SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.5

Mathematics Vision Project

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

/HVVRQ�� Be There or Be Square

A Practice Understanding Task

QuiltsandQuadraticGraphs

Optima’sniece,Jennyworksintheshop,takingordersanddrawingquiltdiagrams.Whentheshopisn’ttoobusy,Jennypullsouthermathhomeworkandworksonit.Oneday,sheisworkingongraphingparabolasandnoticesthattheequationssheisworkingwithlooksalotlikeanorderforaquiltblock.Forinstance, Jennyissupposedtographtheequation:! = (! − 3)! + 4 . Shethinks,“That’sfunny.Thiswouldbeanorderwherethelengthofthestandardsquareisreducedby3andthenweaddalittlepieceoffabricthathasasareaof4.Wedon’tusuallygetorderslikethat,butitstillmakessense.Ibettergetbacktothinkingaboutparabolas.Hmmm…”

1. FullydescribetheparabolathatJennyhasbeenassignedtograph.

2. Jennyreturnstoherhomework,whichisaboutgraphingquadraticfunctions.Muchtoherdismay,shefindsthatshehasbeengiven:! = !! − 6! + 9. “Ohdear”,thinksJenny.“Ican’ttellwherethevertexisoridentifyanyofthetransformationsoftheparabolainthisform.NowwhatamIsupposedtodo?”

“Waitaminute—isthistheareaofaperfectsquare?”UseyourworkfromBuildingthePerfectSquaretoanswerJenny’squestionandjustifyyouranswer.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.5

Mathematics Vision Project

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

3. Jennysays,“IthinkI’vefiguredouthowtochangetheformofmyquadraticequationsothatIcangraphtheparabola.I’llchecktoseeifIcanmakemyequationaperfectsquare.”Jenny’sequationis:! = !! − 6! + 9.

Seeifyoucanchangetheformoftheequation,findthevertex,andgraphtheparabola.

a. ! = !! − 6! + 9 Newformoftheequation:_________________________________

b. Vertexoftheparabola:______________

c. Graph(withatleast3accuratepointsoneachsideofthelineofsymmetry):

4. ThenextquadratictographonJenny’shomeworkis! = !! + 4! + 2.Doesthisexpressionfitthepatternforaperfectsquare?Whyorwhynot?

a. Useanareamodeltofigureouthowtocompletethesquaresothattheequationcanbewritteninvertexform,! = ! ! − ℎ ! + !.

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.5

Mathematics Vision Project

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

b. Istheequationyouhavewrittenequivalenttotheoriginalequation?Ifnot,whatadjustmentsneedtobemade?Why?

c. Identifythevertexandgraphtheparabolawiththreeaccuratepointsonbothsidesofthelineofsymmetry.

5. Jennyhopedthatshewasn’tgoingtoneedtofigureouthowtocompletethesquareonanequationwherebisanoddnumber.Ofcourse,thatwasthenextproblem.HelpJennytofindthevertexoftheparabolaforthisquadraticfunction:

! ! = !! + 7! + 10

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.5

Mathematics Vision Project

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

6. Don’tworryifyouhadtothinkhardabout#5.Jennyhastodoacouplemore:a. ! ! = !! − 5! + 3 b. ! ! = !! − ! − 5

7. Itjustgetsbetter!HelpJennyfindthevertexandgraphtheparabolaforthequadraticfunction:ℎ ! = 2!! − 12! + 17

8. Thisoneisjusttoocute—you’vegottotryit!Findthevertexanddescribetheparabolathatisthegraphof:! ! = !

! !! + 2! − 3

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