem–Ifyoudivideapolynomial,𝑓(𝑥),byafactor𝑥 − 𝑘,theremainderis𝑓(𝑘).
FactorTheorem–If𝑥 − 𝑘isafactorof𝑓(𝑥),then𝑓(𝑘) = 0.
3. Whichexpressionisafactorof𝑓 𝑥 = 𝑥B − 2𝑥& + 1?
i. 𝑥 − 1ii. 𝑥 + 1iii. 𝑥 − 2
NumberofZeroesTheorem–Apolynomialofdegree𝑛hasatmost𝑛distinctzeroes. Example:Apolynomialofdegreefourhasatmostfourdifferentzeroes.
ConjugateZeroesTheorem–Ifoneofthezeroesofapolynomialis𝑎 + 𝑏𝑖,then𝑎 − 𝑏𝑖isanotherzero.Forexample,if5 − 4𝑖isazero,then________isanotherzero.
4. Findapolynomialoflowestdegreewithzeroesof3,2𝑖,0,and𝑓 2 = 8.
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2.05RationalFunctions
Rationalfunction–Aquotientoftwopolynomialfunctionsintheform𝑓 𝑥 = R(S)T(S)
,where
𝑞(𝑥)isnottheconstantpolynomial0.Whencomparedtopolynomialfunctions,rationalfunctionsmayhaveadditionaltraits,likeholesandasymptotes.
Hole–Asinglepointwherethegraphisundefinedandindicatedvisuallybyanopencircle
Todetermineholes:Whenafactor(𝑥 − 𝑘)ispresentinboththenumeratoranddenominator,you“cross”thesefactorsouttowriteanequivalentequation,sothesamefactorisnolongerinthedenominator,givingyouaholeat𝑥 = 𝑘.
Verticalasymptotes
• 𝑥 = 𝑎willbeaverticalasymptotewhen𝑓(𝑥) → ∞,or𝑓 𝑥 → −∞as𝑥 → 𝑎fromtheleftsideorrightsideof𝑥 = 𝑎.
• Tofindtheverticalasymptotes,setthedenominatorequaltozeroafteryousimplifytheequationforholes.
Horizontalasymptotes
• 𝑦 = 𝑏willbeahorizontalasymptotewhen𝑥 → ∞or𝑥 → −∞as𝑓(𝑥) → 𝑏.• Tofindhorizontalasymptotes,comparethedegreeofthepolynomialinthenumerator
tothedegreeofthepolynomialinthedenominator.o Case1:Ifthedegreesarethesame,thehorizontalasymptoteistheratioofthe
leadingcoefficients.o Case2:Ifthedegreeofthedenominatorislarger,thehorizontalasymptotewill
be𝑦 = 0.o Case3:Ifthedegreeofthenumeratorislarger,thereisnohorizontal
asymptote. Rather,thereisanobliqueasymptote,whichcanbefoundusinglongdivision.Theequationoftheobliqueasymptoteistheequationbeforetheremainder.
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1. Findtheholesandasymptotesofeachfunctionbelow.
i. 𝑓 𝑥 = SVBSWHX
ii. 𝑔 𝑥 = &SWHCSSWHSH&
iii. ℎ 𝑥 = SV?SWV?
iv. 𝑗 𝑥 = SZV?SWV?
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Afunctioncancrossortouchitshorizontalasymptote.Tofindthe𝑥valuewherethisoccurs,set𝑓 𝑥 =horizontalasymptote,andsolvefor𝑥.
2. Findwherethefunctionintersectsitshorizontalasymptote.Ifitdoesnotintersect,thenstateso.
i. 𝑓 𝑥 = SVBSWVX
ii. 𝑔 𝑥 = SVCSH&
3. Giventhefunction𝑓 𝑥 = BSWHESSWVSHE
,find(a)thedomain,(b)theintercepts,(c)theasymptotes,(d)theholes,and(e)wherethefunctioncrossesthehorizontalasymptote.Then,graphthefunction.
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2.06Inequalities
Tosolvepolynomialorrationalinequalities:
Step1: Moveeverythingtooneside,equaltheothersidetozero,andfactortheentireexpression.
Step2: Findthedomainandthenreducetothelowestterms.Step3: Setthenumeratoranddenominatorequaltozero.Step4: PlacethevaluesfromStep3andthedomainfromStep2onanumberline.Step5: Assignopenandclosedcircles.
• Inthecaseof>or<,allareopen.• Inthecaseof≥or≤,allareclosedexceptfornumbersnotinthedomain,
whichareopen.Step6: Todeterminethesignbetweeneachnumberonyournumberline:
• Pluginavaluefor𝑥betweeneachintervaltodetermine+or–𝑦value,or• Pluginonenumbertostart,thenusemultiplicities.
o Evenpowerskeepthesamesign.o Oddpowerschangesign.
Step7: Writethesolutiontotheinequalityinintervalnotation.
1. Findtheinterval(s)wheretheinequality𝑥& − 12 > 𝑥istrue.
2. Findtheinterval(s)wheretheinequality H&S[ SHD W
SV? &SH?WZ< 0istrue.
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3. Findthedomainofthefunction𝑓 𝑥 = &SHD \(BHS)Z
BSW SV?.
4. Findtheinterval(s)forwhichtheinequality BSHD
≤ CSVB
istrue.
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