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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)

Copyright 2017 Licensed and Authorized for Use Only by Texas Education Agency 1

2.06Inequalities

• Precalculus(5)(K)• Precalculus(5)(L)

Note:Unlessstatedotherwise,anydataisfictitiousandusedsolelyforthepurposeofinstruction.


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.


2. Findtheequationofaparabolawithavertexof(5, −8)thatpassesthroughthepoint(2, 1).

Whenaskedtofindthemaximumorminimumvalueofafunctioninawordproblemthatinvolvesquadraticfunctions,youarefindingthevertexofthatequation.

3. SupposethepriceandcostofbuyingTexasLonghornsfootballjerseysinbulkcanbeexpressedas𝑝 𝑥 = 400 − ?

&𝑥and𝐶 𝑥 = 20𝑥 + 300respectively.Findthenumberof

jerseysthatneedtobesoldtomaximizeprofit.

4. Therearetwopositivenumbers;thesumofthefirstnumberandtwicethesecondnumberis40.Determinethetwonumbersthatmaximizetheirproduct.


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


Complexconjugatesdifferinthesignbetweentherealandimaginarynumber.

Theconjugateof𝑎 + 𝑏𝑖is𝑎 − 𝑏𝑖.Multiplyingthemtogethergivesus

2. Findthecomplexconjugateforeachofthefollowing:i. 5 − 4𝑖

ii. 7 + 3𝑖

iii. 3𝑖

iv. −7


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


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.


2. Graphthepolynomialfunction𝑓 𝑥 = 2 𝑥 − 1 𝑥 + 3 &.

3. Graphthepolynomialfunction𝑓 𝑥 = 𝑥C − 4𝑥&.


2.04LongDivision

Longdivisionissimilartodividingintegers,butinthecaseofpolynomialdivision,youwilluse0asaplaceholderforanypowermissingbeforethehighestpowerofthepolynomial.

1. Uselongdivisiontowriteanequivalentexpressionfor 9𝑥B − 3𝑥& + 5 ÷ 3𝑥& + 1 .

Syntheticdivisionisashortcutoflongdivisionthatcanbeusedwhenthedivisorisabinomialintheform𝑥 − 𝑘.Onlythecoefficientsareusedforsyntheticdivision.

2. Usesyntheticdivisiontowriteanequivalentexpressionfor 2𝑥B − 4𝑥 + 1 ÷ 𝑥 + 2 .


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.


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.


1. Findtheholesandasymptotesofeachfunctionbelow.

i. 𝑓 𝑥 = SVBSWHX

ii. 𝑔 𝑥 = &SWHCSSWHSH&

iii. ℎ 𝑥 = SV?SWV?

iv. 𝑗 𝑥 = SZV?SWV?


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.


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.


3. Findthedomainofthefunction𝑓 𝑥 = &SHD \(BHS)Z

BSW SV?.

4. Findtheinterval(s)forwhichtheinequality BSHD

≤ CSVB

istrue.


Download - Section 2: Polynomial and Rational Functions - Study Edge · Section 2: Polynomial and Rational Functions The following maps the videos in this section to the Texas Essential Knowledge

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