1

IntegratedMath3Module1Honors

PolynomialFunctionsReady,Set,Go!Homework

Solutions

Adaptedfrom

TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,

TravisLemon,JanetSutorius

©2014MathematicsVisionProject|MVPInpartnershipwiththeUtahStateOfficeofEducation

LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense.

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SDUHSDMath3Honors

Name PolynomialFunctions 1.1HReady,Set,Go!ReadyTopic:Inequalitystatements.Whichisgreater?Foreachproblem,makeatruestatementbyplacingtheappropriateinequalitysymbolbetweenthetwoexpressions.If ,then: If 10,then:1. 3 3 4. 2 2. 5. √ 3. 6. SetTopic:TypesoffunctionsDeterminethetypeoffunctionforeachproblem.Explainhowyouknow.7. 8. 9. 10.

1 3 1 3 1 3 1 32 6 2 6 2 9 2 123 9 3 12 3 18 3 304 12 4 24 4 30 4 605 25 5 48 5 45 5 105

Linear Exponential Quadratic Cubic

11. 2 3 5 12.Quadratic 13.Exponential Cubic

14. 2 4 7 15. 2 ⋅ 3 1 16. log 3 Linear Exponential Logarithmic

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SDUHSDMath3Honors

GoTopic:Combiningfunctions.Usethegivenfunctionstosolveproblems17–21.

17. 18. 19. ⋅ 20. 21. DetermineifthefollowingstatementsareALWAYSorNEVERtrue.IfthestatementisNOTtrue,rewriteitsothatitisALWAYSTRUE.22.Thesumoftwolinearfunctionsisanotherlinearfunction. Always23.Thesumofalinearandaquadraticisacubicfunction.

Never;Theproductofalinearandaquadraticisacubicfunction.ORThesumofalinearandaquadraticfunctionisaquadraticfunction.

24.Thesumofacubicandaquadraticfunctionisacubicfunction. AlwaysTopic:Comparinglogarithmic&exponentialexpressionsOrderthefollowingnumbersfromleasttogreatestwithoutusingacalculator.25. log 49 log √7 log log √343

, √ , √ , 26.8 ln ln

, , , 27. log log 7 log ln ln log 10 log log 32

, , , 28.10 10 log ln log 10

, , ,

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SDUHSDMath3Honors

Name PolynomialFunctions 1.2HReady,Set,Go!ReadyTopic:Combiningpolynomialfunctionsgraphically.Usethegraphsof and tosketchthegraphofthefollowing:1. 2.

3. ⋅

4. Completeeachsentencebelow.

a. Thesumoftwolinearfunctionsis… b. Thedifferenceoftwolinearfunctionsis… linear linearc. Theproductoftwolinearfunctionsis… quadratic

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SDUHSDMath3Honors

SetTopic:OrderingrealnumberexpressionsOrderthefollowingnumbersfromleasttogreatestwithoutusingacalculator.5. 100 √100 log 100 100 , √ , ,

6. 2 √100 log 0

, , , √ 7. 2 √16 log 8 2 , , , √ 8. 2 log 100 √100 100 , √ , , Whichisgreater?Foreachproblem,makeatruestatementbyplacingtheappropriateinequalitysymbolbetweenthetwoexpressions.(Hint:thinkaboutwhatyouknowabouttheexpressionandtheendbehavioraswellasratesofchangeofafunctioninsteadofplugginginvalues).If 100,then: If 100,then:9. 2 12. 2 10. 13. 11. 14. GoTopic:CombiningfunctionsPerformeachoperation.Writeyouranswersinstandardform.15. 3 4 , 3 3 16. 3 4 , 5 ⋅ 17. 6 5 , 2 7 6

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SDUHSDMath3Honors

Grapheachsetoffunctionsonthesameaxes.Labeleachfunctionandstatehowthefunctionsarerelatedtothegraphsoftheirparentfunctions.18. 2 2 2 2

Featuresincommon:All3functionshavebeentranslatedup2unitsfromtheparentfunctions.

19. 3√ 2 3 2 3 2

Featuresincommon:All3functionshavebeentranslatedright2unitsfromtheparentfunctionandhaveastretchfactorof3.

20. | 1| 2

1 2

1 2

Featuresincommon:All3functionshavebeentranslatedright1unitanddown2unitsfromtheparentfunctionandhaveastretchfactorof .

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SDUHSDMath3Honors

Name PolynomialFunctions 1.3HReady,Set,Go!ReadyTopic:FormsoflinearandquadraticfunctionsThedifferentformsoflinearandquadraticfunctionsarelistedbelow.Determinewhatfeaturesofthefunction/graphcanquicklybedeterminedbaseduponthestructureofeachformoflinearandquadraticfunctions.Linear Quadratic1. Standardform: 4. Standardform: x‐andy‐intercepts y‐intercept,directionofopening2. Slope‐interceptform: 5. Factoredform: Slopeandy‐intercept x‐intercepts,directionofopening3. Point‐slopeform: 6. Vertexform: Slopeandpointongraph Vertexandmax/minvalue, directionofopeningForeach,writewhatyouknowaboutthefunction(includingendbehavior)andthengraph.7. Equation: 2 3 Graph:

WhatIknowaboutthisfunction:Answersmayinclude:

and isthelineofsymmetry

Endbehavior:As → ∞, → ∞As → ∞, → ∞.

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SDUHSDMath3Honors

8. Equation: 1 1 2 Graph:WhatIknowaboutthisfunction:Answersmayinclude:y‐interceptis ‐intercepts: , , Endbehavior:As → ∞, → ∞As → ∞, → ∞

9. Equation: 4 6 Graph:

WhatIknowaboutthisfunction:Answersmayinclude:Vertexis , ,opensdown‐intercept: Endbehavior:As → ∞, → ∞As → ∞, → ∞

   

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SDUHSDMath3Honors

10.Equation: √ 2 Graph:WhatIknowaboutthisfunction:Answersmayinclude:Domain: , ∞ Range: , ∞ Endbehavior:As → , → As → ∞, → ∞

SetTopic:EndbehaviorofvarioustypesoffunctionsDeterminethefunctiontypeandstatetheendbehaviorintheformas → , → .11. 12 1 12. 4 ⋅ 2 Quadratic Exponential → ∞, → ∞ → ∞, → → ∞, → ∞ → ∞, → ∞13. 1 14. 3 1 Cubic Quadratic → ∞, → ∞ → ∞, → ∞ → ∞, → ∞ → ∞, → ∞Usetheequationsinquestions11‐14toanswerthefollowing:15.Whichfunctionabovehasthegreatestvalueat 1,000? 16.Whichfunctionaboveisalwaysincreasing? 17.Whichfunctionaboveisalwaysdecreasing? 18.Whichfunctionabovehasamaximumvalue? 19.Whichfunctionabovehasaminimumvalue?

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SDUHSDMath3Honors

GoTopic:SolvingequationsSolveforx.20.27 9 21.2 4 3 0

√ 22. log log 7 3 23. log 3 2 isextraneous24. 4 3 1 0 25.4 16 16 0 , , ,

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SDUHSDMath3Honors

Name PolynomialFunctions 1.4HReady,Set,Go!

ReadyTopic:Solvingequations.Solvefor .1. 5 16 15 4 5 2. 3 2 4 13 3. 8 14 9 0 √ ,

4. 3 2 0 5. 2 1 4 5 2 =0 6. 81

, , , SetTopic:Combiningpolynomialfunctions.Given , ,and ,find:7. 8. 9. ⋅ 10. 11. 12. ⋅

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SDUHSDMath3Honors

Graphsoftheindividualfunctionsaregiven.Graphthesolutiononthesamesetofaxes.

13. 14. 15.

16. ⋅ 17. ⋅ 18. ⋅ ⋅

GoTopic:Simplifyingexpressionscontainingexponents

19. 20. 21.

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SDUHSDMath3Honors

Topic:Solvinglogarithmicandexponentialequations.Solveeachequation.22. log 3 5 log 17 23.64 512

24. log 6 9 5 25. 81

Topic:Multiplyingpolynomials.Multiplyeach.Simplifysolutionsbycombiningliketerms26. 27. 3 3 9 28. 5 5 25 29. 1 1 30. 7 7 49 31. 32.Usingthepatternsfromquestions26‐31,whatdoyouthinkyouarethefactorsof 1?Checkyour

factorizationbymultiplyingthefactorstogether.

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SDUHSDMath3Honors

Name PolynomialFunctions 1.5HReady,Set,Go!ReadyTopic:Describethefeaturesofvariousfunctions.Identifythefeaturesofthefollowingfunctions.(Featuresincludedomain,range,intercepts,andendbehavior).1. 2. 3.

Domain: ∞,∞ Domain:integers Domain: ∞,∞ Range: , ∞ Range: , , , , , … Range: ∞,∞ x‐intercepts: x‐intercepts:NA x‐intercepts: , y‐intercept:4 y‐intercept:8 y‐intercept:0 EndBehavior: EndBehavior: EndBehavior: As → ∞, → ∞ As → ∞, → ∞ As → ∞, → ∞ As → ∞, → ∞ As → ∞, → As → ∞, → ∞Topic:Combinationsandpermutations

Permutations

nPr!

!

Combinations

nCr!

! !

Findthenumberofpermutationsorcombinations.4. 5C2 5. 4P2 6. 6C3 10 12 20

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SDUHSDMath3Honors

SetTopic:FeaturesofpolynomialfunctionsWritethekeyfeaturesofeachfunction(intercepts,endbehavior,andwherethefunctionisincreasing/decreasing),thengraph.7. Equation: 1 Graph:

WhatIknowaboutthisfunction:Vertex: , x‐intercept:1y‐intercept:1increasing: , ∞ decreasing: ∞, Endbehavior:As → ∞, → ∞As → ∞, → ∞

8. Equation: 1 1 Graph:

WhatIknowaboutthisfunction:x‐intercepts: y‐intercept:1increasing: , , , ∞ decreasing: ∞, , , Endbehavior:As → ∞, → ∞As → ∞, → ∞

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SDUHSDMath3Honors

9. Equation: 3 4 1 Graph:WhatIknowaboutthisfunction:x‐intercepts: , , y‐intercept: increasing:Approximately∞, . , . , ∞

decreasing:Approximately . , . Endbehavior:As → ∞, → ∞As → ∞, → ∞

10.Equation: Graph:

WhatIknowaboutthisfunction:x‐intercept:0y‐intercept:0increasing: ∞,∞ decreasing:NAEndbehavior:As → ∞, → ∞As → ∞, → ∞

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SDUHSDMath3Honors

GoTopic:Comparingfunctionsindifferentforms.Usefunctionsa‐htoanswerthequestionsbelow.a. b. c. √ d. e. f. g. h.

11.Whichfunction(s)donothaveadomainofall

realnumbers? b,c

12. Whichfunction(s)donothavearangeofallrealnumbers?

c,f,h

13.Whichfunction(s)haveexactlytwox‐intercepts? g,h

14. Comparea andc:whichhasthegreatestvalueas→ ∞?

c

15.Comparedandf:whichhasthegreatestvalueas→ ∞?

f

16. Comparef andg:whichhasthegreatestvalueas→ ∞?

f

17.Compareeandh:whichhasthegreatestvalueas→ ∞?

h

18. Compareg andh:whichhasthehighestrelativemaximumvalue?

h

19.Comparebandf:whichhasthegreatestaveragerateofchangeovertheinterval 15, 20 ?

f

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SDUHSDMath3Honors

Name PolynomialFunctions 1.6HReady,Set,Go!ReadyTopic:ArithmeticofpolynomialsInthetaskToBeDetermined...wedefinedpolynomialstobeexpressionsofthefollowingform:

⋯ wherealloftheexponentsarepositiveintegersandallofthecoefficients … areconstants.Dothefollowingforeachoftheproblemsbelow:

A. Choosethebestwordtocompleteeachconjecture.B. Afteryouhavemadeaconjecture,createatleastfourexamplestoshowwhyyourconjecture

istrue.C. Ifyoufindacounter‐example,changeyourconjecturetofityourwork.

1. Conjecture#1:Thesumoftwopolynomialsis[always,sometime,never]apolynomial. A. Bestwordchoice: Always B. Atleast4examples: Answersmayvary C. Counter‐example2. Conjecture#2:Thedifferenceoftwopolynomialsis[always,sometime,never]apolynomial. A. Bestwordchoice: Always B. Atleast4examples: Answersmayvary C. Counter‐example3. Conjecture#3:Theproductoftwopolynomialsis[always,sometime,never]apolynomial. A. Bestwordchoice: Always B. Atleast4examples: Answersmayvary C. Counter‐example

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SDUHSDMath3Honors

SetTopic:BinomialexpansionUsePascal’sTriangletohelpyouexpandeachbinomial.4. 3 5. 2 6. 7. 2

8. Findthe3rdtermintheexpansionof 3 9. Findthe2ndtermintheexpansionof 2 3

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SDUHSDMath3Honors

GoTopic:SolvingpolynomialequationsFactoreachpolynomial.Thenusethezeroproductpropertytosolveforthevariable.10. 7 6 0 11.15 72 108 0

, √ , , √

12.27 1 0 13. 7 4 28

, √ ,

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SDUHSDMath3Honors

Name PolynomialFunctions 1.7HReady,Set,Go!ReadyTopic:FactoringspecialproductsFactor.1. 4 25 2. 9 16 3.

FactoringRulefortheSumofCubes:

FactoringRulefortheDifferenceofCubes:

4. 64 125 5. 27 8 6. 1000 SetTopic:Findingzerosofpolynomialfunctions.Findallzerosofeachpolynomial,thensketchthegraph.Usetechnologytocheckyouranswer.7. 25 8. 4 9 9. 5 6 , ,

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SDUHSDMath3Honors

10. 25 11. 4 9 12. 5 6 , ,

Topic:Usingpolynomialdivision13.Theproductoftwopolynomialsis 4 6.Oneofthefactorsis 1.Usetheboxmethodto

findtheotherfactors.

14.Usetheboxmethodtodividethefollowingpolynomials. 10 29 56 7

8

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SDUHSDMath3Honors

GoTopic:MultiplypolynomialsMultiplyeachexpression.Expressyoursolutionsinsimplestformbycombiningliketerms.15. 3 5 3 5 16. 7 4 7 4 17. 2 2 4 18. 1 1 Hint:Binomialexpansion19.Expand: 2 5 20.Findthe5thtermin: 2 5 Topic:MultiplyingandFactorpolynomials21.Factor asthedifferenceofsquares.Thenfactoryouranswerasthedifferenceof2cubesand

thesumofcubes. 22.Factor asthedifferenceofcubes.Thenfactoryouranswerevenfurther. 23.Shouldyouultimatelygetthesameanswersforquestions14&15?Didyou?Explain. Yes,yes,whenyoumultiply youget

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SDUHSDMath3Honors

Name PolynomialFunctions 1.8HReady,Set,Go!ReadyTopic:GraphingpolynomialfunctionsWithoutusingtechnology,sketchagraphofthepolynomialfunctiondescribed(ifpossible).Ifnotpossible,statewhynot.Answerswillvary1. Acubicfunctionwithonenegativezero (multiplicity2)andonepositive.

2. Aquarticfunction(4thdegree)withanegativeleadingcoefficient,apositivey‐intercept,onenegativedoubleroot,onepositivezero,andoneadditionalzero.

3. Acubicfunctionwithzerorealroots.

4. Aquarticfunctionwithzerorealroots,apositiveleadingcoefficient,andapositivey‐intercept.

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SDUHSDMath3Honors

SetTopic:FindingfactorsofpolynomialfunctionsFindalllinearfactorsandsketchthegraphofthepolynomialfunctions(unlessyouseeanothermethodthatallowsforquickergraphing.Ifso,explainmethod).5. 5 6. 25 ⋅ √ √ √ √

7. 1 8. 2 2 Hint:onerootis

√ √

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SDUHSDMath3Honors

UsetheRemainderTheoremtodetermineifthefollowingarerootstothegivenequation.Ifso,findtheotherrootsandgraphtheequation.Thenwritethefunctioninfactoredform.9. 5 2 8; 1 10. 2 ; 1 isaroot,otherroots , isaroot,otherroots ,

Topic:WritingpolynomialfunctionsgivenrootsWritethepolynomialfunctioninstandardformwithleastdegreeusingthegiveninformation.Makesuretoincludeanymissingconjugatepairs.11.Leadingcoefficient:2;roots:2,√2

12.Leadingcoefficient: 1;roots:1, 1 √3

13.Leadingcoefficient:2;roots:4

14. Passesthroughthepoints2, 0 , 3, 0 , 1, 0 , 0, 1

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SDUHSDMath3Honors

GoTopic:ExpandingbinomialsUsePascal’striangletohelpexpandthefollowingbinomials.15. 2 3 16. 3 2 Topic:FindingrootsofpolynomialfunctionsFindtherootsofthepolynomialfunctionsusingthegiveninformation.17. 3 2, 1isadoubleroot(multiplicityof2) , , , 18. 7 3 21, 7 0 , , √ Topic:FactoringpolynomialsFactoreachpolynomialcompletely.19. 7 12 20. 15 16 √ √

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SDUHSDMath3Honors

Name PolynomialFunctions 1.9HReady,Set,Go!ReadyTopic:Solvingpolynomial,logarithmic,andrationalequations.Solveforx.1. 2 2 1 0 2. 6 12 , , 3. 1 0 4. 4 9 0

, √ √ . & . 5. log 9 6. 6 .

Topic:UsingtheRemainderTheoremFind foreachpolynomialandstatewhetherornot isafactor.7. 9 3 8. 9 27 28 9. 2 5 12 27 notafactor notafactor yesitisafactor

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SDUHSDMath3Honors

SetTopic:Graphingpolynomialfunctions.Completetheinformationbelowusingthegraph10.Function:

EndBehavior:As → ∞, → ∞As → ∞, → ∞Roots(withmultiplicity):

(multiplicityof2), ValueofLeadingCoefficient:

Domain:∞,∞

Range:∞,∞

11.Writethepolynomialfunctionwithleastdegree,inbothfactoredandstandardforms,giventhefollowing

rootsandapointthatthefunctionpassesthrough.

Roots: 1, 3,Pointonthegraph: 0, 9 FactoredForm: StandardForm:

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SDUHSDMath3Honors

Withoutusingtechnology,sketchthegraphofthepolynomialfunctiondescribed.12.Acubicfunctionwithaleadingcoefficientof 1,

withonepositivezero. Answerswillvary

13. Aquarticfunctionwithaleadingcoefficientof1,withtwodoublezeros.

Answerswillvary

14.Acubicfunctionwithaleadingcoefficientof 3,withonepositivetripleroot.

Answerswillvary

15. Aquarticfunctionwithaleadingcoefficientof2,withtwonegativezerosandtwoimaginaryroots.

Answerswillvary

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SDUHSDMath3Honors

GoTopic:EndbehaviorCircletheexpressionthathasthegreatestvalueof as → ∞.16. 2 2 10 5 log

17. 2 10 4 3 √

18. 3 ⋅ 2 4 2 3 Topic:Determiningthetypeofafunctionbasedonatableofdata.19.Determinethetypeofeachfunction.Thenfindanexplicitequationforeach.

26 35 27 0.015625 251 17 24 22 0.03125 129 10 15 17 0.0625 55 5 8 12 0.125 17 2 3 7 0.25 3 1 0 2 0.5 1 2 1 3 1 1 5 0 8 2 15 10 3 13 4 53 17 8 18 8 127 26 15 23 16 249

quadratic, quadratic, linear, exponential, ⋅

cubic,