Structures of Expressions - MR. CONGLETON · WCPSS Math 2 Unit 4: MVP MODULE 2 Structures of...

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2017 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

WCPSS Math 2 Unit 4: MVP MODULE 2

Structures of Expressions

SECONDARY

MATH TWO

An Integrated Approach

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SECONDARY MATH 2 // MODULE 2

QUADRATIC FUNCTIONS

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MODULE 2 - TABLE OF CONTENTS

STRUCTURES OF EXPRESSIONS

2.1 Transformers: Shifty y’s – A Develop Understanding Task

Connecting transformations to quadratic functions and parabolas

(NC.M2.F-IF.7, NC.M2.F-BF.3)

READY, SET, GO Homework: Structure of Expressions 2.1

2.2 Transformers: More Than Meets the y’s – A Solidify Understanding Task

Working with vertex form of a quadratic, connecting the components to transformations

(NC.M2.F-IF.7, NC.M2.F-BF.3)


2.3 Building the Perfect Square – A Develop Understanding Task

Visual and algebraic approaches to completing the square (NC.M2.F-IF.8)


2.4 A Square Deal– A Solidify Understanding Task



2.5 Be There or Be Square– A Practice Understanding Task



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SECONDARY MATH 2 // MODULE 2

QUADRATIC FUNCTIONS

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2.8 The Wow Factor – A Solidify Understanding Task

Connecting the factored and expanded forms of a quadratic when a-value is not equal to one.

(NC.M2.F-IF.8, NC.M2.A-SSE.3)


2.9 Lining Up Quadratics – A Solidify Understanding Task Focus on

the vertex and intercepts for quadratics (NC.M2.F-IF.7, NC.M2.F-BF.1)


2.10 I’ve Got a Fill-in – A Practice Understanding Task

Building fluency in rewriting and connecting different forms of a quadratic

(NC.M2.F-IF.8, NC.M2.F-BF.1, NC.M2.A-SSE.3)

READY, SET, GO Homework: Structure of Expressions

3.4 Pascal's Pride – A Solidify Understanding Task

Building fluency in multiplying polynomials. (NC.M2.A-APR.1)

READY, SET, GO Homework: Polynomial Functions

3.3 It All Adds Up – A Develop Understanding Task

Building fluency in adding and subtratcting polynomials. (NC.MA-APR.1)

READY, SET, GO Homework: Polynomial Functions

2.6 Factor Fixin’ – A Solidify Understanding Task Connecting the factored and expanded forms of a quadratic(NC.M2.F-IF.8, NC.M2.F-BF.1, NC.M2.A-SSE.3) READY, SET, GO Homework: Structure of Expressions 2.6

2.7 The x Factor – A Solidify Understanding Task Connecting the factored and expanded or standard forms of a quadratic(NC.M2.F-IF.8, NC.M2.F-BF.1, NC.M2.A-SSE.3) READY, SET, GO Homework: Structure of Expressions 2.7

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SECONDARY MATH II // MODULE 2

STRUCTURES OF EXPRESSIONS – 2.1

Mathematics Vision Project

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2.1 Transformers: Shifty y’s

A Develop Understanding Task

OptimaPrimeisdesigningarobotquiltforhernewgrandson.Sheplansfortherobottohavea

squareface.Theamountoffabricthatsheneedsforthefacewilldependontheareaoftheface,so

Optimadecidestomodeltheareaoftherobot’sfacemathematically.SheknowsthattheareaAofa

squarewithsidelengthxunits(whichcanbeinchesorcentimeters)ismodeledbythe

function, ! ! = !!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)andwonderingabouthowchangestotheequationofthefunctionlikeadding5ormultiplyingby5

affectthegraph.Shedecidedtomakepredictionsabouttheeffectsandthencheckthemout.

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3. Predicthowthegraphsofeachofthefollowingequationswillbethesameordifferentfrom

thegraphof! = !!.Similaritiestothegraphof

! = !!Differencesfromthegraphof

! = !!! = 5!!

! = (! + 5)!

! = (5!)!

! = !! + 5

4. Optimadecidedtotestherideasusingtechnology.Shethinksthatitisalwaysagoodidea

tostartsimple,soshedecidestogowith! = !! + 5.Shegraphsitalongwith! = !!inthesamewindow.Testityourselfanddescribewhatyoufind.

5. Knowingthatthingsmakealotmoresensewithmorerepresentations,Optimatriesafew

moreexampleslike! = !! + 2 and! = !! − 3,lookingatbothatableandagraphforeach.Whatconclusionwouldyoudrawabouttheeffectofaddingorsubtractinganumberto! = !!?Carefullyrecordthetablesandgraphsoftheseexamplesinyournotebookandexplainwhyyour

conclusionwouldbetrueforanyvalueofk,given,! = !! + !.

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6. Afterheramazingsuccesswithadditioninthelastproblem,Optimadecidedtolookatwhat

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

7. Optimathoughtthat#6wasverytrickyandhopedthatmultiplicationwasgoingtobemore

straightforward.Shedecidestostartsimpleandmultiplyby-1,soshebeginswith

! = −!!.Predictwhattheeffectisonthegraphandthentestit.Whydoesithavethiseffect?

8. Optimaisencouragedbecausethatonewaseasy.Shedecidestoendherinvestigationfor

thedaybydeterminingtheeffectofamultiplier,a,intheequation:! = !!!.Usingbothpositiveandnegativenumbers,fractionsandintegers,createatleast4tablesandmatchinggraphsto

determinetheeffectofamultiplier.

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STRUCTURES OF EXPRESSIONS – 2.1 2.1

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READY Topic:Findingkeyfeaturesinthegraphofaquadraticequation

Makeapointonthevertexanddrawadottedlinefortheaxisofsymmetry.Labelthecoordinatesofthevertexandstatewhetherit’samaximumoraminimum.Writetheequationfortheaxisofsymmetry.

1. 2. 3.

4. 5. 6.

7. Whatconnectionexistsbetweenthecoordinatesofthevertexandtheequationoftheaxisofsymmetry?

8. Lookbackat#6.Trytofindawaytofindtheexactvalueofthecoordinatesofthevertex.Testyourmethodwitheachvertexin1-5.Explainyourconjecture.

9. Howmanyx-interceptscanaparabolahave?

10. Sketchaparabolathathasnox-intercepts,thenexplainwhathastohappenforaparabolatohavenox-intercepts.

READY, SET, GO! Name Period Date

8

6

4

2

–2

–5 5

8

6

4

2

5

8

6

4

2

–2

–5 5

6

4

2

–2

–4

–5

4

2

–2

–4

–6

5

6

4

2

–2

–4

5

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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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16a)g(x)= b)

x -3 -2 -1 0 1 2 3g(x) 11 6 3 2 3 6 11

c) Inwhatwaydidthegraphmove?d) Whatpartoftheequationindicatesthismove?

17a)g(x)= b)

x -4 -3 -2 -1 0 1 2g(x) 9 4 1 0 1 4 9


18a)g(x)= b)

x 0 1 2 3 4 5 6g(x) 9 4 1 0 1 4 9


GO Topic:FindingSquareRootsSimplifythefollowingexpressions

19. 49a!b! 20. x + 13 ! 21. x − 16 !

22. 36x + 25 ! 23. 11x − 7 ! 24. 9!! 2!! − ! !

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2.2 Transformers: More Than

Meets the y’s

A Solidify Understanding Task

Writetheequationforeachproblembelow.Useasecond

representationtocheckyourequation.

1. Theareaofasquarewithsidelengthx,wheretheside

lengthisdecreasedby3,theareaismultipliedby2andthen4squareunitsareaddedtothearea.

2.

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3.

4.

x f(x)

-4 7

-3 2

-2 -1

-1 -2

0 -1

1 2

2 7

3 14

4 23

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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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READY Topic:Standardformofquadraticequations

Thestandardformofaquadraticequationisdefinedas ! = !!! + !" + !, (! ≠ !).Identifya,b,andcinthefollowingequations.

Example:Given!!! + !" − !,a=4,b=7,andc=-61. y = 5x! + 3x + 6 2. y = x! − 7x + 3 3. y = −2x! + 3x

a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________

4. y = 6x! − 5 5. y = −3x! + 4x 6. y = 8x! − 5x − 2

a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________

Multiplyandwriteeachproductintheformy=ax2+bx+c.Thenidentifya,b,andc.7. y = x x − 4 8. y = x − 1 2x − 1 9. y = 3x − 2 3x + 2

a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________

10. y = x + 6 x + 6 11. y = x − 3 ! 12. y = − x + 5 !

a=____________ a=____________ a=____________b=____________ b=____________ b=____________c=____________ c=____________ c=____________


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SET

Topic:Graphingastandardy=x2parabola13. Graphtheequationy = x!.Includeatleast3accuratepointsoneachsideoftheaxisofsymmetry.

a. Statethevertexoftheparabola.

b. Completethetableofvaluesfory = x!.! ! !-3

-2

-1

0

1

2

3

Topic:Writingtheequationofatransformedparabolainvertexform.Findavaluefor!suchthatthegraphwillhavethespecifiednumberofx-intercepts.14. ! = !! +! 15. ! = !! +! 16. ! = !! +!

2(x-intercepts) 1(x-intercept) no(x-intercepts)

17. y = −x! + ω 18. y = −x! + ω 19. y = −x! + ω2(x-intercepts) 1(x-intercept) no(x-intercepts)

Graphthefollowingequations.Statethevertex.(Beaccuratewithyourkeypointsandshape!)20. y = x − 1 ! 21. y = x − 1 ! + 1 22. y = 2 x − 1 ! + 1

Vertex?_____________________ Vertex?_____________________ Vertex?_____________________

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23. y = x + 3 ! 24. y = − x + 3 ! − 4 25. y = −0.5 x + 1 ! + 4

Vertex?_____________________Vertex?_____________________ Vertex?_____________________

GO Topic:FeaturesofParabolasUsethetabletoidentifythevertex,theequationfortheaxisofsymmetry(AoS),andstatethe

numberofx-intercept(s)theparabolawillhave,ifany.Statewhetherthevertexwillbea

minimumoramaximum.

26. x y-4-3-2-1012

103-2-5-6-5-2

27. x y-2-101234

49281341413

28. x y-7-6-5-4-3-2-1

-9373-9-29-57

29. x y-8-7-6-5-4-3-2

-9-8-9-12-17-24-33

a.

b.

c.

d.

Vertex:_________

AoS:____________

x-int(s):________

MINorMAX

a.

b.

c.

d.

Vertex:_________

AoS:____________

x-int(s):________

MINorMAX

a.

b.

c.

d.

Vertex:_________

AoS:____________

x-int(s):________

MINorMAX

a.

b.

c.

d.

Vertex:_________

AoS:____________

x-int(s):________

MINorMAX

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2.3 Building the Perfect Square


QuadraticQuilts

Optimahasaquiltshopwhereshesellsmanycolorful

quiltblocksforpeoplewhowanttomaketheirown

quilts.Shehasquiltdesignsthataremadesothattheycanbesizedtofitanybed.She

basesherdesignsonquiltsquaresthatcanvaryinsize,soshecallsthelengthofthe

sideforthebasicsquarex,andtheareaofthebasicsquareisthefunction! ! = !!.Inthisway,shecancustomizethedesignsbymakingbiggersquaresorsmallersquares.

1. IfOptimaadds3inchestothesideofthesquare,whatistheareaofthesquare?

WhenOptimadrawsapatternforthesquareinproblem#1,itlookslikethis:

2. Useboththediagramandtheequation,! ! = (! + 3)!toexplainwhytheareaofthequiltblocksquare,! ! ,isalsoequalto!! + 6! + 9.

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ThecustomerservicerepresentativesatOptima’sshopworkwithcustomerordersandwrite

uptheordersbasedontheareaofthefabricneededfortheorder.Asyoucanseefrom

problem#2therearetwowaysthatcustomerscancallinanddescribetheareaofthequilt

block.Onewaydescribesthelengthofthesidesoftheblockandtheotherwaydescribesthe

areasofeachofthefoursectionsoftheblock.

Foreachofthefollowingquiltblocks,drawthediagramoftheblockandwritetwoequivalent

equationsfortheareaoftheblock.

3. Blockwithsidelength:! + 2.

4. Blockwithsidelength:! + 1.

5. Whatpatternsdoyounoticewhenyourelatethediagramstothetwoexpressionsfor

thearea?

6. Optimalikestohaveherlittledog,Clementine,aroundtheshop.Onedaythedoggota

littlehungryandstartedtochewuptheorders.WhenOptimafoundtheorders,oneof

themwassochewedupthattherewereonlypartialexpressionsforthearea

remaining.HelpOptimabycompletingeachofthefollowingexpressionsforthearea

sothattheydescribeaperfectsquare.Then,writethetwoequivalentequationsforthe

areaofthesquare.

a. !! + 4!

b. !! + 6!

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c. !! + 8!

d. !! + 12!

7. If !! + !" + !isaperfectsquare,whatistherelationshipbetweenbandc?Howdoyouusebtofindc,likeinproblem6?

Willthisstrategyworkifbisnegative?Whyorwhynot?

Willthestrategyworkifbisanoddnumber?Whathappenstocifbisodd?

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READY Topic:Graphinglinesusingtheintercepts

Findthex-interceptandthey-intercept.Thengraphtheequation.

1. 3! + 2! = 12 2. 8! − 12! = −24 3. 3! − 7! = 21a. x-intercept: a. x-intercept: a. x-intercept:b. y-intercept: b. y-intercept: b. y-intercept:

4. 5! − 10! = 20 5. 2! = 6! − 18 6. ! = −6! + 6a. x-intercept: a. x-intercept: a. x-intercept:b. y-intercept: b. y-intercept: b. y-intercept:


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SET

Topic:CompletingthesquarebypayingattentiontothepartsMultiply.Showeachstep.Circlethepairofliketermsbeforeyousimplifytoatrinomial.

7. ! + 5 ! + 5 8. 3! + 7 3! + 7 9. 9! + 1 ! 10. 4! + 11 !

11. Writearuleforfindingthecoefficient“B”ofthex-term(themiddleterm)whenmultiplyingandsimplifying ax + q !.

Inproblems12–17, (a)Fillinthenumberthatcompletesthesquare.(b)Thenwritethetrinomialastheproductoftwofactors.

12.a)x! + 8x + _____ 13.a)x! + 10x + _____ 14.a) x! + 16x + _____

b) b) b)

15.a)x! + 6x + _____ 16.a)x! + 22x + _____ 17.a) x! + 18x + _____

b) b) b)

Inproblems18–26, (a)Findthevalueof“B,”thatwillmakeaperfectsquaretrinomial.(b)Thenwritethetrinomialasaproductoftwofactors.

18. x! + Bx + 16a)b)

19. x! + Bx + 121a)b)

20. x! + Bx + 625a)b)

21. x! + Bx + 225a)b)

22. x! + Bx + 49a)b)

23. x! + Bx + 169a)b)

24. x! + Bx + !"!

a)b)

25. x! + Bx + !!

a)b)

26. x! + Bx + !"!

a)b)

GO Topic:FeaturesofhorizontalandverticallinesFindtheinterceptsofthegraphofeachequation.Statewhetherit’sanx-ory-intercept.

27. y=-4.5 28. x=9.5 29. x=-8.2 30. y=112

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2.4 A Square Deal


QuadraticQuilts,Revisited

RememberOptima’squiltshop?Shebasesherdesignsonquiltsquaresthatcanvaryinsize,soshecallsthelengthofthesideforthebasicsquarex,andtheareaofthebasicsquareisthefunctionA ! = !!.Inthisway,shecancustomizethedesignsbymakingbiggersquaresorsmallersquares.

1. Sometimesacustomerordersmorethanonequiltblockofagivensize.Forinstance,whenacustomerorders4blocksofthebasicsize,thecustomerservicerepresentativeswriteupanorderforA ! = 4!!.Modelthisareaexpressionwithadiagram.

2. Oneofthecustomerservicerepresentativesfindsanenvelopethatcontainstheblockspicturedbelow.Writetheorderthatshowstwoequivalentequationsfortheareaoftheblocks.

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3. Whatequationsfortheareacouldcustomerservicewriteiftheyreceivedanorderfor2blocksthataresquaresandhavebothdimensionsincreasedby1inchincomparisontothebasicblock?Writetheareaequationsintwoequivalentforms.Verifyyouralgebrausingadiagram.

4. Ifcustomerservicereceivesanorderfor3blocksthatareeachsquareswithbothdimensionsincreasedby2inchesincomparisontothebasicblock?Again,show2differentequationsfortheareaandverifyyourworkwithamodel.

5. Clementineisatitagain!Whenisthatdoggoingtolearnnottochewuptheorders?(ShealsochewsOptima’sshoes,butthat’sastoryforanotherday.)Herearesomeoftheordersthathavebeenchewedupsotheyaremissingthelastterm.HelpOptimabycompletingeachofthefollowingexpressionsfortheareasothattheydescribeaperfectsquare.Then,writethetwoequivalentequationsfortheareaofthesquare.

2!! + 8!

3x! + 24!

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Sometimesthequiltshopgetsanorderthatturnsoutnottobemoreorlessthanaperfectsquare.Customerservicealwaystriestofillorderswithperfectsquares,oratleast,theystartthereandthenadjustasneeded.

6. Nowhere’sarealmess!CustomerservicereceivedanorderforanareaA x = 2!! + 12! + 13.Helpthemtofigureoutanequivalentexpressionfortheareausingthediagram.

7. Optimareallyneedstogetorganized.Here’sanotherscrambleddiagram.Writetwoequivalentequationsfortheareaofthisdiagram:

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8. Optimarealizedthatnoteveryoneisinneedofperfectsquaresandnotallordersarecominginasexpressionsthatareperfectsquares.Determinewhetherornoteachexpressionbelowisaperfectsquare.Explainwhytheexpressionisorisnotaperfectsquare.Ifitisnotaperfectsquare,findtheperfectsquarethatseems“closest”tothegivenexpressionandshowhowtheperfectsquarecanbeadjustedtobethegivenexpression.

A. ! ! = !! + 6! + 13 B. ! ! = !! − 8! + 16

C. ! ! = !! − 10! − 3 D. ! ! = 2!! + 8! + 14

E. ! ! = 3!! − 30! + 75 F. ! ! = 2!! − 22! + 11

9. Nowlet’sgeneralize.Givenanexpressionintheform!"! + !" + ! (! ≠ 0),describeastep-by-stepprocessforcompletingthesquare.

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READY

Topic:Findy-interceptsinparabolas

Statethey-interceptforeachofthegraphs.Thenmatchthegraphwithitsequation.

1. 2. 3.

4. 5. 6.

a. ! ! = −!! + 2! − 1 b. ! ! = −. 25!! − 2! + 5 c. ! ! = !! + 3! − 5

d. ! ! =. 5!! + ! − 7 e. ! ! = −.25!! + 3! + 1 f. ! ! = !! − 4! + 4


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SET

Topic:Completingthesquarewhena>1.Fillinthemissingconstantsothateachexpressionrepresents5perfectsquares.Thenstatethedimensionsofthesquaresineachproblem.

7. 5!! + 30! + _______ 8. 5!! − 50! + _______ 9. 5!! + 10! + _______

10. Giventhescrambleddiagrambelow,writetwoequivalentequationsforthearea.

Determineifeachexpressionbelowisaperfectsquareornot.Ifitisnotaperfectsquare,find

theperfectsquarethatseems“closest”tothegivenexpressionandshowhowtheperfectsquare

canbeadjustedtobethegivenexpression.

11. ! ! = !! + 10! + 14 12. ! ! = 2!! + 16! + 613.! ! = 3!! + 18! − 12

GO Topic:Evaluatingfunctions.

Findtheindicatedfunctionvaluewhen! ! = !!! − !! − !" !"# ! ! = −!!! + ! − !.14. ! 1 15.! 5 16. ! 10 17. ! −5 18. ! 0 + ! 0

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2.5 Be There or Be Square

A Practice Understanding Task

QuiltsandQuadraticGraphs

Optima’sniece,Jennyworksintheshop,takingordersanddrawingquiltdiagrams.Whenthe

shopisn’ttoobusy,Jennypullsouthermathhomeworkandworksonit.Oneday,sheis

workingongraphingparabolasandnoticesthattheequationssheisworkingwithlooksalot

likeanorderforaquiltblock.Forinstance, Jennyissupposedtographtheequation:! = (! − 3)! + 4 . Shethinks,“That’sfunny.Thiswouldbeanorderwherethelengthofthestandardsquareisreducedby3andthenweaddalittlepieceoffabricthathasasareaof4.

Wedon’tusuallygetorderslikethat,butitstillmakessense.Ibettergetbacktothinking

aboutparabolas.Hmmm…”

1. FullydescribetheparabolathatJennyhasbeenassignedtograph.

2. Jennyreturnstoherhomework,whichisaboutgraphingquadraticfunctions.Muchtoher

dismay,shefindsthatshehasbeengiven:! = !! − 6! + 9. “Ohdear”,thinksJenny.“Ican’ttellwherethevertexisoridentifyanyofthetransformationsoftheparabolainthis

form.NowwhatamIsupposedtodo?”

“Waitaminute—isthistheareaofaperfectsquare?”UseyourworkfromBuildingthePerfectSquaretoanswerJenny’squestionandjustifyyouranswer.

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3. Jennysays,“IthinkI’vefiguredouthowtochangetheformofmyquadraticequationso

thatIcangraphtheparabola.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. Useanareamodeltofigureouthowtocompletethesquaresothattheequation

canbewritteninvertexform,! = ! ! − ℎ ! + !.

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b. Istheequationyouhavewrittenequivalenttotheoriginalequation?Ifnot,what

adjustmentsneedtobemade?Why?

c. Identifythevertexandgraphtheparabolawiththreeaccuratepointsonbothsides

ofthelineofsymmetry.

5. Jennyhopedthatshewasn’tgoingtoneedtofigureouthowtocompletethesquareonan

equationwherebisanoddnumber.Ofcourse,thatwasthenextproblem.HelpJennytofindthevertexoftheparabolaforthisquadraticfunction:

! ! = !! + 7! + 10

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6. Don’tworryifyouhadtothinkhardabout#5.Jennyhastodoacouplemore:

a. ! ! = !! − 5! + 3 b. ! ! = !! − ! − 5

7. Itjustgetsbetter!HelpJennyfindthevertexandgraphtheparabolaforthequadratic

function:ℎ ! = 2!! − 12! + 17

8. Thisoneisjusttoocute—you’vegottotryit!Findthevertexanddescribetheparabola

thatisthegraphof:! ! = !! !

! + 2! − 3

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READY

Topic:RecognizingQuadraticEquationsIdentifywhetherornoteachequationrepresentsaquadraticfunction.Explainhowyouknewitwasaquadratic.

1. x! + 13x − 4 = 0 2. 3x! + x = 3x! − 2 3. x 4x − 5 = 0Quadraticorno?

Justification:

Quadraticorno?

Justification:

Quadraticorno?

Justification:

4. 2x − 7 + 6x = 10 5. 2! + 6 = 0 6. 32 = 4x!

Quadraticorno?

Justification:

Quadraticorno?

Justification:

Quadraticorno?

Justification:

SET

Topic:Changingfromstandardformofaquadratictovertexform.Changetheformofeachequationtovertexform:! = ! ! − ! ! + !.Statethevertexandgraphtheparabola.Showatleast3accuratepointsoneachsideofthelineofsymmetry.

7. y = x! − 4x + 1

vertex:

8. y = x! + 2x + 5

vertex:


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9. y = x! + 3x + !"!

vertex:

10. y = !! x

! − x + 5

vertex:

11. Oneoftheparabolasinproblems9–10shouldlook“wider”thantheothers.Identifytheparabola.Explainwhythisparabolalooksdifferent.

Fillintheblankbycompletingthesquare.Leavethenumberthatcompletesthesquareasanimproperfraction.Thenwritethetrinomialinfactoredform.

12. 13. 14.

15. 16. 17.

x2 −11x + _________ x2 + 7x + _________ x2 +15x + _________

x2 + 23x + _________ x2 − 1

5x + _________ x2 − 3

4x + _________

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GO Topic:Writingrecursiveequationsforquadraticfunctions.

Identifywhetherthetablerepresentsalinearorquadraticfunction.Ifthefunctionislinear,writeboththeexplicitandrecursiveequations.Ifthefunctionisquadratic,writeonlytherecursiveequation.

18. 19.

Typeoffunction:

Equation(s):

! ! !1 02 33 64 95 12

Typeoffunction:

Equation(s):

! ! !1 72 103 164 255 37

20. 21.

Typeoffunction:

Equation(s):

! ! !1 82 103 124 145 16

Typeoffunction:

Equation(s):

! ! !1 282 403 544 705 88

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2.6 Factor Fixin’


Atfirst,Optima’sQuiltsonlymadesquareblocksforquiltersandOptimaspenthertimemaking

perfectsquares.Customerservicerepresentativesweretrainedtoaskforthelengthoftheside

oftheblock,x,thatwasbeingordered,andtheywouldletthecustomerknowtheareaoftheblock

tobequiltedusingtheformula! ! = !!.Optimafoundthatmanycustomersthatcameintothestoreweremakingdesignsthatrequireda

combinationofsquaresandrectangles.So,Optima’sQuiltshasdecidedtoproduceseveralnew

linesofrectangularquiltblocks.Eachnewlineisdescribedintermsofhowtherectangularblock

hasbeenmodifiedfromtheoriginalsquareblock.Forexample,onelineofquiltblocksconsistsof

startingwithasquareblockandextendingonesidelengthby5inchesandtheothersidelengthby

2inchestoformanewrectangularblock.Thedesigndepartmentknowsthattheareaofthisnew

blockcanberepresentedbytheexpression:A(x)=(x+5)(x+2),buttheydonotfeelthatthis

expressiongivesthecustomerarealsenseofhowmuchbiggerthisnewblockis(e.g.,howmuch

moreareaithas)whencomparedtotheoriginalsquareblocks.

1. Canyoufindadifferentexpressiontorepresenttheareaofthisnewrectangularblock?You

willneedtoconvinceyourcustomersthatyourformulaiscorrectusingadiagram.

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HerearesomeadditionalnewlinesofblocksthatOptima’sQuiltshasintroduced.Findtwodifferent

algebraicexpressionstorepresenteachrectangle,andillustratewithadiagramwhyyour

representationsarecorrect.

2. Theoriginalsquareblockwasextended3inchesononesideand4inchesontheother.

3. Theoriginalsquareblockwasextended4inchesononlyoneside.

4. Theoriginalsquareblockwasextended5inchesoneachside.

5. Theoriginalsquareblockwasextended2inchesononesideand6inchesontheother.

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Customersstartorderingcustom-madeblockdesignsbyrequestinghowmuchadditionalareathey

wantbeyondtheoriginalareaof!!.Onceanorderistakenforacertaintypeofblock,customerserviceneedstohavespecificinstructionsonhowtomakethenewdesignforthemanufacturing

team.Theinstructionsneedtoexplainhowtoextendthesidesofasquareblocktocreatethenew

lineofrectangularblocks.

Thecustomerservicedepartmenthasplacedthefollowingordersonyourdesk.Foreach,describe

howtomakethenewblocksbyextendingthesidesofasquareblockwithaninitialsidelengthof

!.Yourinstructionsshouldincludediagrams,writtendescriptionsandalgebraicdescriptionsoftheareaoftherectanglesinusingexpressionsrepresentingthelengthsofthesides.

6. !! + 5! + 3! + 15

7. !! + 4! + 6! + 24

8. !! + 9! + 2! + 18

9. !! + 5! + ! + 5

Someoftheordersarewritteninanevenmoresimplifiedalgebraiccode.Figureoutwhatthese

entriesmeanbyfindingthesidesoftherectanglesthathavethisarea.Usethesidesoftherectangle

towriteequivalentexpressionsforthearea.

10. !! + 11! + 10

11. !! + 7! + 10

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12. !! + 9! + 8

13. !! + 6! + 8

14. !! + 8! + 12

15. !! + 7! + 12

16. !! + 13! + 12

17. Whatrelationshipsorpatternsdoyounoticewhenyoufindthesidesoftherectanglesfora

givenareaofthistype?

18. Acustomercalledandaskedforarectanglewithareagivenby:!! + 7! + 9.Thecustomerservicerepresentativesaidthattheshopcouldn’tmakethatrectangle.Doyouagreeordisagree?

Howcanyoutellifarectanglecanbeconstructedfromagivenarea?

Page 36





READY

Topic:CreatingBinomialQuadraticsMultiply.(Usethedistributiveproperty,writeinstandardform.)

1. x 4x − 7 2. 5x 3x + 8 3. 3x 3x − 2

4. Theanswerstoproblems1,2,&3arequadraticsthatcanberepresentedinstandardformax2+bx+c.Whichcoefficient,a,b,orcequals0foralloftheexercisesabove?

Factorthefollowing.(Writetheexpressionsastheproductoftwolinearfactors.)

5. !! + 4! 6. 7!! − 21! 7. 12!! + 60! 8. 8!! + 20!

Multiply9. ! + 9 ! − 9 10. ! + 2 ! − 2 11. 6! + 5 6! − 5 12. 7! + 1 7! − 1

13. Theanswerstoproblems9,10,11,&12arequadraticsthatcanberepresentedinstandardformax2+bx+c.Whichcoefficient,a,b,orcequals0foralloftheexercisesabove?

SET

Topic:FactoringTrinomialsFactorthefollowingquadraticexpressionsintotwobinomials.14. !! + 14! + 45 15. !! + 18! + 45 16. !! + 46! + 45

17. !! + 11! + 24 18. !! + 10! + 24 19. !! + 14! + 24

20. !! + 12! + 36 21. !! + 13! + 36 22. !! + 20! + 36

23. !! − 15! − 100 24. !! + 20! + 100 25. !! + 29! + 100


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26. Lookbackateach“row”offactoredexpressionsinproblems14to25above.Explainhowitispossiblethatthecoefficient(b)ofthemiddletermcanbedifferentnumbersineachproblemwhenthe“outside”coefficients(a)and(c)arethesame.(Recallthestandardformofaquadraticis!!! + !" + !.)

GO Topic:Takingthesquarerootofperfectsquares.

Onlysomeoftheexpressionsinsidetheradicalsignareperfectsquares.Identifywhichonesareperfectsquaresandtakethesquareroot.Leavetheonesthatarenotperfectsquaresundertheradicalsign.Donotattempttosimplifythem.(Hint:Checkyouranswersbysquaringthem.Youshouldbeabletogetwhatyoustartedwith,ifyouareright.)

27. 17xyz ! 28. 3x − 7 ! 29. 121a!b!

30. x! + 8x + 16 31. x! + 14x + 49 32. x! + 14x − 49

33. x! + 10x + 100 34. x! + 20x + 100 35. x! − 20x + 100

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2.7 The x Factor


NowthatOptima’sQuiltsisacceptingordersforrectangularblocks,theirbusinessingrowingby

leapsandbounds.Manycustomerwantrectangularblocksthatarebiggerthanthestandard

squareblockononeside.Sometimestheywantonesideoftheblocktobethestandardlength,!,withtheothersideoftheblock2inchesbigger.

1. Drawandlabelthisblock.Writetwodifferentexpressionsfortheareaoftheblock.

Sometimestheywantblockswithonesidethatisthestandardlength,x,andonesidethatis2

incheslessthanthestandardsize.

2. Drawandlabelthisblock.Writetwodifferentexpressionsfortheareaoftheblock.Useyour

diagramandverifyalgebraicallythatthetwoexpressionsareequivalent.

Therearemanyothersizeblocksrequested,withthesidelengthsallbasedonthestandardlength,

!.Drawandlabeleachofthefollowingblocks.Useyourdiagramstowritetwoequivalentexpressionsforthearea.Verifyalgebraicallythattheexpressionsareequal.

3. Onesideis1”lessthanthestandardsizeandtheothersideis2”morethanthestandardsize.

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4. Onesideis2”lessthanthestandardsizeandtheothersideis3”morethanthestandardsize.

5. Onesideis2”morethanthestandardsizeandtheothersideis3”lessthanthestandardsize.



8. Anexpressionthathas3termsintheform:!!! + !" + !iscalledatrinomial.Lookbackatthetrinomialsyouwroteinquestions3-7.Howcanyoutellifthemiddleterm(!")isgoingtobepositiveornegative?

9. Onecustomerhadanunusualrequest.Shewantedablockthatisextended2inchesononeside

anddecreasedby2inchesontheother.Oneoftheemployeesthinksthatthisrectanglewillhave

thesameareaastheoriginalsquaresinceonesidewasdecreasedbythesameamountastheother

sidewasincreased.Whatdoyouthink?Useadiagramtofindtwoexpressionsfortheareaofthis

block.

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10. Theresultoftheunusualrequestmadetheemployeecurious.Isthereapatternorawayto

predictthetwoexpressionsforareawhenonesideisincreasedandtheothersideisdecreasedby

thesamenumber?Trymodelingthesetwoproblems,lookatyouranswerto#8,andseeifyoucan

findapatternintheresult.

a. (! + 1)(! − 1)

b. (! + 3)(! − 3)

11. Whatpatterndidyounotice?Whatistheresultof(! + !)(! − !)?

12. Somecustomerswantbothsidesoftheblockreduced.Drawthediagramforthefollowing

blocksandfindatrinomialexpressionfortheareaofeachblock.Usealgebratoverifythetrinomial

expressionthatyoufoundfromthediagram.

a. (! − 2)(! − 3)

b. (! − 1)(! − 4)

13. Lookbackoveralltheequivalentexpressionsthatyouhavewrittensofar,andexplainhowto

tellifthethirdterminthetrinomialexpression!!! + !" + !willbepositiveornegative.

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14. Optima’squiltshophasreceivedanumberofordersthataregivenasrectangularareasusinga

trinomialexpression.Findtheequivalentexpressionthatshowsthelengthsofthetwosidesofthe

rectangles.

a. !! + 9! + 18

b. !! + 3x − 18

c. !! − 3! − 18

d. !! − 9! + 18

e. !! − 5! + 4

f. !! − 3! − 4

g. !! + 2! − 15

15. Writeanexplanationofhowtofactoratrinomialintheform:!! + !" + !.

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READY

Topic:Exploringthedensityofthenumberline.Findthreenumbersthatarebetweenthetwogivennumbers.

1. 5 !! !"# 6 !! 2. −2 !! !"# − 1!! 3. !! !"# !! 4. 3 !"# 5

5. 4 !"# 23 6. −9 !! !"# − 8.5 7. !! !"# !

!8. 13 !"# 14

SET

Topic:FactoringQuadraticsTheareaofarectangleisgivenintheformofatrinomialexpression.Findtheequivalentexpressionthatshowsthelengthsofthetwosidesoftherectangle.

9. x! + 9x + 8 10. x! − 6x + 8 11. x! − 2x − 8 12. x! + 7x − 8

13. x! − 11x + 24 14. x! − 14x + 24 15. x! − 25x + 24 16. x! − 10x + 24

17. x! − 2x − 24 18. x! − 5x − 24 19. x! + 5x − 24 20. x! − 10x + 25

21. x! − 25 22. x! − 2x − 15 23. x! + 10x − 75 24. x! − 20x + 51

25. x! + 14x − 32 26. x! − 1 27. x! − 2x + 1 28. x! + 12x − 45


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GO Topic:GraphingParabolas

Grapheachparabola.Includethevertexandatleast3accuratepointsoneachsideoftheaxisofsymmetry.Thendescribethetransformationinwords.

29. ! ! = !!

Description:

30. ! ! = !! − 3

Description:

31. ℎ ! = ! − 2 !

Description:

32. ! ! = − ! + 1 ! + 4

Description:

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2.8 The Wow Factor


Optima’sQuiltssometimesgetsordersforblocksthataremultiplesofagivenblock.Forinstance,

Optimagotanorderforablockthatwasexactlytwiceasbigastherectangularblockthathasaside

thatis1”longerthanthebasicsize,!, andonesidethatis3”longerthanthebasicsize.1. Drawandlabelthisblock.Writetwoequivalentexpressionsfortheareaoftheblock.

2. Ohdear!Thisorderwasscrambledandthepiecesareshownhere.Putthepiecestogetherto

makearectangularblockandwritetwoequivalentexpressionsfortheareaoftheblock.

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3. Whatdoyounoticewhenyoucomparethetwoequivalentexpressionsinproblems#1and#2?

4. Optimahasalotofneworders.Usediagramstohelpyoufindequivalentexpressionsforeachof

thefollowing:

a. 5!! + 10!

b. 3!! + 21! + 36

c. 2!! + 2! − 4

d. 2!! − 10! + 12

e. 3!! − 27

Becausesheisagreatbusinessmanager,Optimaoffershercustomerslotsofoptions.Oneoptionis

tohaverectanglesthathavesidelengthsthataremorethanone!.Forinstance,Optimamadethiscoolblock:

5. Writetwoequivalentexpressionsforthisblock.Usethedistributivepropertytoverifythat

youransweriscorrect.

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6. Herewehavesomepartialorders.Wehaveoneoftheexpressionsfortheareaoftheblock

andweknowthelengthofoneofthesides.Useadiagramtofindthelengthoftheothersideand

writeasecondexpressionfortheareaoftheblock.Verifyyourtwoexpressionsfortheareaofthe

blockareequivalentusingalgebra.

a. Area: 2!! + 7! + 3 Side: (! + 3)

Equivalentexpressionforarea:

b. Area: 5!! + 8! + 3 Side: (! + 1)


c. Area: 2!! + 7! + 3 Side: (2! + 1)


7. Whataresomepatternsyouseeinthetwoequivalentexpressionsforareathatmighthelp

youtofactor?

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8. Businessisbooming!Moreandmoreordersarecomingin!Usediagramsornumber

patterns(orboth)towriteeachofthefollowingordersinfactoredform:

a. 3!! + 16! + 5

b. 2!! − 13! + 15

c. 3!! + ! − 10

d. 2!! + 9! − 5

9. InThe !Factor,youwrotesomerulesfordecidingaboutthesignsinsidethefactors.Dothoserulesstillworkinfactoringthesetypesofexpressions?Explainyouranswer.

10. ExplainhowOptimacantelliftheblockisamultipleofanotherblockorifonesidehasa

multipleof!inthesidelength.

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11. There’sonemoretwistonthekindofblocksthatOptimamakes.Thesearethetrickiestof

allbecausetheyhavemorethanone!inthelengthofbothsidesoftherectangle!Here’sanexample:

Writetwoequivalentexpressionsforthisblock.Usethedistributivepropertytoverifythatyour

answeriscorrect.

12. Allright,let’strythetrickyones.Theymaytakealittlemessingaroundtogetthefactored

expressiontomatchthegivenexpression.Makesureyoucheckyouranswerstobesurethat

you’vegotthemright.Factoreachofthefollowing:

a. 6!! + 7! + 2

b. 10!! + 17! + 3

c. 4!! − 8! + 3

d. 4!! + 4! − 3

e. 9!! − 9! − 10

12. Writea“recipe”forhowtofactortrinomialsintheform,!"! + !" + !.

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READY

Topic:ComparingarithmeticandgeometricsequencesThefirstandfifthtermsofeachsequencearegiven.Fillinthemissingnumbers.

1.

Arithmetic 3 1875

Geometric 3 1875

2.

Arithmetic -1458 -18

Geometric -1458 -18

3.

Arithmetic 1024 4

Geometric 1024 4


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SET

Topic:Writinganareamodelasaquadraticexpression

Writetwoequivalentexpressionsfortheareaofeachblock.Letxbethesidelengthofeachofthelargesquares.

4. 5.

6.

7. Problems4,5,and6allcontainthesamenumberofsquaresmeasuring!!and1!.A.Whatisdifferentaboutthem?

B.Howdoesthisdifferenceaffectthequadraticexpressionthatrepresentsthem?

C.Describehowthearrangementofthesquaresandrectanglesaffectsthefactoredform.

Topic:Factoringquadraticexpressionswhen! > !Factorthefollowingquadraticexpressions.

8. 4x! + 7x − 2 9. 2x! − 7x − 15 10. 6x! + 7x − 3 11. 4x! − x − 3

Page 51





12. 4x! + 19x − 5 13. 3x! − 10x + 8 14. 6x! + x − 2 15. 3x! − 14x − 24

16. 2x! + 9x + 10 17. 5x! + 31x + 6 18. 5x! + 7x − 6 19. 4x! + 8x − 5

20. 3x! − 75 21. 3x! + 7x + 2 22. 4x! + 8x − 5 23. 2x! + x − 6

GO

Topic:Findingtheequationofthelineofsymmetryofaparabola

Giventhex-interceptsofaparabola,writetheequationofthelineofsymmetry.

24. x-intercepts:(-3,0)and(3,0) 25. x-intercepts:(-4,0)and(16,0)

26. x-intercepts:(-2,0)and(5,0) 27. x-intercepts:(-14,0)and(-3,0)

28. x-intercepts:(17,0)and(33,0) 29. x-intercepts: −0.75, 0 and 2.25, 0

Page 52

Whenever we’re thinking about algebra and working

with variables, it is useful to consider how it relates to

the number system and operations on numbers. Right

now, polynomials are on our minds, so let’s see if we

can make some useful comparisons between whole

numbers and polynomials.

Let’s start by looking at the structure of numbers and polynomials. Consider the number 132. The

way we write numbers is really a shortcut because:

132 = 100 + 30 + 2

1. Compare 132 to the polynomial 𝑥2 + 3𝑥 + 2. How are they alike? How are they different?

2. Write a polynomial that is analogous to the number 2,675.

When two numbers are to be added together, many people use a procedure like this:

132

+ 451

583

3. Write an analogous addition problem for polynomials and find the sum of the two polynomials.

4. How does adding polynomials compare to adding whole numbers?

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5. Use the polynomials below to find the specified sums in a-f.

𝑓(𝑥) = 𝑥3 + 3𝑥2 − 2𝑥 + 10 𝑔(𝑥) = 2𝑥 − 1 ℎ(𝑥) = 2𝑥2 + 5𝑥 − 12 𝑘(𝑥) = −𝑥2 − 3𝑥 + 4

a) ℎ(𝑥) + 𝑘(𝑥) b) 𝑔(𝑥) + 𝑓(𝑥) c) 𝑓(𝑥) + 𝑘(𝑥)

______________________________ _______________________________ ______________________________

d) 𝑙(𝑥) + 𝑚(𝑥) e) 𝑚(𝑥) + 𝑛(𝑥) f) 𝑙(𝑥) + 𝑝(𝑥)

𝑝(𝑥) = x2 - 7xy + 4𝑛(𝑥) = 4xy + 2x2𝑙(𝑥) = 4x2 - 3y2 + 5xy 𝑚(𝑥) = 8xy + 3y2

______________________________ ______________________________ ______________________________

Page 54

6. What patterns do you see when polynomials are added?

Subtraction of whole numbers works similarly to addition. Some people line up subtraction

vertically and subtract the bottom number from the top, like this:

368

-157

211

7. Write the analogous polynomials and subtract them.

8. Is your answer to #7 analogous to the whole number answer? If not, why not?

9. Subtracting polynomials can easily lead to errors if you don’t carefully keep track of your

positive and negative signs. One way that people avoid this problem is to simply change all the

signs of the polynomial being subtracted and then add the two polynomials together. There are two

common ways of writing this:

(𝑥3 + 𝑥2 − 3𝑥 − 5) − (2𝑥3 − 𝑥2 + 6𝑥 + 8)

Step 1: = (𝑥3 + 𝑥2 − 3𝑥 − 5) + (−2𝑥3 + 𝑥2 − 6𝑥 − 8)

Step 2: = (−𝑥3 + 2𝑥2 − 9𝑥 − 13)

Or, you can line up the polynomials vertically so that Step 1 looks like this:

Step 1: 𝑥3 + 𝑥2 − 3𝑥 − 5

+(−2𝑥3 + 𝑥2 − 6𝑥 − 8)

Step 2: −𝑥3 + 2𝑥2 − 9𝑥 − 13

The question for you is: Is it correct to change all the signs and add when subtracting? What

mathematical property or relationship can justify this action?

Page 55

10. Use the given polynomials to find the specified differences in a-d.

𝑓(𝑥) = 𝑥3 + 2𝑥2 − 7𝑥 − 8 𝑔(𝑥) = −4𝑥 − 7 ℎ(𝑥) = 4𝑥2 − 𝑥 − 15 𝑘(𝑥) = −𝑥2 + 7𝑥 + 4

a) ℎ(𝑥) − 𝑘(𝑥) b) 𝑓(𝑥) − ℎ(𝑥) c) 𝑓(𝑥) − 𝑔(𝑥)

______________________________ _______________________________ ______________________________

d) 𝑘(𝑥) − 𝑓(𝑥) e) 𝑙(𝑥) − 𝑚(𝑥)

______________________________

11. List three important things to remember when subtracting polynomials.

𝑙(𝑥) = 5x2 - 7y2 + 4xy 𝑚(𝑥) = -10x2 + 9y2 -12xy + 4

______________________________

Page 56

SECONDARY MATH III // MODULE 3

POLYNOMIAL FUNCTIONS –


3.3


READY Topic:Usingthedistributiveproperty

Multiply.

1. 2𝑥(5𝑥% + 7) 2. 9𝑥(−𝑥% − 3) 3. 5𝑥%(𝑥, + 6𝑥.)

4. −𝑥(𝑥% − 𝑥 + 1) 5. −3𝑥.(−2𝑥% + 𝑥 − 1) 6. −1(𝑥% − 4𝑥 + 8)

SET Topic:Addingandsubtractingpolynomials

Add.Writeyouranswersindescendingorderoftheexponents.(Standardform)

7. (3𝑥, + 5𝑥% − 1) + (2𝑥. + 𝑥) 8. (4𝑥% + 7𝑥 − 4) + (𝑥% − 7𝑥 + 14)

9. (2𝑥. + 6𝑥% − 5𝑥) + (𝑥4 + 3𝑥% + 8𝑥 + 4) 10. (−6𝑥4 − 2𝑥 + 13) + (4𝑥4 + 3𝑥% + 𝑥 − 9)

Subtract.Writeyouranswersindescendingorderoftheexponents.(Standardform)

11. (5𝑥% + 7𝑥 + 2) − (3𝑥% + 6𝑥 + 2) 12. (10𝑥, + 2𝑥% + 1) − (3𝑥, + 3𝑥 + 11)

13. (7𝑥. − 3𝑥 + 7) − (4𝑥% − 3𝑥 − 11) 14. (𝑥, − 1) − (𝑥, + 1)

READY, SET, GO! Name PeriodDate

GO Topic:Usingexponentrulestocombineexpressions

Simplify.

19. 20. 21.x-5 ∙ x4 ∙ x-2 x3 ∙ x-7 ∙ x-2 x4 ∙ x4 ∙ x-1

Page 57

Multiplying polynomials can require a bit of skill in the

algebra department, but since polynomials are

structured like numbers, multiplication works very

similarly. When you learned to multiply numbers, you may have learned to use an area

model. To multiply 12 × 15 the area model and the related procedure probably looked like this:

You may have used this same idea with quadratic expressions. Area models help us think about

multiplying, factoring, and completing the square to find equivalent expressions. We modeled

(𝑥 + 2)(𝑥 + 5) = 𝑥2 + 7𝑥 + 10 as the area of a rectangle with sides of length 𝑥 + 2 and 𝑥 + 5. The

various parts of the rectangle are shown in the diagram below:

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𝑥 + 5

𝑥2 𝑥 𝑥 𝑥 𝑥 𝑥

1

𝑥 1

𝑥

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1 1 1 1 𝑥+

2

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50

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Some people like to shortcut the area a model a little bit to just have sections of area that

correspond to the lengths of the sides. In this case, they might draw the following.

= 𝑥2 + 7𝑥 + 10

1. What is the property that all of these models are based upon?

2. Now that you’ve been reminded of the happy past, you are ready to use the strategy of your

choice to find equivalent expressions for each of the following:

a) (𝑥 + 3)(𝑥 + 4) b) (𝑥 + 7)(𝑥 − 2)

Maybe now you remember some of the different forms for quadratic expressions—factored form

and standard form. These forms exist for all polynomials, although as the powers get higher, the

algebra may get a little trickier. In standard form polynomials are written so that the terms are in

order with the highest-powered term first, and then the lower-powered terms. Some examples:

Quadratic: 𝑥2 − 3𝑥 + 8 or 𝑥2 − 9

Cubic: 2𝑥3 + 𝑥2 − 7𝑥 − 10 or 𝑥3 − 2𝑥2 + 15

Quartic: 𝑥4 + 𝑥3 + 3𝑥2 − 5𝑥 + 4

Hopefully, you also remember that you need to be sure that each term in the first factor is

multiplied by each term in the second factor and the like terms are combined to get to standard

form. You can use area models or boxes to help you organize, or you can just check every time to

be sure that you’ve got all the combinations. It can get more challenging with higher-powered

polynomials, but the principal is the same because it is based upon the mighty Distributive

Property.

𝑥 +5

𝑥 𝑥2 5𝑥

+2 2𝑥 10

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3. Tia’s favorite strategy for multiplying polynomials is to make a box that fits the two factors. She

sets it up like this: (𝑥 + 2)(𝑥2 − 3𝑥 + 5)

Try using Tia’s box method to multiply these two factors together and then combining like terms to

get a polynomial in standard form.

4. Try checking your answer by graphing the original factored polynomial, (𝑥 + 2)(𝑥2 − 3𝑥 + 5)

and then graphing the polynomial that is your answer. If the graphs are the same, you are right

because the two expressions are equivalent! If they are not the same, go back and check your work

to make the corrections.

5. Tehani’s favorite strategy is to connect the terms he needs to multiply in order like this:

(𝑥 − 3)(𝑥2 + 4𝑥 − 2)

Try multiplying using Tehani’s strategy and then check your work by graphing. Make any

corrections you need and figure out why they are needed so that you won’t make the same mistake

twice!

6. Use the strategy of your choice to multiply each of the following expressions. Check your work

by graphing and make any needed corrections.

a) (𝑥 + 5)(𝑥2 − 𝑥 − 3) b) (𝑥 − 2)(2𝑥2 + 6𝑥 + 1)

c) (𝑥 + 2)(𝑥 − 2)(𝑥 + 3)

𝑥2 −3𝑥 +5

𝑥

+2

d) (x + xy - 2y)(𝑥 − y)

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When graphing, it is often useful to have a perfect square quadratic or a perfect cube. Sometimes it

is also useful to have these functions written in standard form. Let’s try re-writing some related

expressions to see if we can see some useful patterns.

7H. Multiply and simplify both of the following expressions using the strategy of your choice:

a) 𝑓(𝑥) = (𝑥 + 1)2 b) 𝑓 (𝑥) = (𝑥 + 1)3

Check your work by graphing and make any corrections needed.

8H.Some enterprising young mathematician noticed a connection between the coefficients of the

in the polynomial and the number pattern known as Pascal’s Triangle. Put your answers from

problem 5 into the table. Compare your answers to the numbers in Pascal’s Triangle below and

describe the relationship you see.

(𝑥 + 1)0 1 1

(𝑥 + 1)1 𝑥 + 1 1 1

(𝑥 + 1)2 1 2 1

(𝑥 + 1)3 1 3 3 1

(𝑥 + 1)4

9H. It could save some time on multiplying the higher power polynomials if we could use Pascal’sTriangle to get the coefficients. First, we would need to be able to construct our own Pascal’s Triangle and add rows when we need to. Look at Pascal’s Triangle and see if you can figure out how to get the next row using the terms from the previous row. Use your method to find the terms in the next row of the table above.

10H. Now you can check your Pascal’s Triangle by multiplying out (𝑥 + 1)4 and comparing thecoefficients. Hint: You might want to make your job easier by using your answers from #5 in some way. Put your answer in the table above.

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11H. Make sure that the answer you get from multiplying (𝑥 + 1)4 and the numbers in Pascal’sTriangle match, so that you’re sure you’ve got both answers right. Then describe how to get the next row in Pascal’s Triangle using the terms in the previous row.

12H. Complete the next row of Pascal’s Triangle and use it to find the standard form of (𝑥 + 1)5.Write your answers in the table on #6.

13H. Pascal’s Triangle wouldn’t be very handy if it only worked to expand powers of 𝑥 + 1. Theremust be a way to use it for other expressions. The table below shows Pascal’s Triangle and the expansion of 𝑥 + 𝑎.

(𝑥 + 𝑎)0 1 1

(𝑥 + 𝑎)1 𝑥 + 𝑎 1 1

(𝑥 + 𝑎)2 𝑥2 + 2𝑎𝑥 + 𝑎2 1 2 1

(𝑥 + 𝑎)3 𝑥3 + 3𝑎𝑥2 + 3𝑎2𝑥 + 𝑎3 1 3 3 1

(𝑥 + 𝑎)4 𝑥4 + 4𝑎𝑥3 + 6𝑎2𝑥2 + 3𝑎3𝑥 + 𝑎4 1 4 6 4 1

What do you notice about what happens to the 𝑎 in each of the terms in a row?

14H. Use the Pascal’s Triangle method to find standard form for (𝑥 + 2)3. Check your answer bymultiplying.

15H. Use any method to write each of the following in standard form:

a) (𝑥 + 3)3 b) (𝑥 − 2)3 c) (𝑥 + 5)4

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3.4

Need help? Visit www.rsgsupport.org

READY Topic: Solve Quadratic Equations

1. 2. 3.

4. 5.

SET Topic: Multiplying polynomials

Multiply. Write your answers in standard form.

6. 𝑎 + 𝑏 𝑎 + 𝑏 7. 𝑥 − 3 𝑥! + 3𝑥 + 9

8. 𝑥 − 5 𝑥! + 5𝑥 + 25 9. 𝑥 + 1 𝑥! − 𝑥 + 1

10. 𝑥 + 7 𝑥! − 7𝑥 + 49 11. 𝑎 − 𝑏 𝑎! + 𝑎𝑏 + 𝑏!


Solve the following quadratic equations.

x2 = 121 x2 + 8x = -15(4x + 5)(x + 1) = 0

3x2 - 16r - 7 = 5 8x2 - 4x - 18 = 0

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3.4


( 𝑥 + 𝑎)! 1 1

(𝑥 + 𝑎)! 𝑥 + 𝑎 1 1

(𝑥 + 𝑎)! 𝑥! + 2𝑎𝑥 + 𝑎! 1 2 1

(𝑥 + 𝑎)! 𝑥! + 3𝑎𝑥! + 3𝑎!𝑥 + 𝑎! 1 3 3 1

(𝑥 + 𝑎)! 𝑥! + 4𝑎𝑥! + 6𝑎!𝑥! + 3𝑎!𝑥 + 𝑎! 1 4 6 4 1

Use the table above to write each of the following in standard form.

12. 𝑥 + 1 ! 13. 𝑥 − 5 ! 14. 𝑥 − 1 !

15, 𝑥 + 4 ! 16. 𝑥 + 2 ! 17. 3𝑥 + 1 !

GO Topic: Examining transformations on different types of functions

Graph the following functions.

18. 𝑔(𝑥) = 𝑥 + 2 19. ℎ(𝑥) = 𝑥! + 2 20. 𝑓(𝑥) = 2! + 2

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3.4


21. 𝑔(𝑥) = 3(𝑥 − 2) 22. ℎ(𝑥) = 3(𝑥 − 2)!

24. 𝑔 𝑥 = !! 𝑥 − 1 − 2 25. ℎ 𝑥 = !

! 𝑥 − 1 ! − 2

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2.9 Lining Up Quadratics


Grapheachfunctionandfindthevertex,they-interceptandthe!-intercepts.Besuretoproperlywritetheinterceptsaspoints.

1. ! = (! − 1)(! + 3)

LineofSymmetry_________

Vertex__________

!-intercepts_________ _________

y-intercept________

2. ! ! = 2(! − 2)(! − 6)

LineofSymmetry_________

Vertex__________

!-intercepts_________ _________

y-intercept________C

C B

Y R

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Jnag

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3. ! ! = −!(! + 4)

LineofSymmetry___________

Vertex__________

!-intercepts_________ _________

y-intercept________

4. Basedontheseexamples,howcanyouuseaquadraticfunctioninfactoredformto:a. Findthelineofsymmetryoftheparabola?

b. Findthevertexoftheparabola?

c. Findthe!-interceptsoftheparabola?

d. Findthey-interceptoftheparabola?

e. Findtheverticalstretch?

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5. Chooseanytwolinearfunctionsandwritethemintheform:!(!) = ! ! − ! , wheremistheslopeoftheline.Graphthetwofunctions.

Linearfunction1:

Linearfunction2:

6. Onthesamegraphas#5,graphthefunctionP(x)thatistheproductofthetwolinearfunctionsthatyouhavechosen.Whatshapeiscreated?

7. Describetherelationshipbetweenx-interceptsofthelinearfunctionsandthex-interceptsofthefunctionP(x).Whydoesthisrelationshipexist?

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8. Describetherelationshipbetweeny-interceptsofthelinearfunctionsandthey-interceptsofthefunctionP(x).Whydoesthisrelationshipexist?

9. Giventheparabolatotheright,sketchtwolinesthatcouldrepresentitslinearfactors.

10. Writeanequationforeachofthesetwolines.

11. Howdidyouusethexandyinterceptsoftheparabolatoselectthetwolines?

12. Arethesetheonlytwolinesthatcouldrepresentthelinearfactorsoftheparabola?Ifso,explainwhy.Ifnot,describetheotherpossiblelines.

13. Useyourtwolinestowritetheequationoftheparabola.Isthistheonlypossibleequationoftheparabola?

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READY

Topic:MultiplyingBinomialsUsingaTwo-WayTable

Multiplythefollowingbinomialsusingthegiventwo-waytabletoassistyou.Example:(2x+3)(5x–7)

=10x2+x–21

1. (3x–4)(7x–5) 2. (9x+2)(x+6) 3. (4x–3)(3x+11)

4. (7x+3)(7x–3) 5. (3x–10)(3x+10) 6. (11x+5)(11x–5)

7. (4x+5)2 8. (x+9)2 9. (10x–7)2

10. The“like-term”boxesin#’s7,8,and9revealaspecialpattern.Describetherelationshipbetween

themiddlecoefficient(b)andthecoefficients(a)and(c).


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SET

Topic:FactoredFormofaQuadraticFunction

Giventhefactoredformofaquadraticfunction,identifythevertex,intercepts,andverticalstretchoftheparabola.

11. ! = 4 ! − 2 ! + 6 12. ! = −3 ! + 2 ! − 6 13. ! = ! + 5 ! + 7a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________

b. x-inter(s)____________ b. x-inter(s)____________ b. x-inter(s)___________

c. y-inter________________ c. y-inter:_______________ c. y-inter_______________

d. Stretch_______________ d. Stretch_______________ d. Stretch_______________

14. ! = !! ! − 7 ! − 7 15. ! = − !

! ! − 8 ! + 4 16. ! = !! ! − 25 ! − 9

a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________




17. ! = !! ! − 3 ! + 3 18. ! = − ! − 5 ! + 5 19. ! = !

! ! + 10 ! + 10a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________




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GO Topic:VertexFormofaQuadraticEquation

Giventhevertexformofaquadraticfunction,identifythevertex,intercepts,andverticalstretchoftheparabola.

20. ! = ! + 2 ! − 4 21. ! = −3 ! + 6 ! + 3 22. ! = 2 ! − 1 ! − 8a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________




23. ! = 4 ! + 2 ! − 64 24. ! = −3 ! − 2 ! + 48 25. ! = ! + 6 ! − 1a. Vertex:_______________ a. Vertex:_______________ a. Vertex:_______________




26. Didyounoticethattheparabolasinproblems11,12,&13arethesameastheonesinproblems23,

24,&25respectively?Ifyoudidn’t,gobackandcomparetheanswersinproblems11,12,&13and

problems23,24,&25.

Provethat a. 4 ! − 2 ! + 6 = 4 ! + 2 ! − 64

b. −3 ! + 2 ! − 6 = −3 ! − 2 ! + 48

c. ! + 5 ! + 7 = ! + 6 ! − 1

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2.10 I’ve Got a Fill-in


Foreachproblembelow,youaregivenapieceofinformationthattellsyoualot.Usewhatyou

knowaboutthatinformationtofillintherest.

1. Yougetthis: Fillinthis:

! = !! − ! − 12Factoredformoftheequation:

Graphoftheequation:

CC

BY

One

Poi

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! = !! − 6! + 3Vertexformoftheequation:

Graphoftheequation:

3. Yougetthis: Fillinthis:Vertexformoftheequation:

Standardformoftheequation:

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4. Yougetthis: Fillinthis:Factoredformoftheequation:

Standardformoftheequation:


! = −!! − 6! + 16Eitherformoftheequationotherthanstandardform.

Vertexoftheparabola

x-interceptsandy-intercept

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! = 2!! + 12! + 13Eitherformoftheequationotherthanstandardform.

Vertexoftheparabola



! = −2!! + 14! + 60Eitherformoftheequationotherthanstandardform.

Vertexoftheparabola


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2.10



READY

Agolf-propracticeshisswingbydrivinggolfballsofftheedgeofacliffintoalake.Theheightoftheballabovethelake(measuredinmeters)asafunctionoftime(measuredinsecondsandrepresentedbythevariablet)fromthe

instantofimpactwiththegolfclubis

58.8 + 19.6t − 4.9t!.

Theexpressionsbelowareequivalent:

−4.9t! + 19.6t + 58.8 standardform

−4.9 t − 6 t + 2 factoredform

−4.9 t − 2 ! + 78.4 vertexform

1. Whichexpressionisthemostusefulforfindinghowmanysecondsittakesfortheballtohitthewater? Why?

2. Whichexpressionisthemostusefulforfindingthemaximumheightoftheball? Justifyyouranswer.

3. Ifyouwantedtoknowtheheightoftheballatexactly3.5seconds,whichexpressionwouldhelpthemosttofindtheanswer?Why?

4. Ifyouwantedtoknowtheheightofthecliffabovethelake,whichexpressionwouldyouuse?Why?

SET Topic:FindingmultiplerepresentationsofaquadraticOneformofaquadraticfunctionisgiven.Fill-inthemissingforms.5a.StandardForm b. VertexForm c. FactoredForm

! = ! + 5 ! − 3

d. Table(Includethevertexandatleast2pointsoneachsideofthevertex.)

x y

Showthefirstdifferencesandtheseconddifferences.

e. Graph


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2.10



6a.StandardForm b. VertexForm! = −3 ! − 1 ! + 3

c. FactoredForm


x y


e. Graph

7a.StandardForm! = −!! + 10! − 25

b. VertexForm c. FactoredForm


x y


e. Graph

8a.StandardForm b. VertexForm c. FactoredForm


x y


e. Graph

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2.10



9a.StandardForm b. VertexForm c. FactoredFormSkipthisfornow

d. Tablex y0123456

122-4-6-4212


e. Graph

GO Topic:FactoringQuadratics

Verifyeachfactorizationbymultiplying.

10. x! + 12x − 64 = x + 16 x − 4 11. x! − 64 = x + 8 x − 8

12. x! + 20x + 64 = x + 16 x + 4 13. x! − 16x + 64 = x − 8 x − 8

Factorthefollowingquadraticexpressions,ifpossible.(Somewillnotfactor.)

14. x! − 5x + 6 15. x! − 7x + 6 16. x! − 5x − 36

17. m! + 16m + 63 18. s! − 3s − 1 19. x! + 7x + 2

20. x! + 14x + 49 21. x! − 9 22. c! + 11c + 3

23. Whichquadraticexpressionabovecouldrepresenttheareaofasquare?Explain.

24. WouldanyoftheexpressionsaboveNOTbetheside-lengthsforarectangle?Explain.

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Structures of Expressions - MR. CONGLETON · WCPSS Math 2 Unit 4: MVP MODULE 2 Structures of...

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