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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
, , ,
4
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
5
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
6
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 .
7
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 → ∞, → ∞.
8
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 → ∞, → ∞
9
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?
10
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.
14
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
15
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 → ∞, → ∞
16
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 → ∞, → ∞
17
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
23
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,