Gr12AdvancedFunction.docx
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Gr12AdvancedFunction.docx
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Gr12AdvancedFunction
MHF4UGrade12AdvancedFunctions
Unit1:
PolynomialFunctions
PolynomialExpressionhastheform:
anxn+an-1xn-1+an-2xn-1+…+a3x3+a2x2+a1x+a0
n:
wholenumber
x:
variable
a:
coefficientXER
Degree:
thehighestexponentonvariablex,whichisn.
LeadingCoefficient:
anxn
PowerFunctions:
y=a*xn,nEI
Evendegreepowerfunctionsmayhavelinesymmetryalongy-axis.
Odddegreepowerfunctionsmayhavepointsymmetryaboutorigin.
FiniteDifferences:
a(n!
)
PropertiesforOddFunctions:
PositiveLeadingCoefficients:
GoesfromQ3toQ1
NegativeLeadingCoefficients:
GoesfromQ2toQ4
Domain{XER},Range{YER}
PropertiesforEvenFunctions:
PositiveLeadingCoefficients:
GoesfromQ2toQ1
NegativeLeadingCoefficients:
GoesfromQ3toQ4
Havelocalmaximum/minimumpoints
Domain{XER},Rangeabove/beyondmaximum/minimumpoints
SketchingGraphs
Sketchgraphsaccordingtoitsdegree
Polynomialfunctionshave0tomaximumofnx-intercepts,wherenisthedegree.
Polynomialfunctionscanalsohaveatmostn–1localmax/mins,wherenisthedegree.
X–Interceptsaretherootsofthefunction.
Factorsofafactoredformequationarealsotheroots.
Rootscanhavedifferentorders,andaregrapheddifferently:
Order1root[ie(x-1)]
passesthroughthex-intercept
Order2root[iex2]
passesintangenttothex-intercept
Order3root[iex3]
passesthroughinterceptlikey=x3attheorigin
EvenFunctionTest:
f(-x)=f(x)
OddFunctionTest:
f(-x)=-f(x)
Transformationsofpolynomialfunctionsoftheformy=a[k(x-d)]n+c:
a>=1:
verticalstretchbyafactorofa.
0<=a<=1:
verticalcompressionbyfactorofa.
k>=1:
horizontalcompressionbyafactorof1/k
0<=a<=1:
horizontalstretchbyafactorof1/k.(becarefultofactoroutsometimes).
d:
horizontaltranslationbydunitsleftorright(becarefultofactoroutsometimes).
c:
verticaltranslationbycunitsupordown.
AverageRatesofChange:
DeltaY/DeltaX
Slopeofsecantof2pointsgivetheaveragerateofchange
InstantaneousRatesofChange:
Tocalculatewithoutcalculus,substituteanumberthatsreallyclosetothetimewanted.
Usethatvaluetoachieveatheslopeofatangentatthatpoint
Instantaneousratesofchangeisthetangentatthosepoints
Unit2:
PolynomialEquationsandInequalities
RemainderTheorem
Longdivisioncanbeusedtodividepolynomials
Thedivisionstatementcanbesaidtobe:
P(x)=(x-b)(Q(x))+RwhereRistheremainder,x-bisafactor,Q(x)isthedivisor
TheorystatesthatwhenP(x)isdividedby(x-b),theremainderisP(b),whereifP(b)=0,thenbisarootoftheequation.
FactorTheorem
Statesthat(x-b)isafactorofanequationifP(b)=0
(ax–b)isafactorifP(b/a)=0
IntegralZeroTheorem
If(x–b)isafactor,andifthepolynomialequationhadaleadingcoefficientof1,thenbmustbeafactoroftheconstantterm
RationalZeroTheorem
If(ax–b)isafactor,andthepolynomialequationhadaleadingcoefficientgreaterthan1,thenx=b/aisarationalzeroofP(x),suchthat
bisafactoroftheconstantterm
aisafactoroftheleadingcoefficient
(ax–b)isafactor
FamiliesofPolynomialEquations
Afamilyisagroupofequationswiththesamecharactoristics
Theyhavethesamex-intercepts
Theytaketheform:
y=k(x-a)(x-b)(x-c)
Familiescanbeconvertedintoanequationwithagivenpoint
SolvingInequalities
Putalltermsononeside
Factorifpossible
UseIntervalTable/graphtosatisfytheinequality(areasthatare
Unit3:
RationalEquations
Thereciprocalofalinearfunctionhastheform:
f(x)=1/kx–c
Therestrictiononadomainofareciprocallinearfunctioncanbedeterminedbyfindingthevalueofxthatmakesthedenominatorequaltozero,thatisx=c/k.
TheVerticalAsymptoteofareciprocallinearfunctionhasanequationoftheformx=k/c.
Thehorizontalasymptoteofareciprocallinearfunctionhasequationy=0.
Ifk>0,theleftbranchofareciprocallinearfunctionhasanegative,decreasingslope,andtherightbranchhasanegative,increasingslope.
BasicallyoccupiesQ3andQ1.
Ifk<0,theleftbranchofareciprocallinearfunctionhasapositive,increasingslow,andtherightbranchhasapositive,decreasingslope.
BasicallyoccupiesQ2andQ4.
Rationalquadraticfunctionscanbeanalyzedusingkeyfeatures:
asymptotes,intercepts,slope(positiveornegative,increasingordecreasing),domain,range,andpositiveandnegativeintervals.
Reciprocalofquadraticfunctionswithtwozeroshavethreeparts,withthemiddleonereachingamaximumorminimumpoints.Thispointisequidistantfromthetwoverticalasymptotes.
Thebehaviornearasymptotesissimilartothatofreciprocalsoflinearfunctions.
Allofthebehaviorslistedabovecanbepredictedbyanalyzingtherootsofthequadraticrelationtothedenominator.
Arationalfunctionoftheformf(x)=(ax+b)/(cx+d)hasthefollowingkeyfeatures:
Theverticalasymptotecanbefoundbysettingthedenominatorequaltozeroandsolvingforx,providedthenumeratordoesnothavethesamezero.
Thehorizontalasymptotecanbefoundbydividingeachterminboththenumeratorandthedenominatorbyxandinvestigatingthebehaviorofthefunctionasx->positiveornegativeinfinity.
Thecoefficientbactstostretchthecurvebuthasnoeffectontheasymptotes,domain,orrange.
Thecoefficientdshiftstheverticalasymptote.
Thetwobranchesofthegraphofthefunctionareequidistantfromthepointofintersectionoftheverticalandhorizontalasymptotes.
AnalysisofEndBehavior
Forverticalasymptote
SubstituteanumberveryclosetotheVAfromtheright,andanumberfromtheleft
Analyzetheresultofthatnumberandexpresstheendbehavior
WhetherAsx->VA+/-,y->+/-infinity
Forhorizontalasymptote
Substituteaverylargenegativeandpositivenumberforxandanalyzethebehaviorofy.
Expresstheendbehaviorwiththeresultsfromthatsubstitution
Asx->+/-Infinity,y->HAfromabove/below
Tosolverationalequationsalgebraically,startbyfactoringtheexpressionsinthenumeratoranddenominatortofindasymptotesandrestrictions.
Next,multiplybothsidesbythefactoreddenominators,andsimplifytoobtainapolynomialequation.Thensolve.
ForRationalinequalities
Settherightsideoftheequationzero.
Factortheexpressiontofindrestrictions
Basedontheassumptionthatx=a/bistrueifandonlyifa*b=x.
Ontheleftsideoftheequation,takethedenominatorandmultiplyitbythenumerator.
Sincetheequationisalreadyfactored,therootsareclearlyshown.GraphoruseIntervalTablemethodtofindtheintervalsxwhichsatisfytheequation
Unit4:
Trigonometry
RadianMeasure:
AngleXisdefinedasthelength,a,thearcthatextendstheangledividedbytheradius,r,ofthecircle:
X=a/r.
2Pirad=360degreesorPirad=180degrees.
Toconvertdegreestoradians,multiplydegreemeasurewithPi/180.
Toconvertradianmeasuretodegrees,multiplyradianmeasurewith180/Pi.
Youcanuseacalculatortocalculatetrigonometricratiosandspecialanglesforanangleexpressedinradianmeasurebysettingtheanglemodetoradians.
Youcandeterminethereciprocaltrigonometricratiosforanangleexpressedinradianmeasurebyfirstcalculatingtheprimarytrigonometricratiosthenusingthereciprocalkeyonthecalculator.
Youcanalsousetheunitcircleandspecialtrianglestodetermineexactvaluesforthetrigonometricratiosofthespecialangles0,Pi/6,Pi/3.Pi/4,andPi/2.
YoucanusetheunitcirclealongwiththeCASTruletodetermineexactvaluesforthetrigonometricratiosofmultiplesofthespecialangles.
ForEquivalentTrigonometricExpressions,youcanusearighttriangletoderiveequivalenttrigonometricexpressionsthatformthecofunctionidentities,suchassinx=cos(Pi/2–x).
Youcanusetheunitcirclealongwithtransformationstoderiveequivalenttrigonometricexpressionsthatformothertrigonometricidentities,suchascos(Pi/2+X)=-sinx
Givenatrigonometricexpressionofaknownangle,youcanusetheequivalenttrigonometricexpressionstoevaluatetrigonometricexpressionsofotherangles.
Youcanusegraphingtechnologytodemonstratethattwotrigonometricexpressionsareequivalent.SomeofwhichareknownastheCo-FunctionsIdentities.
CompoundAngleFormulas
Youcandevelopcompoundanglesformulasusingalgebraandtheunitcircle.
Onceyouhavedevelopedonecompoundangleformula,youcandevelopothersbyapplyingequivalenttrigonometricexpressions.
Thecompoundangle,oradditionandsubtraction,formulasforsineandcosineare:
sin(X+Y)=sinXcosY+cosXsinY
sin(X-Y)=sinXcosY-cosXsinY
cos(X+Y)=cosXcosY-sinXsinY
cos(X-Y)=cosXcosY+sinXsinY
tan(X+Y)=tanX+tanY/1-tanXtanY
tan(X-Y)=tanX-tanY/1+tanXtanY
Theseidentitiescanalsobemadeintomoreidentites:
cos2X=cos^2X–sin^2X
=1–2sin^2X
=2sin^2X–1
sin2X=2sinXcosX
tan2X=2tanX/1-tan^2X
ProvingTrigonometricIdentities:
ATrigonometricidentityisanequationwhichtrigonometricexpressionsthatistrueforallanglesinthedomainoftheexpressionsonbothsides.
Onewaytoshowthatanequationisnotanidentityistodetermineacounterexample
Toprovethatanequationisanidentity,treateachsideoftheequationindependentlyandtransformtheexpressionononesideintotheexactformoftheexpressionontheotherside.
ThebasictrigonometricidentitiesarethePythagoreanidentity,thequotientidentity,thereciprocalidentities,thecompoundangleformulas.Youcanusetheseidentitiestoprovemore
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