L2–1.2–CharacteristicsofPolynomialFunctionsLessonMHF4UJensenInsection1.1welookedatpowerfunctions,whicharesingle-termpolynomialfunctions.Manypolynomialfunctionsaremadeupoftwoormoreterms.Inthissectionwewilllookatthecharacteristicsofthegraphsandequationsofpolynomialfunctions.NewTerminology–LocalMin/Maxvs.AbsoluteMin/MaxLocalMinorMaxPoint–Pointsthatareminimumormaximumpointsonsomeintervalaroundthatpoint.AbsoluteMaxorMin–Thegreatest/leastvalueattainedbyafunctionforALLvaluesinitsdomain.Investigation:GraphsofPolynomialFunctionsThedegreeandtheleadingcoefficientintheequationofapolynomialfunctionindicatetheendbehavioursofthegraph.Thedegreeofapolynomialfunctionprovidesinformationabouttheshape,turningpoints(localmin/max),andzeros(x-intercepts)ofthegraph.Completethefollowingtableusingtheequationandgraphsgiven:

Inthisgraph,(-1,4)isalocalmaxand(1,-4)isalocalmin.Thesearenotabsoluteminandmaxpointsbecausethereareotherpointsonthegraphofthefunctionthataresmallerandgreater.Sometimeslocalminandmaxpointsarecalledturningpoints.

Onthegraphofthisfunction…Thereare3localmin/maxpoints.2arelocalminand1isalocalmax.Oneofthelocalminpointsisalsoanabsolutemin(itislabeled).

EquationandGraph DegreeEvenorOdd

Degree?

LeadingCoefficient EndBehaviour

Numberofturningpoints

Numberofx-intercepts

𝑓 𝑥 = 𝑥$ + 4𝑥 − 5

𝑓 𝑥 = 3𝑥* − 4𝑥+ − 4𝑥$ + 5𝑥 + 5

𝑓 𝑥 = 𝑥+ − 2𝑥

𝑓 𝑥 = −𝑥* − 2𝑥+ + 𝑥$ + 2𝑥

𝑓 𝑥 = 2𝑥- − 12𝑥* + 18𝑥$ + 𝑥 − 10

𝑓 𝑥 = 2𝑥1 + 7𝑥* − 3𝑥+ − 18𝑥$ + 5

SummaryofFindings:

• Apolynomialfunctionofdegree𝑛hasatmost𝒏 − 𝟏localmax/minpoints(turningpoints)• Apolynomialfunctionofdegree𝑛mayhaveupto𝒏distinctzeros(x-intercepts)• Ifapolynomialfunctionisodddegree,itmusthaveatleastonex-intercept,andanevennumberofturning

points• Ifapolynomialfunctionisevendegree,itmayhavenox-intercepts,andanoddnumberofturningpoints

• Anodddegreepolynomialfunctionextendsfrom…o 3rdquadrantto1stquadrantifithasapositiveleadingcoefficiento 2ndquadrantto4thquadrantifithasanegativeleadingcoefficient

• Anevendegreepolynomialfunctionextendsfrom…o 2ndquadrantto1stquadrantifithasapositiveleadingcoefficiento 3rdquadrantto4thquadrantifishasanegativeleadingcoefficient

EquationandGraph DegreeEvenorOdd

Degree?

LeadingCoefficient EndBehaviour

Numberofturningpoints

Numberofx-intercepts

𝑓 𝑥 = 5𝑥1 + 5𝑥* − 2𝑥+ + 4𝑥$ − 3𝑥

𝑓 𝑥 = −2𝑥+ + 4𝑥$ − 3𝑥 − 1

𝑓 𝑥 = 𝑥* + 2𝑥+ − 3𝑥 − 1

Note:OdddegreepolynomialshaveOPPOSITEendbehaviours

Note:EvendegreepolynomialshaveTHESAMEendbehaviour.

Example1:Describetheendbehavioursofeachfunction,thepossiblenumberofturningpoints,andthepossiblenumberofzeros.Usethesecharacteristicstosketchpossiblegraphsofthefunctiona)𝑓 𝑥 = −3𝑥1 + 4𝑥+ − 8𝑥$ + 7𝑥 − 5ThedegreeisoddandtheleadingcoefficientisnegativesothefunctionmustextendfromQ2toQ4As𝑥 → −∞, 𝑦 → ∞As𝑥 → ∞, 𝑦 → −∞Thefunctioncanhaveatmost5𝑥-intercepts(1,2,3,4,or5)Thefunctioncanhaveatmost4turningpoints(0,2,or4)Possiblegraphsof5thdegreepolynomialfunctionswithanegativeleadingcoefficient:b)𝑔 𝑥 = 2𝑥* + 𝑥$ + 2Thedegreeisevenandtheleadingcoefficientispositivesothefunctionmustextendfromthesecondquadranttothefirstquadrant.As𝑥 → −∞, 𝑦 → ∞As𝑥 → ∞, 𝑦 → ∞Thefunctioncanhaveatmost4𝑥-intercepts(0,1,2,3,or4)Thefunctioncanhaveatmost3turningpoints(1,or3)Possiblegraphsof4thdegreepolynomialfunctionswithapositiveleadingcoefficient:Example2:Filloutthefollowingchart

Degree Possible#of𝒙-intercepts Possible#ofturningpoints1 1 02 0,1,2 13 1,2,3 0,24 0,1,2,3,4 1,35 1,2,3,4,5 0,2,4

Note:Odddegreefunctionsmusthaveanevennumberofturningpoints.

Note:Evendegreefunctionsmusthaveanoddnumberofturningpoints.

Example3:Determinethekeyfeaturesofthegraphofeachpolynomialfunction.Usethesefeaturestomatcheachfunctionwithitsgraph.Statethenumberof𝑥-intercepts,thenumberoflocalmax/minpoints,andthenumberofabsolutemax/minpointsforthegraphofeachfunction.Howarethesefeaturesrelatedtothedegreeofeachfunction?a)𝑓 𝑥 = 2𝑥+ − 4𝑥$ + 𝑥 + 1b)𝑔 𝑥 = −𝑥* + 10𝑥$ + 5𝑥 − 4c)ℎ 𝑥 = −2𝑥1 + 5𝑥+ − 𝑥d)𝑝 𝑥 = 𝑥- − 16𝑥$ + 3a)Thefunctioniscubicwithapositiveleadingcoefficient.ThegraphextendsfromQ3toQ1.They-interceptis1.Graphiv)correspondstothisequation.Thereare3x-interceptsandthedegreeis3.Thefunctionhasonelocalmaxandonelocalmin,whichisatotaloftwoturningpoints,whichisonelessthanthedegree.Thereisnoabsolutemaxorminpoint.b)Thefunctionisquarticwithanegativeleadingcoefficient.Thegraphextendsfromquadrant3to4.They-interceptis-4.Graphi)correspondstothisequation.Thereare4x-interceptsandthedegreeis4.Thefunctionhastwolocalmaxandonelocalmin,whichisatotalof3turningpoints,whichisonelessthanthedegree.Thefunctionhasoneabsolutemaxpoint.c)Thefunctionisquinticwithanegativeleadingcoefficient.Thegraphextendsfromquadrant2to4.They-interceptis0.Graphiii)correspondstothisequation.Thereare5x-interceptsandthedegreeis5.Thefunctionhastwolocalmaxandtwolocalmin,whichisatotalof4,whichisonelessthanthedegree.Thefunctionhasnoabsolutemaxorminpoints.d)Thefunctionisdegree6withapositiveleadingcoefficient.Thegraphextendsfromquadrant2to1.They-interceptis3.Graphii)correspondstothisequation.Thereare4x-interceptsandthedegreeis6.Thefunctionhastwolocalmaxandonelocalmin,whichisatotalof3,whichisthreelessthanthedegree.Thefunctionhastwoabsoluteminpoints.FiniteDifferencesForapolynomialfunctionofdegree𝑛,where𝑛isapositiveinteger,the𝑛?@differences…

• areequal• havethesamesignastheleadingcoefficient• areequalto𝑎 ∙ 𝑛!,where𝑎istheleadingcoefficient

Note:𝑛!isreadas𝑛factorial.𝑛! = 𝑛×(𝑛 − 1)×(𝑛 − 2)×…×2×15! = 5×4×3×2×1 = 120

Example4:Thetableofvaluesrepresentsapolynomialfunction.Usefinitedifferencestodetermine

a) thedegreeofthepolynomialfunctionb) thesignoftheleadingcoefficientc) thevalueoftheleadingcoefficient

a)Thethirddifferencesareconstant.So,thetableofvaluesrepresentsacubicfunction.Thedegreeofthefunctionis3.b)Theleadingcoefficientispositivesince6ispositive.c) 6 = 𝑎 ∙ 𝑛! 6 = 𝑎 ∙ 3! 6 = 𝑎 ∙ 6 -

-= 𝑎

𝑎 = 1Therefore,theleadingcoefficientofthepolynomialfunctionis1.Example5:Forthefunction2𝑥* − 4𝑥$ + 𝑥 + 1whatisthevalueoftheconstantfinitedifferences?𝐹𝑖𝑛𝑖𝑡𝑒𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑐𝑒𝑠 = 𝑎 ∙ 𝑛! = 2 ∙ 4! = 2 ∙ 24 = 48Therefore,thefourthdifferenceswouldallbe48forthisfunction.

𝒙 𝒚 Firstdifferences Seconddifferences Thirddifferences

-3 -36 -2 -12 −12 − −36 = 24 -1 -2 −2 − −12 = 10 10 − 24 = −14 0 0 0 − −2 = 2 2 − 10 = −8 −8 − −14 = 61 0 0 − 0 = 0 0 − 2 = −2 −2 − −8 = 62 4 4 − 0 = 4 4 − 0 = 4 4 − −2 = 63 18 18 − 4 = 14 14 − 4 = 10 10 − 4 = 64 48 48 − 18 = 30 30 − 14 = 16 16 − 10 = 6