ThingstoKnowChapter1:Functions

ShapesofQuadraticEquationsShapesofquadraticequationsare‘smiley’or‘mountain’shaped.Alwaysconsiderthe‘a’ofboth𝑎𝑥! + 𝑏𝑥 + 𝑐and𝑎 𝑥 + 𝑏 ! + 𝑐formats.1.Whena>0(aisapositivenumber)thenthegraphhasa‘smiley’shape.SmileyshapegraphswillhaveaminimumpointExample:𝑥! − 8𝑥 + 15(a=1thusa>0)

2.Whena<0(anegativenumber)thenthegraphhasa‘mountain’shape.MountainshapegraphswillhaveamaximumpointExample:−𝑥! − 8𝑥 − 15(a=-1thusa<0)

TheabovetipscanbeusedforquestionsaskingaboutRangeandDomain.HowtofindStationaryPointsofQuadraticEquations1.Whentheequationisin𝑎𝑥! + 𝑏𝑥 + 𝑐format,thenthex–coordinateofthestationarypointwillbe𝑥 = − !

!!theninputthisxvalueintotheequationtoget

they–coordinate.Example:𝑥! − 8𝑥 + 15Thisequationisintheform𝑎𝑥! + 𝑏𝑥 + 𝑐.Thereforea=1,b=-8,andc=15.Thex–coordinateofthestationarypointis𝑥 = − !

!!

à𝑥 = − !!

! != 4

àPutx=4intotheequationtogetthey–coordinateà 4 ! − 8 4 + 15 = −1sothestationarypointis(4,-1)2.Whentheequationisinitscompletedsquareform𝑎 𝑥 + 𝑏 ! + 𝑐thenthecoordinatesofthestationarypointis(-b,c)Example:𝑥 − 4 ! + 1Thestationarypointis(4,1)2 𝑥 + 5 ! − 6Thestationarypointis(-5,-6)3.Youcandifferentiateanyequationanduse!"

!"= 0togetthestationarypoint

becausestationarypointshaveagradientof0.Forquadraticequations,methods1.and2.areeasierandfaster

TheabovetipscanbeusedforquestionsaskingaboutRangeandDomain.Using"𝐛𝟐 − 𝟒𝐚𝐜""b! − 4ac"isusedinthefollowingsituations:1.Tocheckwhethertheequation(mostlyquadraticequations)istangenttothex–axis,lyingaboveorbelowthex–axis,hasequalrootsorhastwodifferentroots.2.Tocheckifalineisintersecting,tangent,intersectingattwodistinctpointsornotintersectingacurveatall.Inthecaseofaquadraticequation:

- Thegraphislyingaboveorbelowthex–axis- Nointersectionwiththex–axis

- Thegraphistangenttothex–axis- Theequationhasequalroots

• Theequationhastworoots• Intersectsthex–axisattwodistinctpoints

Inthecaseofalineandacurve:Combinetheequationbyequatingthey’sandthenusingthefinalequationintheformof𝑎𝑥! + 𝑏𝑥 + 𝑐

• Thelineandcurvedonotintersect• Noroots(realroots)

• Oneequalroot• Lineistangenttothecurve• Lineintersectsthecurve

• Twodistinctroots• Thelineintersectsthecurveattwodistinctpoints

**Somequestionsaskforthecasewhere“linemeetsthecurve”or“lineintereststhecurve”.Inbothcases,bothb! − 4ac = 0andb! − 4ac > 0workfortheaboveconditionssoyoucancombinethembywritingb! − 4ac ≥ 0**Doesnotworkforthe𝑎 𝑥 + 𝑏 ! + 𝑐formsoexpandallequationstogetthe𝑎! + 𝑏𝑥 + 𝑐formCompositeandInverseFunctionsCompositeFunctions:Forcompositefunctionslike𝑔 ∘ 𝑓 (𝑥)or𝑓 ∘ 𝑔 (𝑥),writedownthemainfunctionwithitsxreplacedbybrackets,theninputtheotherfunctionintothebrackets*Theletterinfrontisthemainfunction*Example:Find𝑓 ∘ 𝑔 (𝑥)for𝑓 𝑥 = 2𝑥 + 6,𝑔 𝑥 = !

!!!

Sincef(x)isthemainfunctionà𝑓 𝑥 = 2 + 6

à𝑓 ∘ 𝑔 𝑥 = 2 !!!!

+ 6Example:Find𝑔 ∘ 𝑓 (𝑥)for𝑓 𝑥 = 2𝑥 + 6,𝑔 𝑥 = !

!!!

Since𝑔 (𝑥)isthemainfunctionà𝑔 ∘ 𝑓 𝑥 = !

!!!! !!

InverseFunctions:

• Inversefunctionandtheoriginalfunctionaremirrorimagesaboutthey=xline

• Substituteyasf(x)

o Swapxandyo Rearrangethefunctionwithyasthesubjecto Changeyto𝑓!! 𝑥

• Domainof𝑓 𝑥 becomestherangeof𝑓!! 𝑥

• Rangeof𝑓 𝑥 becomesthedomainof𝑓!! 𝑥

àDomainof𝑓 𝑥 =rangeof𝑓!! 𝑥 àRangeof𝑓 𝑥 =domainof𝑓!! 𝑥 Example:Find𝑓!! 𝑥 of𝑓 𝑥 = 2𝑥 + 6𝑓 𝑥 = 𝑦 = 2𝑥 + 6à𝑥 = 2𝑦 + 6à𝑥 − 6 = 2𝑦à !!!

!= 𝑦

à𝑓!! 𝑥 = !!!

!

Example:

Find𝑔!! 𝑥 of𝑔 𝑥 = !!!!

𝑔 𝑥 = 𝑦 =5

𝑥 − 3à𝑥 = !

!!!

à𝑦 − 3 = !

!

à𝑦 = !

!+ 3

Theabovetipsareusedforaquadraticequationsinverse.Alwaysusethecompletedsquareform𝑎 𝑥 + 𝑏 ! + 𝑐InequalitiesofQuadraticEquations(a.k.aQuadraticInequalities)Quadraticinequalitiesonlyworkforequationswithtwodistinctroots.Considercheckthefollowing:

• Thegraphhasa‘smiley’shapeora‘mountain’shape.Dothisbycheckingthe‘a’of𝑎𝑥! + 𝑏𝑥 + 𝑐or𝑎 𝑥 + 𝑏 ! + 𝑐

• Thesignoftheinequality>,< ,≥ ,≤Thensimpledrawthegraphtomakesureoftheanswerortestanyvaluesbyinputtingthevaluesoftheanswerintotheequation

**For > 0 , ≥ 0,colourabovey=0**For < 0 , ≥ 0,colourbelowy=0

**pandqarerootsoftheequationsExample:𝑥! − 8𝑥 + 15 > 0,sincea=1thusithasa‘smiley’shape.Gettherootsoftheequation:

Simplydrawthegraphjustnotingtheroots:

Biggerthan0thusoutsideoftheroots(colourabovey=o)Answer:x<3,5<x

Example:−2𝑥! − 4𝑥 + 16 ≤ 0,since𝑎 = −2thenithasa‘mountain’shape.Gettherootsoftheequation:

àSimplydrawthegraphàSmallerorequalto0thusthecolourisbelowy=0

Answer:𝑥 ≤ −4 , 2 ≤ 𝑥TrigonometryGraphsFortrigonometryquestionsinfunctions,equationsareintheformofasin 𝑏𝑥 + 𝑐oracos 𝑏𝑥 + 𝑐aistheamplitudeofthegraphi.e.howfarthegraphrisesanddecreasesfromthestartinglinebistheperiodofthewave.!"#

!or!!

!givestheperiod,whichisthetotalangleto

give1cycleofawave

cisthestartinglineorstartingpointofthegraphonySignofaifa<0,thenthegraphisflippedupsidedownOriginalGraphs

y=sin(x)

a=1,a>0,b=1,c=0

y=cos(x)

a=1,a>0,b=1,c=0

*Sinceb=1thentheperiodis!"#

!= 360°or!!

!= 2𝜋

*Beawareofthedomainsgiven.Youonlyneedtodrawthegraphuntilthedomain’slimits.Example:

2 sin 𝑥 + 5

a=2,a>0,b=1,c=5

−3 sin 𝑥 + 2

a=-3,a<0,b=1,c=2

4 sin 2(𝑥)+ 1

a=4,a>0,b=2,c=1àb=2thus!!

!= 𝜋onecycleis𝜋thusupuntil2𝜋,thereare2cycles:

Example:

2 cos 𝑥 + 5

a=2,a>0,b=1,c=5

−3 cos 𝑥 + 2

a=-3,a<0,b=1,c=2

4 cos12 𝑥 + 1

a=4,a>0,b=!

!,c=1

àb=!!thus!!!

!= 4𝜋onlyhalfacyclewillbetherefor2𝜋

à