SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

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2.1 Transformers: Shifty y’s

A Develop Understanding Task

OptimaPrimeisdesigningarobotquiltforhernewgrandson.Sheplansfortherobottohaveasquareface.Theamountoffabricthatsheneedsforthefacewilldependontheareaoftheface,soOptimadecidestomodeltheareaoftherobot’sfacemathematically.SheknowsthattheareaAofasquarewithsidelengthxunits(whichcanbeinchesorcentimeters)ismodeledbythefunction, ! ! = !!squareunits.1. Whatisthedomainofthefunction! ! inthiscontext?

2. Matcheachstatementabouttheareatothefunctionthatmodelsit:MatchingEquation

(A,B,C,orD)

Statement FunctionEquation

Thelengthofeachsideisincreasedby5units.

A) AB) !(!) = 5!!

Thelengthofeachsideismultipliedby5units.

C) BD) ! = (! + 5)!

Theareaofasquareisincreasedby5squareunits.

E) CF) ! = (5!)!

Theareaofasquareismultipliedby5. G) DH) ! = !! + 5

Optimastartedthinkingaboutthegraphof! = !!(inthedomainofallrealnumbers)andwonderingabouthowchangestotheequationofthefunctionlikeadding5ormultiplyingby5affectthegraph.Shedecidedtomakepredictionsabouttheeffectsandthencheckthemout.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

3. Predicthowthegraphsofeachofthefollowingequationswillbethesameordifferentfromthegraphof! = !!.

Similaritiestothegraphof! = !!

Differencesfromthegraphof! = !!

! = 5!!

! = (! + 5)!

! = (5!)!

! = !! + 5

4. Optimadecidedtotestherideasusingtechnology.Shethinksthatitisalwaysagoodideatostartsimple,soshedecidestogowith! = !! + 5.Shegraphsitalongwith! = !!inthesamewindow.Testityourselfanddescribewhatyoufind.

5. Knowingthatthingsmakealotmoresensewithmorerepresentations,Optimatriesafewmoreexampleslike! = !! + 2 and! = !! − 3,lookingatbothatableandagraphforeach.Whatconclusionwouldyoudrawabouttheeffectofaddingorsubtractinganumberto! = !!?Carefullyrecordthetablesandgraphsoftheseexamplesinyournotebookandexplainwhyyourconclusionwouldbetrueforanyvalueofk,given,! = !! + !.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

6. Afterheramazingsuccesswithadditioninthelastproblem,Optimadecidedtolookatwhathappenswithadditionandsubtractioninsidetheparentheses,orasshesaysit,“addingtothe!beforeitgetssquared”.Usingyourtechnology,decidetheeffectofhintheequations:! = (! + ℎ)!and! = (! − ℎ)!.(Choosesomespecificnumbersforh.)Recordafewexamples(bothtablesandgraphs)inyournotebookandexplainwhythiseffectonthegraphoccurs.

7. Optimathoughtthat#6wasverytrickyandhopedthatmultiplicationwasgoingtobemorestraightforward.Shedecidestostartsimpleandmultiplyby-1,soshebeginswith! = −!!.Predictwhattheeffectisonthegraphandthentestit.Whydoesithavethiseffect?

8. Optimaisencouragedbecausethatonewaseasy.Shedecidestoendherinvestigationforthedaybydeterminingtheeffectofamultiplier,a,intheequation:! = !!!.Usingbothpositiveandnegativenumbers,fractionsandintegers,createatleast4tablesandmatchinggraphstodeterminetheeffectofamultiplier.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1 2.1

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Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

SET

Topic:TransformationsonquadraticsMatching:Choosetheareamodelthatisthebestmatchfortheequation.

______11.x! + 4 ______12. x + 4 ! ______13. 4x ! _______14.4x!a. b.

c. d.

x -3 -2 -1 0 1 2 3f(x)=x2 9 4 1 0 1 4 9

15a)g(x)= b)x -3 -2 -1 0 1 2 3g(x) 2 -3 -6 -7 -6 -3 2

c) Inwhatwaydidthegraphmove?d) Whatpartoftheequationindicatesthismove?

Atableofvaluesandthegraphforf(x)=x2isgiven.Comparethevaluesinthetableforg(x)tothoseforf(x).Identifywhatstaysthesameandwhatchanges.a)Usethisinformationtowritethevertexformoftheequationofg(x).b)Graphg(x).c)Describehowthegraphchangedfromthegraphoff(x).Usewordssuchasright,left,up,anddown.d)Answerthequestion.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.2

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

2.2 Transformers: More Than

Meets the y’s

A Solidify Understanding Task

Writetheequationforeachproblembelow.Useasecondrepresentationtocheckyourequation.1. Theareaofasquarewithsidelengthx,wherethesidelengthisdecreasedby3,theareaismultipliedby2andthen4squareunitsareaddedtothearea.

2.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.2

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

3.

4.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.2

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

Grapheachequationwithoutusingtechnology.Besuretohavetheexactvertexandatleasttwocorrectpointsoneithersideofthelineofsymmetry.

5. ! ! = −!! + 3

6. ! ! = (! + 2)! − 5

7. ℎ ! = 3(! − 1)! + 2

8. Given:! ! = !(! − ℎ)! + !a. Whatpointisthevertexoftheparabola?

b. Whatistheequationofthelineofsymmetry?

c. Howcanyoutelliftheparabolaopensupordown?

d. Howdoyouidentifythedilation?

9. Doesitmatterinwhichorderthetransformationsaredone?Explainwhyorwhynot.

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