Transformation Quiz Review - Mrs. Pfrommer’s Calculus · Answers: 1. a. Horizontally compressed...
Transcript of Transformation Quiz Review - Mrs. Pfrommer’s Calculus · Answers: 1. a. Horizontally compressed...
Name:_____________________ CP1Math2Unit4Extension:Transformations
TransformationsQuizReviewHerearetherulesforallthetypesoftransformations:translation,scaling,andreflectionthatwehavestudied.Thestartingfunctionforalltheexamplesisthefunctiony=log(x).
graphicaltransformation equationchangeverticaltranslation Addontothefunction.Example:y=log(x)+k.
Ifk>0,thegraphmovesup.Ifk<0,thegraphmovesdown.
horizontaltranslation Replaceeveryxwith(x+n).Example:y=log(x+n).Ifn>0,thegraphmovesleft.Ifn<0,thegraphmovesright.
verticalscaling
Multiplythefunctionbya.Example:y=alog(x).Ifa>1,thegraphstretchesvertically(getstaller)If0<a<1,thegraphshrinksvertically(getsshorter)
horizontalscaling
Replaceeveryxwith(bx).Example:y=log(bx).Ifb>1,thegraphshrinkshorizontally(getsthinner)If0<b<1,thegraphstretcheshorizontally(getswider)
x-axisreflection Applya–signtothewholefunction.Example:y=–log(x).
y-axisreflection Replaceeveryxwith(–x).Example:y=log(–x).
ReviewProblems:1.Thefunction𝑦 = 𝑥! − 𝑥!istransformedinvariousways.Foreachresultingequation,describehowthe
graphof𝑦 = 𝑥! − 𝑥!isaffected.
a. 𝑦 = (4𝑥)! − (4𝑥)! b.𝑦 = (𝑥 − 2)! − (𝑥 − 2)! − 2
c.𝑦 = !!𝑥! − !
!𝑥! d.𝑦 = (−𝑥)! − (−𝑥)!
e. 𝑦 = 𝑥! − 𝑥! + 5 f.𝑦 = (!!𝑥)! − (!
!𝑥)! − 8
g.𝑦 = −2((𝑥 + 1)! − (𝑥 + 1)!) h.𝑦 = !!((−4𝑥)! − (−4𝑥)!)
i.𝑦 = −3(− !!𝑥)! + 3(− !
!𝑥)!
Hint:factoroutthecommonfactor
2.Writetheequationthatresultsfromeachtransformationdescribed.
a. Thefunction𝑓(𝑥) = 𝑥!istranslated2unitsleftand1unitup.
b. Thefunction𝑔(𝑥) = 𝑥 istranslated4unitsdownandis3timestallerthanthebasicfunction.
c. Thefunction ℎ(𝑥) = 𝑥!isreflectedacrossthexaxis.
d. Thefunction𝑔(𝑥) = 𝑥isshrunktobe!
! aswideandtranslateddown2units.
e. Thefunction𝑓(𝑥) = !!!isstretchedtobe3timesastallandtranslated1unitright.
f. Thefunctionℎ(𝑥) = !!!!!
isreflectedacrossthex-axisandtranslateddown5units
g. Thefunction𝑓(𝑥) = 𝑥! isreflectedacrosstheboththex-andy-axes.
h. Thefunction𝑔(𝑥) = !
!isstretchedtobe6timesaswide.
i. Thefunction𝑓(𝑥) = 𝑥! − 𝑥! + 𝑥isreflectedacrossthey-axisandshrunktobe!
!aswide.
j. Thefunctionℎ(𝑥) = !!isshrunktobe!
! astall,reflectedacrossthex-axis,andtranslatedleftby9units
3. Considerthefunction𝑦 = 𝑥! − 𝑥.Eachrowinthetablerepresentsadifferenttransformationorcombinationoftransformationstothisfunction.Fillintheemptyboxes.
Description Equation Graph
Theoriginalgraphwithnotransformations. 𝑦 = 𝑥! − 𝑥
Thegraphisreflectedoverthey-axis.
𝑦 = 𝑥! − 𝑥 − 2
5
4
3
2
1
–1
–2
–3
–4
–5
–6 –4 –2 2 4 6
4
3
2
1
–1
–2
–3
–4
–4 –2 2 4
4
3
2
1
–1
–2
–3
–4
–4 –2 2 4
4
3
2
1
–1
–2
–3
–4
–4 –2 2 4
Description Equation Graph
𝑦 = (3𝑥)! − (3𝑥)
Thegraphis4timesastall.
Thegraphisreflectedoverthex-axisanditis4timesas
wide.
4
3
2
1
–1
–2
–3
–4
–4 –2 2 4
4
3
2
1
–1
–2
–3
–4
–4 –2 2 4
4
3
2
1
–1
–2
–3
–4 –2 2 4
4
3
2
1
–1
–2
–3
–4
–4 –2 2 4
4.Thegraphofafunctionℎ(𝑥)isshownbelow.Ontheaxesprovided,sketchagraphof−ℎ(𝑥)+ 2.Thendescribethetransformationinwords.
Description:5.Howisthegraphof!
!= (𝑥 − 9)!relatedtothegraphof𝑦 = 𝑥!?
6.Supposeyoumakethegraphof𝑦 = |𝑥|shorterbyafactorof½andthentranslatethegraph3unitsleftand7unitsdown.Writetheresultingequation.7.Whichequationdescribesthegraphof𝑦 = !
!iftheoriginalgraphisstretchedtobe4timesaswide?
A.𝑦 = !
! B.𝑦 = !
! C.𝑦 = 4𝑥 D.𝑦 = 𝑥 + 4
Answers:1.a.Horizontallycompressedbyafactorof4
b.Translatedright2unitsanddown2unitsc.Verticallycompressedsothatitishalfofitsoriginalheightd.Reflectedoverthey-axise.Translatedup5unitsf.Horizontallystretchedsoitistwiceaswideandtranslateddown8unitsg.Translatedleft1unit,verticallystretchedsoitistwiceastall,andreflectedoverthex-axish.Horizontallycompressedbyafactorof4,reflectedoverthey-axis,andverticallycompressedsoitis1/3
astalli.Horizontallystretchedsoitis7timesaswide,reflectedoverthey-axis,verticallystretchedtobethree
timesastall,andreflectedoverthex-axis2.a.𝑓(𝑥) = (𝑥 + 2)! + 1
b.𝑔(𝑥) = 3 𝑥 − 4c.ℎ(𝑥) = −𝑥!d.𝑔(𝑥) = 4𝑥 − 2e.𝑓(𝑥) = !
(!!!)!
f.ℎ(𝑥) = !!!!!!
− 5,whichisthesamethingasℎ(𝑥) = − !!!!!
− 5g.𝑓(𝑥) = − −𝑥! h.𝑔(𝑥) = !
!
i.𝑓(𝑥) = (−5𝑥)! − (−5𝑥)! + (−5𝑥)j.ℎ(𝑥) = !!
!∙ !!!
3.(CheckgraphsoncalculatororDesmos)
a.𝑦 = (−𝑥)! − (−𝑥)b.Thegraphistranslateddown2unitsc.Thegraphistranslatedleft3units.𝑦 = (𝑥 + 3)! − (𝑥 + 3)d.Thegraphishorizontallycompressedtobe1/3aswidee.𝑦 = 4(𝑥! − 𝑥)f.Thegraphisreflectedacrossthey-axisandtranslatedup1unit.(ifyousaidreflectedacrossx-axis,that
isalsocorrect).𝑦 = (−𝑥)! − (−𝑥)+ 1 = −𝑥! + 𝑥 + 1g.𝑦 = −(!
!𝑥)! + (!
!𝑥)
4. Graphshouldbereflectedoverthex-axisandTHENshiftedup2units.5. Thegraphistranslated9unitsrightandtwiceastall6. 𝑦 = !
!|(𝑥 + 3)|− 7
7. A