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