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©2012MathematicsVisionProject|MVPInpartnership withtheUtahStateOfficeofEducation
LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense
IntegratedMath1Module5
FeaturesofFunctions
AdaptedFrom
TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,
TravisLemon,JanetSutoriuswww.mathematicsvisionproject.org
InpartnershipwiththeUtahStateOfficeofEducation
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SDUHSDMath1CollegePrep
Module5OverviewPrerequisiteConcepts&Skills:
Solvemulti‐stepequations Graphlinearandexponentialfunctions Identifydomainandrangeofafunction Comparelinearandexponentialmodels Transitionfromarithmeticandgeometricsequencestolinearandexponentialmodels Distinguishbetweencontinuousvs.discrete Applylinearandexponentialfunctiontomodelsituations Solvelinearandexponentialequations Analyzerateofchangeforagivencontext Representlinearequationsusingslope‐intercept,standard,andpoint‐slopeform
SummaryoftheConcepts&SkillsinModule5:
Useacontexttographanddescribekeyfeaturesoffunctions Usetablesandgraphstointerpretkeyfeaturesoffunctions Interpretfunctionsusingnotation Combinefunctionsandanalyzecontextsusingfunctions Usegraphstosolveproblemsgiveninfunctionnotation Definefunction Identifywhetherornotarelationisafunctiongivenvariousrepresentations
ContentStandardsandStandardsofMathematicalPracticeCovered:
ContentStandards:F.IF.1,F.IF.2,F.IF.3,F.IF.4,F.IF.5,F.IF.7,F.BF.1B,A.REI.11,A.CED.3,A.CED.4, StandardsofMathematicalPractice:
1. Makesenseofproblems&persevereinsolvingthem.2. Attendtoprecision3. Reasonabstractly&quantitatively4. Constructviablearguments&critiquethereasoningofothers5. Modelwithmathematics6. Useappropriatetoolsstrategically7. Lookfor&makeuseofstructure8. Lookfor&expressregularityinrepeatedreasoning
Materials:
GraphingUtility
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Module5Vocabulary: Domain Range Function Functionnotation x‐andy‐intercepts Intervalnotation Linearmodel/equation/function Exponentialmodel/equation/function Continuousfunction Discretefunction Rateofchange Recursivenotation Explicitnotation
ConceptsUsedintheNextModule:
Developdefinitionsofrigid‐motiontransformations:translations,rotations,andreflections Examineslopeofperpendicularandparallellines Examinewhichrigidmotiontransformationcarryoneimageontoanothercongruentimage Writeandapplyformaldefinitionsoftherigid‐motiontransformations Findrotationalsymmetryandlinesofsymmetryinquadrilaterals Examinecharacteristicsofregularpolygonsthatemergefromrotationalsymmetryandlinesofsymmetry Makeandjustifypropertiesofquadrilateralsusingsymmetrytransformations Describeasequenceoftransformationsthatwillcarrycongruentimagesontoeachother EstablishtheASA,SAS,andSSScriteriaforcongruenttriangles Explorecompassandstraightedgeconstructions Writeproceduresforcompassandstraightedgeconstructionsandwhyitcreatesthedesiredobject(s)
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Module5–FeaturesofFunctions5.1UsingastorycontexttographanddescribekeyfeaturesoffunctionsandUsingtablesandgraphstointerpretkeyfeaturesoffunctions(F.IF.4,F.IF.5)WarmUp:GettingReadyforaPoolParty‐ADevelopUnderstandingTaskClassroomTask:FloatingDowntheRiver–ASolidifyUnderstandingTaskReady,Set,GoHomework:FeaturesofFunctions5.15.2Featuresoffunctionsusingvariousrepresentations(F.IF.4,F.IF.5)WarmUp:DanMeyer’s3Acts:WaterlineORGraphingLinearandExponentialFunctionsClassroomTask:FeaturesofFunctions–APracticeUnderstandingTaskReady,Set,GoHomework:FeaturesofFunctions5.25.3Interpretingfunctionsusingnotation(F.IF.2,F.IF.4,F.IF.5,F.IF.7,A.REI.11,A.CED.3)WarmUp:WritingDomainandRangeinIntervalNotationClassroomTask:TheWaterPark–ASolidifyUnderstandingTaskReady,Set,GoHomework:FeaturesofFunctions5.35.4Combiningfunctionsandanalyzingcontextsusingfunctions(F.BF.1b,F.IF.2,F.IF.4,F.IF.5,F.IF.7,A.REI.11,A.CED.3)WarmUp:UsingGraphicalRepresentationsofFunctionstoFindSolutionsClassroomTask:PoolingitTogether–ASolidifyUnderstandingTaskSet,GoHomework:FeaturesofFunctions5.45.5Usinggraphstosolveproblemsgiveninfunctionnotation(F.BF.1b,F.IF.2,F.IF.4,F.IF.5,F.IF.7,A.REI.11,A.CED.3)ClassroomTask:InterpretingFunctions–APracticeUnderstandingTaskReady,Set,GoHomework:FeaturesofFunctions5.55.6DefiningFunctionandIdentifyingwhetherornotarelationisafunctiongivenvariousrepresentations(F.IF.1,F.IF.3)WarmUp:ToFunctionorNottoFunction–APracticeUnderstandingTaskClassroomTask:AWaterFunction–ASolidifyUnderstandingTaskReady,Set,GoHomework:FeaturesofFunctions5.65.7Matchingfeaturesandrepresentationsofaspecificfunction(F.IF.2,F.IF.4,F.IF.5,F.IF.7,A.REI.11,A.CED.3)WarmUp:KeyFeaturesofFunctionsClassroomTask:MatchthatFunction–APracticeUnderstandingTaskReady,Set,GoHomework:FeaturesofFunctions5.7Homework:IntroducingTransformations
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© www.flickr.com/photos/jensm
ith826 5.1WarmUp
GettingReadyforaPoolParty–ADevelopUnderstandingTaskSylviahasasmallpoolfullofwaterthatneedstobeemptiedandcleaned,thenrefilledforapoolparty.Duringtheprocessofgettingthepoolready,Sylviadidallofthefollowingactivities,eachduringadifferenttimeinterval.
Removedwaterwithasinglebucket Filledthepoolwithahose(samerateasemptyingpool)
Drainedwaterwithahose(samerateasfillingpool)
Cleanedtheemptypool
Sylviaandhertwofriendsremovedwaterwiththreebuckets
Tookabreak
1. Sketchagraphshowingtheheightofthewaterlevelinthepoolovertime.Besuretoincludeallofactivities
Sylviadidtopreparethepoolfortheparty.Rememberthatonlyoneactivityhappenedatatime.Thinkcarefullyabouthoweachsectionofyourgraphwilllook,labelingwhereeachactivityoccurs.
2. WriteastoryconnectingSylvia’sprocessforemptying,cleaning,andthenfillingthepooltothegraphyouhave
created.Doyourbesttouseappropriatemathvocabulary. 3. a. Doesyourgraphrepresentafunction?Whyorwhynot?
b. Wouldallgraphscreatedforthissituationrepresentafunction?
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5.1FloatingDowntheRiverASolidifyUnderstandingTask Alonzo,Maria,andSierrawerefloatingininnertubesdownariver,enjoyingtheirday.Alonzonoticedthatsometimesthewaterlevelwashigherinsomeplacesthaninothers.Marianoticedthereweretimestheyseemedtobemovingfasterthanatothertimes.Sierralaughedandsaid“Mathiseverywhere!”Tolearnmoreabouttheriver,AlonzoandMariacollecteddatathroughoutthetrip.Alonzocreatedatableofvaluesbymeasuringthedepthofthewatereverytenminutes.
Time(inminutes)
0 10 20 30 40 50 60 70 80 90 100 110 120
Depth(infeet) 4 6 8 10 6 5 4 5 7 12 9 6.5 5
1. UsethedatacollectedbyAlonzotointerpretthekeyfeaturesofthisrelationship.MariacreatedagraphbycollectingdataonaGPSunitthattoldherthedistanceshehadtraveledoveraperiodoftime.
2. UsingthegraphcreatedbyMaria,describethekeyfeaturesofthisrelationship.
©www.flickr.com
/photos/hamiltonca
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3. Sierralookedatthedatacollectedbyhertwofriendsandmadeseveralofherownobservations.Explainwhyyoueitheragreeordisagreewitheachobservationmade.
a. Thedepthofthewaterincreasesanddecreasesthroughoutthe120minutesoffloatingdowntheriver.
Agree/Disagree Why?
b. Thedistancetraveledisalwaysincreasing.
Agree/Disagree Why?
c. Thedistancetraveledisafunctionoftime.
Agree/Disagree Why?
d. Thedistancetraveledisgreatestduringthelasttenminutesofthetripthanduringanyothertenminuteintervaloftime.
Agree/Disagree Why?
e. Thedomainofthedistancevs.timegraphisallrealnumbers.
Agree/Disagree Why?
f. They‐interceptofthedepthofwaterovertimefunctionis 0, 0 .
Agree/Disagree Why?
g. Thedistancetraveledincreasesanddecreasesovertime.
Agree/Disagree Why?
h. Thewaterlevelisafunctionoftime.
Agree/Disagree Why?
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i. Therangeofthedistancevs.timegraphisfrom 0, 15000 .
Agree/Disagree Why?
j. Thedomainofthedepthofwaterwithrespecttotimeisfrom 0, 120
Agree/Disagree Why?
k. Therangeofthedepthofwaterovertimeisfrom 4, 5 .
Agree/Disagree Why?
l. Thedistancevs.timegraphhasnomaximumvalue.
Agree/Disagree Why?
m. Thedepthofwaterreachedamaximumat30minutes.
Agree/Disagree Why?
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Name: FeaturesofFunctions 5.1Ready,Set,Go!ReadyTopic:SolvesystemsbygraphingGrapheachsystemoflinearequationsandfindwhere .1. 5 2
2 1
2. 4
1
3. 3
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SetTopic:Describingattributesofafunctionbasedonthegraphicalrepresentation.Foreachgraphgiven,matchittothecontextualdescriptionthatfitsbest.Thenlabeltheindependentanddependentaxesoneachgraphwiththepropervariables. Graphs ContextualDescriptions
4.
a. Theamountofmoneyinasavingsaccountwhereregulardepositsandsomewithdrawalsaremade.
5.
b. Thetemperatureoftheovenonadaythatmombakesseveralbatchesofcookies.
6.
c. Theamountofgasolineonhandatthegasstationbeforeatankertruckdeliversmore.
7.
d. Thenumberofwatermelonsavailableforsaleatthefarmer’smarketonThursday.
8.
e. Theamountofmileagerecordedontheodometerofadeliverytruckoveratimeperiod.
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Giventhepairofgraphsoneachcoordinategrid,createalistofsimilaritiesthetwographsshareandalistofdifferences.Considerattributeslike,continuous,discrete,increasing,decreasing,linear,exponential,restrictionsondomainorrange,etc.9.
Similarities:
Differences:10.
Similarities:Differences:
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GoTopic:SolvingequationsFindthevalueofxineachequation.11.10 100,000 12.3 7 5 21 13. 6 15 4 3514.5 8 37 15.3 81 16.3 12 4 2317.10 2 22 18.243 8 3 19.5 7 118
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5.2AlternateWarmUpGraphingLinearandExponentialFunctions Grapheachofthefunctions.1. 2 5 2. 4 3 3. 5 3
4. 4 2 5. 2.5 4 6. 8 3
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©www.flickr.com
/photos/intercontinentalhongkong
5.2FeaturesofFunctionsAPracticeUnderstandingTaskForeachgraph,determineiftherelationshiprepresentsafunction.Ifso,statethekeyfeaturesofthefunction(intervalswherethefunctionisincreasingordecreasing,themaximumorminimumvalueofthefunction,domainandrange,andx‐andy‐intercepts.).1. 2.
3. 4.
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5.
6. 7.
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Thefollowingrepresentsacontinuousfunctiondefinedontheintervalfrom , .
8. Determinethedomain,range,x‐ andy‐intercepts.9. Basedonthetable,identifytheminimumvalueandwhereitislocated.
0 21 32 03 24 65 126 20
Thefollowingrepresentsadiscretefunctiondefinedontheintervalfrom , .
10.Determinethedomain,range,x‐ andy‐intercepts.11.Basedonthetable,identifytheminimumvalueandwhereitislocated.
1 42 103 54 85 3
Describethekeyfeaturesforeachsituation.12.Theamountofdaylightdependentonthetimeofyear.13.Thefirstterminasequenceis12.Eachconsecutivetermisexactly ofthepreviousterm.14.Marcusboughta$900couchonasixmonths,interestfree,paymentplan.Hemakes$50paymentstotheloan
eachweek.15.Thefirstterminasequenceis36.Eachconsecutivetermis lessthanthepreviousterm.16.Anempty15gallontankisbeingfilledwithgasolineatarateof2gallonsperminute.
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Foreachequation,sketchagraphandshowkeyfeaturesofthegraph.17. 2 4,when 0
18. 3
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Name: FeaturesofFunctions 5.2Ready,Set,Go!ReadyTopic:SolvesystemsbygraphingGrapheachsystemoflinearequationsandfindwhere .1. 2 7
4 5
2. 22 8
3. 5
Topic:Creatinggraphicalrepresentationsandnamingthedomain.Sketchagraphtorepresenteachfunction,andthenstatethedomainofthefunction.4 Asequenceoftermssuchthat 5. Asequenceoftermssuchthat
0 1, 1 7 0 8, 1
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SetTopic:Describefeaturesofafunctionfromitsgraphicalrepresentation.Foreachgraphgivenprovideadescriptionofthefunctionwhichincludesthetypeoffunction(ifknown),intervalswherethegraphisdecreasing&increasing,domain/range,andminimumandmaximumvalues(iftheyexist).
6. Descriptionoffunction: 7. Descriptionoffunction: Typeoffunction: Typeoffunction: Decreasing: Decreasing: Increasing: Increasing: Domain: Domain: Range: Range: Max: Max: Min: Min:Topic:Attributesoflinearandexponentialfunctions.Determineifthestatementistrueorfalse,thenjustifywhy.8. Alllinearfunctionsareincreasing.
True/FalseWhy?9. Arithmeticsequencesareanexampleoflinearfunctions.
True/FalseWhy?10.Exponentialfunctionshaveadomainthatincludesallrealnumbers.
True/FalseWhy?11.Geometricsequenceshaveadomainthatincludesallintegers.
True/FalseWhy?12.Therangeforanexponentialfunctionincludesallrealnumbers.
True/FalseWhy?13.Alllinearrelationshipsarefunctionswithadomainandrangecontainingallrealnumbers.
True/FalseWhy?.
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GoTopic:Createequationsusingbothexplicitandrecursivenotation.Writeequationsforthegiventablesinbothrecursiveandexplicitform.14. 15. 16.
n 1 62 123 24
n0 132 53 1
n1 54 115 13
Explicit: Explicit: Explicit:Recursive: Recursive: Recursive:Topic:Determinethedomainofafunctionfromthegraphicalrepresentation.Foreachgraphdeterminethedomainofthefunction.17. 18.
Domain: Domain:
19. 20.
Domain: Domain:
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5.3WarmUpWritingDomainandRangeinIntervalNotationGiventhegraphs,identifythedomainandrangeinintervalnotation.1.
Domain:Range:
2.
Domain:Range:
3.
Domain:Range:
4.
Domain:Range:
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©www.flickr.com
/photos/ableman/ 5.3TheWaterPark
ASolidifyUnderstandingTaskAlyandDayneworkatawaterparksupervisingdifferentrides.Theyhavetodrainthewaterattheendofeachmonthfortheridetheysupervise.Eachusesapumptoremovethewaterfromthesmallpoolatthebottomoftheirride.ThegraphbelowrepresentstheamountofwaterinAly’spool, ,andDayne’spool, ,overtime.
PartI1. Listasmanyobservations,aspossible,withtheinformationgiveninthegraphabove.
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PartIIDaynefiguredoutthatthepumpheusesdrainswateratarateof1000gallonsperminuteandtakes24minutestodrain.2. WritetheequationtorepresentthedrainingofDayne’spool, .Whatdoeseachpartoftheequation
represent?3. Basedonthisnewinformation,correctlyscaletheaxesonthegraphabovequestion1.4. Whatvaluesofxmakesenseinthissituation?Useintervalnotationtowritethedomainofthesituation.5. Determinetherange,oroutputvalues,thatmakesenseinthissituation.Writetherangeinintervalnotation.6. WritetheequationusedtorepresentthedrainingofAly’spool, .Usingintervalnotation,statethedomain
andrangeforthefunction, ,aswellasthedomainandrangeofthesituation.Comparethetwodomainsbydescribingtheconstraintsmadebythesituation.
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PartIIIBasedonthegraphandcorrespondingequationsforeachpool,answerthefollowingquestions.7. Whenis ?Whatdoesthismeanintermsofthesituation?8. Find 10 .Whatdoesthismeanintermsofthesituation?9. If 2000,then .Whatdoesthismeanintermsofthesituation?10.Whenis ?Whatdoesthismeanintermsofthesituation?
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Name: FeaturesofFunctions 5.3Ready,Set,Go!ReadyTopic:Attributesoflinearandexponentialfunctions.1. Writeawell‐developedparagraphcomparingandcontrastinglinearandexponentialfunctions.Besureto
includeasmanycharacteristicsofeachfunctionaspossibleandbeclearaboutthesimilaritiesanddifferencesthesefunctionshave.
SetTopic:Identifyingattributesofafunctionfromitsgraphicalrepresentation.Basedonthegraphgivenineachproblembelow,identifythedomain,range,intervalsofincreaseanddecrease.2.
Domain:Range:Increasing:Decreasing:
3.
Domain:Range:Increasing:Decreasing:
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4.
Domain:Range:Increasing:Decreasing:
5.
Domain:Range:Increasing:Decreasing:
6.
Domain:Range:Increasing:Decreasing:
7.
Domain:Range:Increasing:Decreasing:
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8.
Domain:Range:Increasing:Decreasing:
9.
Domain:Range:Increasing:Decreasing:
GoTopic:FindingequationsandrulesforfunctionsFindboththeexplicitandtherecursiveequationsforeachtableofvaluesbelow.10. 11. 12.
n 1 32 53 74 9
n2 43 84 165 32
n6 237 198 159 11
Explicit: Explicit: Explicit:Recursive: Recursive: Recursive:
13. 14. 15.
n 1 12 33 9
n3 84 45 2
n6 79 1312 19
Explicit: Explicit: Explicit:Recursive: Recursive: Recursive:
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16. 17. 18.
n 2 404 328 16
n2 163 44 1
n17 520 1026 20
Explicit: Explicit: Explicit:Recursive: Recursive: Recursive:
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5.4WarmUpUsingGraphicalRepresentationsofFunctionstoFindSolutionsUsethegraphofeachfunctionprovidedtofindthevaluesindicated.
1.
a. 4 b. 4 c. 4, d. 7,
2.
a. 1 b. 3 c. 4, d. 1,
3.
a. 0 b. 3 c. 1, d. 2,
4.
a. 5 b. 4 c. 4, d. 0,
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©www.flickr.com
/photos/neilt
5.4PoolingItTogether ASolidifyUnderstandingTaskAlyandDayneworkatawaterparkandhavetodrainthewaterattheendofeachmonthfortheridetheysupervise.Eachusesapumptoremovethewaterfromthesmallpoolatthebottomoftheirride.ThegraphbelowrepresentstheamountofwaterinAly’spool,
,andDayne’spool, ,overtime.Inthisscenario,theydecidedtoworktogethertodraintheirpoolsandcreatedtheequation ).Thefirstgraphbelowshows and separately.Ontheblankaxesbelow,drawthegraphof andscaletheaxesappropriately.
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Answerthefollowingquestionsabout .1. Whatdoes represent?2. Namethefeaturesof andexplainwhateachmeans(x‐andy‐intercepts,domainandrangeforthis
situationandfortheequation,maximaandminima,whetherornot isafunction,etc.)3. Writetheequationfor usingtheinterceptsfromthegraph.Comparethisequationtothesumofthe
equationscreatedfor and fromTheWaterParktask.Shouldtheybeequivalent?Whyorwhynot?4. Whatdoyounoticeabouthowthefunctions , and relatetotheirgraphs?(Hint:Thinkaboutthe
keyfeaturesofthesefunctions,suchasintercepts,slope,etc…andmaybestartwithatable).ForATwist:5. IfAlyandDayne’sbossstartedtodrainthewaterbeforetheyarrivedandwhentheygotthere,therewas
already5,000lessgallonsofwatertobedrained,howwouldthisimpacttheequation?6. Writethenewequationrepresentinghowlongitwilltakethemtodrainthetwopools.
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Name: FeaturesofFunctions 5.4Set,Go!SetTopic:Givencontextofafunctionfindsolutions.Foreachsituationeithercreateafunctionorusethegivenfunctiontofindandinterpretsolutions.1. Francollecteddataonthenumberoffeetshecouldwalkeachsecondandwrotethefollowingruletomodel
herwalkingrate 4 .
a. WhatisFranlookingforifshewrites“ 12 ”?
b. Inthissituationwhatdoes 100tellyou?
c. Howcanthefunctionrulebeusedtoindicateatimeof16secondswaswalked?
d. Howcanthefunctionrulebeusedtoindicatethatadistanceof200feetwaswalked?2. Ms.Callahanworkshardtobudgetandpredicthercostsforeachmonth.Sheiscurrentlyattemptingto
determinehowmuchhercellphonecompanywilllikelychargeherforthemonth.Sheispayingaflatfeeof$80amonthforaplanthatallowsforunlimitedcallingbutcostsheranadditionaltwentycentspertextmessage.a. Writeafunction, ,forMs.Callahan’scurrentcellplanthatwillcalculatethecostforthemonthbasedon
thenumberoftextmessagesshemakes. b. Find 20 c. Find 100 d. Find 45 e. Find 90
f. For$20shecanaddunlimitedtextingtohercurrentplan.Atwhatnumberoftextswouldthisoptionbe
lessexpensivethanhercurrentplan?
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3. Mr.Multbankhasdevelopedapopulationgrowthmodelfortherodentsinthefieldbyhishouse.Hebelievesthatstartingeachspringthepopulationcanbemodeledbasedonthenumberofweekswiththefunction
8 2 a. Find 128 b. Find 4 c. Find 10
d. Findthenumberofweeksitwilltakeforthepopulationtobeover20,000.e. Inayearwith16weeksofsummer,howmanyrodentswouldheexpectbytheendofthesummerusing
Mr.Multbank’smodel?f. Whataresomefactorsthatcouldchangetheactualresultfromyourestimate?
GoTopic:DiscreteandcontinuousForeachcontextorrepresentationdeterminewhetheritisdiscreteorcontinuousorcouldbemodeledbestinadiscreteorcontinuouswayandstatewhy.4. Susanhasasavingsplanwheresheplaces$5aweekinherpiggybank.
Discrete/ContinuousWhy?
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5. 6.
Discrete/Continuous Discrete/Continuous
Why? Why?7. Marshaltracksthenumberofhitshegetseachbaseballgameandisrecordinghistotalnumberofhitsforthe
seasoninatable.
Discrete/ContinuousWhy?
8. Thedistanceyouhavetraveledsincethedaybegan.
Discrete/ContinuousWhy?
9.
NumberofGumBalls
Cost
5 110 215 320 4
Discrete/ContinuousWhy?
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©www.flickr.com
/photos/alanenglish 5.5InterpretingFunctions
APracticeUnderstandingTaskGiventhegraphof ,answerthefollowingquestions.Unlessotherwisespecified,restrictthedomainofthefunctiontowhatyouseeinthegraphbelow.Approximationsareappropriateanswers.1. Whatis 2 ?_____________________2. Forwhatvalues,ifany,does 3?___________________3. Whatisthex‐intercept?_____________________4. Whatisthedomainof ?_____________________5. Onwhatintervalsis 0?_____________________6. Onwhatintervalsis increasing?_____________________7. Onwhatintervalsis decreasing?____________________8. Forwhatvalues,ifany,is 3?_____________________Considerthelineargraphof andthenonlineargraphof toanswerquestions9‐14.Approximationsareappropriateanswers.9. Whereis ?_____________________10.Whereis ?_____________________11.Whatis 0 0 ?_____________________12.Whatis 1 1 ?_____________________13.Whichisgreater: 0 or 3 ?_____________________14.Graph: from 1, 3 ____________________
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Thefollowingtableofvaluesrepresentstwocontinuousfunctions, and .Usethetabletoanswerthefollowingquestions:
15.Whatis 3 ?16.Forwhatvalue(s)is 0?17.Forwhatvaluesis increasing?18.Onwhatintervalis 19.Whichfunctionischangingfasterintheinterval 5, 0 ?Why?
Usethefollowingrelationshipstoanswerthequestionsbelow.
2 3 2 5 4 5 120.Whichoftheaboverelationsarefunctions?Explain.21.Find: 2 _____________________ 2 _____________________ 2 _____________________22.Writetheequationfor .23.Whereis ?24.Whereis increasing?25.Whichoftheabovefunctionshasthefastestgrowthrate?
5 42 134 30 93 20 52 12 11 6 30 2 71 0 112 0 153 2 194 6 235 12 276 20 31
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Name: FeaturesofFunctions 5.5Ready,Set,Go!ReadyTopic:SolvesystemsofequationsSolveeachsystemofequationseitherbysubstitutionorelimination.Useeachmethodatleastonce.Grapheachtoshowthesolution.1. 3
1
2. 6 32
3. 4
1
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SetTopic:ConnectingcontexttographicalrepresentationsForeachgraphcreateacontext,provideindependentanddependentvariablesthatwillfitthecontextyouchoose(labeltheaxeswithyourchoices).Thencreateastorythatdescribeswhatishappeningonthegraph.4.
Descriptionofcontextandastoryforthegraph:5.
Descriptionofcontextandastoryforthegraph:
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GoTopic:Describefeaturesofafunctionfromitsgraphicalrepresentation.Foreachgraphgivenprovideadescriptionofthefunction.Besuretoconsiderthefollowing:decreasing/increasing,min/max,domain/range,etc.6.
Decreasing:Increasing:Min:Max:Domain:Range:
7. Decreasing:Increasing:Min:Max:Domain:Range:
8.
Decreasing:Increasing:Min:Max:Domain:Range:
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9.
Decreasing:Increasing:Min:Max:Domain:Range:
Topic:Solvingliteralequationsforaspecifiedvariable.Rewriteeachequationinslope‐interceptform( ).10.12 3 6 11. 5 3 2 12. 9 4 2Writeanexplicitfunctionforthelinearfunctionthatgoesthroughthegivenpointwiththegivenslopesimplifiedintoslope‐interceptform.13. 3 1, 2 14. 4, 2
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5.6WarmUpToFunctionorNottoFunction–APracticeUnderstandingTaskDetermineifthefollowingrelationshipsarefunctions(bystatingyesorno).Besuretojustifyyourchoice. 1. Aperson’sfirstnameversustheirstudentidnumber.
2. Aperson’sstudentidnumberversustheirfirstname.
3. Thecostofgasversustheamountofgaspumped.
4. 3,6 , 4,10 , 8,12 , 4,10
5. ThetemperatureindegreesFahrenheitinSanDiegowithrespecttothetimeofday.
6.Distance Days
6 210 46 59 8
7. Theareaofacircleasitrelatestotheradius.
8.
3 55 57 7
9. Thelengthoffenceneededwithrespecttotheamountofareatobeenclosed
10.Thesizeoftheradiusofacircledependentonthearea.
11. Studentslettergradedependentonthepercentearned.
12. 4 62 78 36 41
13.Theexplicitformulafortherecursivesituationbelow:
1 3and1 4
14. Ifx is arationalnumber,then1
Ifxisanirrationalnumber,then 0
15.Time Money14 7018 8222 8225 90
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5.6AWaterFunctionASolidifyUnderstandingTaskAndrewwalkedaroundthewaterparktakingphotosofhisfamilywithhisphone.Later,hediscoveredhisphonewasmissing.Sothatotherscouldhelphimlookforhislostphone,hedrewapicturethat‘retracedhissteps’showingwherehehadwalked.
1. IfwewantedtodetermineAndrew’slocationintheparkwithrespecttotime,wouldhislocationbeafunction
oftime?Whyorwhynot?Explain.
©www.flickr.com
/photos/battlecreekcvb
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Refertothepicturewiththearrowsabovequestion1toanswerthefollowingquestions.2. SituationA:SketchagraphofthetotaldistanceAndrewwalkedifhewalkedataconstantratetheentiretime
(labelyouraxes).
3. SituationB:SketchagraphofAndrew’sdistancefromtheentrance(hisstartingpoint)asafunctionoftime
(labelyouraxes).
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4. a. HowwouldthegraphofeachsituationchangeifAndrewstoppedattheslideforaperiodoftime?Showthechangesonthegraphsbelow.
SituationA: SituationB:
b. Wouldthischangewhetherornotthissituationisafunction?
5. Howdidyoudecideonkeyfeaturesofthegraphs?Discussthesedecisionsbelow.Includeinyourdiscussion:
intervalsofincrease,intervalsofdecrease,domain,range,maximumsandminimums.
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SDUHSDMath1CollegePrep
Name: FeaturesofFunctions 5.6Ready,Set,Go!ReadyTopic:Determinedomainandrange,andwhetherarelationisafunctionornotafunction.Determineifeachsetoforderedpairsisafunctionornotthenstatethedomainandrange.1. 7, 2 , 3, 5 , 8, 4 , 6, 5 , 2, 3 Function:Yes/No Domain: Range:2. 9, 2 , 0, 4 , 4, 0 , 5, 3 , 2, 7 , 0, 3 , 3, 1 Function:Yes/No Domain: Range:3. 1, 2 , 2, 3 , 3, 4 , 4, 5 , 5, 6 , 6, 7 , 7, 8 , 8, 9 Function:Yes/No Domain: Range:Fortherepresentationofthefunctiongivendeterminethedomainandrange.4. 5.
Domain: Domain:
Range: Range:
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SDUHSDMath1CollegePrep
SetTopic:ComparingfunctionsfromdifferentrepresentationsUsethegivenrepresentationofthefunctionstoanswerthequestions.9.
a. Wheredoes ? b. Whatis 4 4 ? c. Whatis 2 2 ? d. Onwhatintervalis ? e. Sketch onthegraphprovided.
10.Thefunctions and aredefinedinthetablebelow.Eachfunctionisasetofexactlyfiveorderedpairs.
3 1 1
1 7 5
0 3 10
2 8 2
7 3 3 a. Whatis 3 3 ? b. Whatis 1 1 ? c. Whatis 0 0 ?d. Completetheadditionalcolumnsinthetablewith inonecolumnand intheother
column.
6. 2 7 Domain: Range:
7. 3 5 Domain: Range:
8. Theelementsinthetabledefinetheentiretyofthefunction.
x1 92 983 9874 9876
Domain: Range:
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SDUHSDMath1CollegePrep
11. a. Whereis ? b. Whatis 1 1 ? c. Whatis 0 0 ? d. Createanexplicitrulefor andfor . e. Sketch onthegraph
12.
a. Wheredoes ? b. Whatis 4 4 ? c. Whatis 2 2 ? d. Onwhatintervalis ? e. Sketch onthegraphprovided.
Topic:Determinewhetherornottherelationshipisafunction.Determineiftherelationshippresentedisafunctionornotandprovideajustification.13.ThedistanceapersonisfromthegroundrelatedtotimeastheyrideaFerrisWheel.14.Theamountofdaylightduringadaythroughoutthecalendaryear.15.ThevalueofaVolkswagenBugconvertiblefromtimeoffirstpurchasein1978tonow.16.Aperson’snameandtheirphonenumber.
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SDUHSDMath1CollegePrep
17.Thestadiuminwhichafootballplayerisplayingrelatedtotheoutcomeofthegame.GoTopic:Determiningfeaturesoffunctionsandfindingsolutionsusingfunctions.18.Forthegraphgivenbelow,describetheintervalsofdecrease/increase,min/max,anddomain/range.
Decreasing: Increasing: Min: Max:Domain: Range:
19.Forthegivensituationusethefunctiontofindandinterpretsolutions.
Hopehasbeentrackingtheprogressofherfamilyastheytravelacrossthecountryduringtheirvacationandshehascreatedafunction, 78 tomodeltheprogresstheyaremaking,wheretrepresentsthenumberofdaysand representsthemilestraveled.
a. WhatwouldHopebeattemptingtofindifshewrites: 4 78 4 ?
b. Whatwould 450meaninthissituation?
c. Whatwould 3.5 meaninthissituation?
d. HowcouldHopeusethefunctiontofindthetimeitwouldtaketotravel800miles?
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SDUHSDMath1CollegePrep
Usethegivenrepresentationofthefunctionstoanswerthequestions.20. a. Wheredoes ?
b. Whatis 0 0 ? c. Onwhatinterval(s)is ? d. Whatis 8 8 ?
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SDUHSDMath1CollegePrep
5.7WarmUpKeyFeaturesofFunctionUsethegivendescriptionofseveralofthekeyfeaturesofthefunctiontosketchapossiblegraphofthefunction.1. Domainis 2, 3 .
Rangeis 3, 7 .
Thefunctionisincreasingovertheinterval 2, 0 anddecreasingafter0.
Thefunctionisnotcontinuousateverypoint.
2. Thedomainofthefunctionis 5,∞
Therangeofthefunctionis 0,∞
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SDUHSDMath1CollegePrep
Name: FeaturesofFunctions 5.7Ready,Set,GoReadyTopic:Findtheoutputorinputbasedonwhatisgiven.Foreachfunctionfindthedesiredsolutions.1. 2 5
a. 4 b. 23, c. 13 d. 33,2. 3.
a. 2 a. 1 b. 3, b. 4, c. 0 c. 2 d. Whatistheexplicitrulefor d. Whatistheexplicitrulefor
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SDUHSDMath1CollegePrep
SetTopic:Describingthekeyfeaturesoffunctionsandcreatingarepresentationofafunctiongiventhekeyfeatures.Usethegivendescriptionofseveralofthekeyfeaturesofthefunctiontosketchapossiblegraphofthefunction.4. Thefunctionhasaminimumat 5.
Thefunctionhasamaximumat8.
Thefunctionhastwointervalsonwhichitisdecreasingandoneintervalonwhichitisincreasing.
ThedomainofthefunctionscontainsallRealnumbersfrom1to9.
5. Thisfunctionisnotcontinuousanywhere.
Thefunctioncontainsonlysevenelementsinitsdomain.
Thevaluesofthedomainarebetween 9and2.
Thevaluesoftherangearefrom 1and1.6. Thefunctionhasthreeintervalsonwhichitsslopeiszero.
Thefunctionhasamaximumandaminimum.
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SDUHSDMath1CollegePrep
GoTopic:FeaturesoffunctionsDescribethefollowingfeaturesofthefunction:Domain,range,whetherornotthefunctioniscontinuous,intervalswherethefunctionisincreasinganddecreasing,minimumandmaximumvalues.7. 8.
Domain: Domain:Range: Range:Continuous:Yes/No Continuous:Yes/NoIncreasing: Increasing:Decreasing: Decreasing:Minimum: Minimum:Maximum: Maximum:
9. 10.
Domain: Domain:Range: Range:Continuous:Yes/No Continuous:Yes/NoIncreasing: Increasing:Decreasing: Decreasing:Minimum: Minimum:Maximum: Maximum:
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SDUHSDMath1CollegePrep
11. 0 2, 1 ⋅ 3 12. 9 4 Domain: Domain:Range: Range:Continuous:Yes/No Continuous:Yes/NoIncreasing: Increasing:Decreasing: Decreasing:Minimum: Minimum:Maximum: Maximum:
13. | |
Domain:Range:Continuous:Yes/NoIncreasing:Decreasing:Minimum:Maximum:
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SDUHSDMath1CollegePrep
Module5TestDayHomework:IntroducingTransformationsAdaptedfromCPMGeometryUnit5CG14Inyourownwords,describehowyouwouldmovethefigureonthelefttocoincidewiththeoneontheright.Imagineyouarewritingasetofdirectionsforafriendtofollowthatwillusetheoriginalfiguretogetthefigureontheright.Beprecise!1.
2.
3.
4.
5.
6.