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©2012MathematicsVisionProject|MVPInpartnership withtheUtahStateOfficeofEducation
LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense
IntegratedMath1Module4
LinearandExponentialFunctions
AdaptedFrom
TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,
TravisLemon,JanetSutoriuswww.mathematicsvisionproject.org
InpartnershipwiththeUtahStateOfficeofEducation
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SDUHSDMath1CollegePrep
Module4OverviewPrerequisiteConcepts&Skills:
Writerecursiveandexplicitrules/formulasforarithmeticandgeometricsequences Usefunctionnotationtoevaluatefunctions Modelsequenceswithatableofvaluesandgraph
SummaryoftheConcepts&SkillsinModule4:
Transitionfromarithmeticandgeometricsequencestolinearandexponentialmodels. Distinguishbetweencontinuousvdiscrete Comparelinearandexponentialmodels Applylinearandexponentialfunctiontomodelsituations(population) Solvelinearandexponentialequations Developandusesimpleandcompoundinterestformulas Analyzerateofchangeforagivencontext Representlinearequationsusingslope‐intercept,standard,andpoint‐slopeformandidentifythebenefits
andidealusesofeachformContentStandardsandStandardsofMathematicalPracticeCovered:
ContentStandards:F.IF.3,F.IF.6,F.IF.7,F.LE.1,F.LE.2,F.LE.3,F.LE.5,F.BF.1,F.BF.2,A.SSE.1,A.SSE.3,A.CED.2,A.REI.3
StandardsofMathematicalPractice:1.Makesenseofproblemsandpersevereinsolvingthem. 2.Reasonabstractlyandquantitatively. 3.Constructviableargumentsandcritiquethereasoningofothers. 4.Modelwithmathematics. 5.Useappropriatetoolsstrategically. 6.Attendtoprecision. 7.Lookforandmakeuseofstructure. 8.Lookforandexpressregularityinrepeatedreasoning.
Materials:
GraphingutilityModule4Vocabulary:
Linearmodel/equation/function Exponentialmodel/equation/function Domain DiscreteFunction ContinuousFunction ChangeFactor AveragerateofChange SecantLine TangentLine Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Point‐slopeform Simpleinterest Compoundinterest Principal
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ConceptsUsedIntheNextModule: Useacontexttographanddescribekeyfeaturesoffunctions Usetablesandgraphstointerpretkeyfeaturesoffunctions Interpretfunctionsusingnotation Combinefunctionsandanalyzecontextsusingfunctions Usegraphstosolveproblemsgiveninfunctionnotation Definefunction Identifywhetherornotarelationisafunctiongivenvariousrepresentations
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SDUHSDMath1CollegePrep
Module4–LinearandExponentialFunctions4.1Introducingcontinuouslinearandexponentialfunctions(F.IF.3)Warm‐Up:Writeafunction/graphthesituationClassroomTask:ConnectingtheDots:PiggiesandPools–ADevelopUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.14.2DefininglinearandexponentialfunctionsbaseduponthepatternofchangeandIdentifyingratesofchangeinlinearandexponentialfunctions(F.LE.1,F.LE.2)Warm‐Up:RepresentationsoflinearandexponentialfunctionsClassroomTask:
SortingOuttheChange–ASolidifyUnderstandingTask Where’sMyChange–APracticeUnderstandingTask
Ready,Set,GoHomework:LinearandExponentialFunctions4.24.3ComparingthegrowthoflinearandexponentialfunctionsandComparinglinearandexponentialmodelsofpopulation(F.BF.1,F.BF.2,F.LE.1,F.LE.2,F.LE.3,F.LE.5,F.IF.7)Warm‐Up:GraphingLinearEquationsinSlope‐InterceptFormClassroomTask:Growing,Growing,Gone–ASolidifyUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.34.4Interpretingequationsthatmodellinearandexponentialfunctions(A.SSE.1,A.CED.2,F.LE.5)ClassroomTask:MakingMyPoint–ASolidifyUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.44.5Evaluatingtheuseofvariousformsoflinearandexponentialequations(A.SSE.1,A.SSE.3,A.CED.2,F.LE.5)Warm‐Up:The4formsofalinearequationClassroomTask:EfficiencyExperts–ASolidifyUnderstandingTaskReady,GoHomework:LinearandExponentialFunctions4.54.6Understandingandinterpretingformulasforexponentialgrowthanddecay(A.SSE.1,A.CED.2,F.LE.5,F.IF.7)Warm‐Up:InterestClassroomTask:UpaLittle,DownaLittle–ASolidifyUnderstandingTaskSet,GoHomework:LinearandExponentialFunctions4.64.7Solvingexponentialandlinearequations(A.REI.3)Warm‐Up:Writingalinearequationinslope‐interceptformClassroomTask:XMarkstheSpot–APracticeUnderstandingTaskReady,Set,GoHomework:LinearandExponentialFunctions4.7&OptionalModule4ReviewHomework
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4.1WarmUpRepresentationsofexponentialfunctionsBillyhasawarrenof16rabbits.Eachmonththereare1.5timesmorerabbits.1. Completethetablebelowshowingthenumberofrabbitsasafunctionoftime.
Month #ofRabbits0 1 2 3 4
2. Drawagraphofthesituation,clearlylabelingandscalingtheaxes.
3. Writeafunctionthatmodelsthesituation.
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www.flickr.com/photos/teegardin
4.1ConnectingtheDots:PiggiesandPoolsADevelopUnderstandingTask1. Mylittlesister,Savannah,isthreeyearsold.Shehasapiggybankthatshewantsto
fill.ShestartedwithfivepenniesandeachdaywhenIcomehomefromschool,sheisexcitedwhenIgiveherthreepenniesthatareleftoverfrommylunchmoney.Createamathematicalmodelforthenumberofpenniesinthepiggybankondayn.
2. Ourfamilyhasasmallpoolforrelaxinginthesummerthatholds1500gallonsofwater.Idecidedtofillthe
poolforthesummer.WhenIhad5gallonsofwaterinthepool,IdecidedthatIdidn’twanttostandoutsideandwatchthepoolfill,soIhadtofigureouthowlongitwouldtakesothatIcouldleave,butcomebacktoturnoffthewaterattherighttime.Icheckedtheflowonthehoseandfoundthatitwasfillingthepoolatarateof2gallonseveryminute.Createamathematicalmodelforthenumberofgallonsofwaterinthepoolattminutes.
3. I’mmoresophisticatedthanmylittlesistersoIsavemymoneyinabankaccountthatpaysme3%intereston
themoneyintheaccountattheendofeachmonth.(IfItakemymoneyoutbeforetheendofthemonth,Idon’tearnanyinterestforthemonth.)Istartedtheaccountwith$50thatIgotformybirthday.CreateamathematicalmodeloftheamountofmoneyIwillhaveintheaccountaftermmonths.
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4. Attheendofthesummer,Idecidetodraintheswimmingpool.Inoticedthatitdrainsfasterwhenthereismorewaterinthepool.Thatwasinterestingtome,soIdecidedtomeasuretherateatwhichitdrains.Ifoundthatitwasdrainingatarateof3%everyminute.Createamathematicalmodelofthegallonsofwaterinthepoolattminutes.
5. Compareproblems1and3.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?6. Compareproblems1and2.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?7. Compareproblems3and4.Whatsimilaritiesdoyousee?Whatdifferencesdoyounotice?
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Name: LinearandExponentialFunctions 4.1Ready,Set,Go!ReadyTopic:RecognizingarithmeticandgeometricsequencesPredictthenext2termsinthesequence.Statewhetherthesequenceisarithmetic,geometric,orneither.Justifyyouranswer.1. 4, 20, 100, 500,⋯ 2. 3, 5, 8, 12,⋯3. 64, 4, 3,27,⋯ 4. 1.5, 0.75, 0, 0.75,⋯5. 40, 10, , , ⋯ 6. 1, 11, 111, 1111,⋯7. 3.6, 5.4, 8.1, 12.15,⋯ 8. 64, 47, 30, 13,⋯9. Createapredictablesequenceofatleast4numbersthatisNOTarithmeticorgeometric.
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SetTopic:DiscreteandcontinuousrelationshipsIdentifywhetherthefollowingstatementsrepresentadiscreteoracontinuousrelationship.10.Thehaironyourheadgrows inchpermonth.11.Foreverytonofpaperthatisrecycled,17treesaresaved.12.Approximately3.24billiongallonsofwaterflowoverNiagaraFallsdaily.13.Theaveragepersonlaughs15timesperday.14.ThecityofBuenosAiresadds6,000tonsoftrashtoitslandfillseveryday.15.DuringtheGreatDepression,stockmarketpricesfell75%.GoTopic:SlopesoflinesDeterminetheslopeofthelinethatpassesthroughthefollowingpoints.16. 15, 9 , 10, 4 17. 0.5, 4 , 3, 3.5 18. 50, 85 , 60, 80 19.
5 204 173 14
20.
1 1
0
1 2
21.
5 330 305 27
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4.2WarmUpRepresentationsofLinearandExponentialFunctionsIneachofthefollowingproblems,youaregivenoneoftherepresentationsofafunction.Completetheremaining3representations.Identifytherateofchangefortherelation.1. Equation:
Table:
CreateaContext:Youandyourfriendsgotothestatefair.Itcosts$5togetintothefairand$3eachtimeyougoonaride.
Graph
2. Equation:
Table:Time Amount1 182 533 1624 4865 14586 4374
CreateaContext:
Graph
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© 2012 www.flickr.com/photos/ 401(K) 2012
4.2SortingOuttheChangeASolidifyUnderstandingTask1A. Whichofthefollowingappliestothegiveninformation?
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
B. Bepreparedtodescribethepatternofchangeandtotellhowyoufoundit.1.
30 5725 4720 3715 2710 175 70 3
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
2. 0 3, 1 ⋅
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
3.
3 53 103 203 153 353 50
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
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Usethefigurebelowtoanswerquestions4and5.Step1 Step2 Step3 Step4
4. Thepatternofchangeintheperimeterofthefiguresfromonesteptothenext.
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
5. Thepatternofchangeinthe
areaofthefiguresfromonesteptothenext.
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
6.
0 54 82 3.58 16 0.5
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
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7. Thealgaepopulationincreasesby3%eachyear.
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
8.
Typeofpatternofchange(circleone):
Equaldifferencesoverequalintervals Equalfactorsoverequalintervals Neither
HowIfoundthepatternofchange:
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SDUHSDMath1CollegePrep
© 2012 www.flickr.com/photos/JennuineC
aptures
4.2Where’sMyChange?APracticeUnderstandingTask2LookthroughtheproblemsthatyouworkedwithintheSortingOuttheChangetask.Chooseoneproblemfromyourlinearcategory(equaldifferencesoverequalintervals)andoneproblemfromyourexponentialcategory(equalfactorsoverequalintervals).Addasmanyrepresentationsasyoucantotheproblemyouselected.Forinstance,ifyouchooseproblem#1whichisatable,youshouldtrytorepresentthefunctionwithagraph,anexplicitequation,arecursiveequation,andastorycontext.Identifytherateofchangeinthefunction.Ifthefunctionislinear,identifytheconstantrateofchange.Ifthefunctionisexponential,identifythefactorofchange.Linear:Scenario/Context:RecursiveEquation: ExplicitEquation:Table: Graph:
Explainhowtherateofchangeappearsineachofyourrepresentations.
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Exponential:Scenario/Context:RecursiveEquation: ExplicitEquation:Table: Graph:
Explainhowtherateofchangeappearsineachofyourrepresentations.
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4.2ResourcePage
LinearFunctions:EqualDifferencesOverEqualIntervals
ExponentialFunctions:
EqualFactorsOverEqualIntervals
Representation: Wecanidentifytherateofchangeby: Representation: Wecanidentifythechangefactorsby:
Tables Tables
Graphs Graphs
Equations Equations
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Name: LinearandExponentialFunctions 4.2Ready,Set,Go!ReadyTopic:RatesofchangeinlinearmodelsSaywhichsituationhasthegreatestrateofchange1. Theamountofstretchinashortbungeecordstretches6incheswhenstretchedbya3poundweight.Aslinky
stretches3feetwhenstretchedbya1poundweight.2. Asunflowerthatgrows2incheseverydayoranamaryllisthatgrows18inchesinoneweek.3. Pumping25gallonsofgasintoatruckin3minutesorfillingabathtubwith40gallonsofwaterin5minutes.4. Ridingabike10milesin1hourorjogging3milesin24minutes.Topic:Recognizingthegreaterrateofchangewhencomparinglinearfunctionsandexponentialfunctions.Identifywhethersituation“a”orsituation“b”hasthegreaterrateofchange.5. a.
10 489 438 387 33
b.
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6. a.
b.
7. a. Leehas$25withheldeachweekfromhissalarytopayforhissubwaypass.
b. Joseoweshisbrother$50.Hehaspromisedtopayhalfofwhatheoweseachweekuntilthedebtispaid.
8. a.x 6 10 14 18y 13 15 17 19
b. Thenumberofrhombiineachshape.
Figure1 Figure2 Figure3
9. a. 2 5 b. Inthechildren’sbook,TheMagicPot,everytimeyouputoneobjectintothepot,twoofthesameobjectscomeout.Imaginethatyouhave5magicpots.
10. a. Examinethegraphattheleftfrom0to1.Whichgraphdoyouthinkisgrowingfaster?
b. Nowlookatthegraphfrom2to3.Whichgraphis
growingfasterinthisinterval?
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SDUHSDMath1CollegePrep
SetTopic:linearratesofchangeDeterminetherateofchangeineachtablebelow.11.
3 131 50 13 11
12.
4 40 42 86 16
13.10 145 825 050 10
14.Completethetablesbelowtocreatetheindicatedtypeoffunction.
a. LinearFunction b. ExponentialFunction
5 5
4 4
3 3
2 2
1 1
0 0 Topic:Recognizinglinearandexponentialfunctions.Foreachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.15.Thepopulationofatownisdecreasingatarateof
1.5%peryear.
16. Joanearnsasalaryof$30,000peryearplusa4.25%commissiononsales
17.Thenumberofgiftsreceivedeachdayof“The12DaysofVacation”asafunctionoftheday.(“Onthe4th dayofVacationmytruelovegavetome,4callingbirds,3Frenchhens,2turtledoves,andapartridgeinapeartree.”)
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GoTopic:Recursiveandexplicitequationsofgeometricsequences.Writetherecursiveandexplicitequationsforeachgeometricsequence.18.Marissahassaved$1000inajar.Sheplanstowithdrawhalfofwhat’sremaininginthejarattheendofeach
month.Recursiveequation: ExplicitEquation:
19.
Time(Days)
NumberofBacteria
1 102 1003 10004 10000
RecursiveEquation:ExplicitEquation:
10.Foldsinpaper
Numberofrectangles
0 11 22 43 8
RecursiveEquation:ExplicitEquation:
21. 1024, 256, 64, 16, …
RecursiveEquation:
ExplicitEquation:
22. 3, 9, 27, 81, …
RecursiveEquation:ExplicitEquation:
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Topic:GeometricmeansForeachgeometricsequencebelow,findthemissingtermsinthesequence.23.
x 1 2 3 4 5
y 2 162
24.
x 1 2 3 4 5
y 3
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4.3ResourcePageLinearModel:
1910 1930 1950 1970 1990 2010 2030
Year#
Population 92 250
ExplicitFormula:ExponentialModel:
Year Year# PercentGrowth
5% 10%
1910 92 92 92 92 92 92 92
1930
1950
1970
1990 250
2010
2030
Percenttoolowortoohigh?
ExplicitFormula:
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4.3Warm‐UpGraphinglinearequationsinslope‐interceptform.Graphthefollowingequations.1. 3 1 2. 5 4
3. 1 2 3 4. 2 4
5. – 6 6. 5 6
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©www.flickr.com/photos/arenam
ont
4.3Growing,Growing,GoneASolidifyUnderstandingTask21. TheU.S.populationin1910was92millionpeople.In1990thepopulationwas250
million.Createbothalinearandanexponentialmodelofthepopulationfrom1910to2030,withprojecteddatapointsatleastevery20years,startingin1910.
LinearModel: ExponentialModel: Equationfromdatapoints: Equationfromdatapoints:
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2. TheactualU.S.populationdata(inmillions)was:1930: 122.81950: 152.31970: 204.9
WhichmodelprovidesabetterforecastoftheU.S.populationfortheyear2030?Explainyouranswer.
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Name: LinearandExponentialFunctions 4.3Ready,Set,Go!ReadyTopic:Comparingarithmeticandgeometricsequences.The1stand5thtermsofasequencearegiven.Fillinthemissingnumbersforanarithmeticsequence.Thenfillinthenumbersforageometricsequence.1.
Arithmetic 3 48
Geometric 3 48
2.
Arithmetic 12 0.75
Geometric 12 0.75
Topic:ComparinglinearandexponentialmodelsInquestions3‐8,comparedifferentcharacteristicsofeachtypeoffunctionbyfillinginthecellsofeachtableascompletelyaspossible. 4 3 4 3
3. Typeofgrowth
4. Whatkindofsequencecorrespondstoeachmodel?
5. Makeatableofvalues
x y x y
6. Findtherateofchange
7. Grapheachequation.
Comparethegraphs.Whatisthesame?Whatisdifferent?
8. Findthey‐interceptforeachfunction.
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9. Findthey‐interceptsforthefollowingequations a. 3
b. 3
10.Explainhowyoucanfindthey‐interceptofalinearequationandhowthatisdifferentfromfindingthe
y‐interceptofageometricequation.SetTopic:Comparinglinearandexponentialfunctions.11.Calcu‐ramahadanetincomeof5milliondollarsin2010,whileasmallcompetingcompany,Computafest,hada
netincomeof2milliondollars.ThemanagementofCalcu‐ramadevelopsabusinessplanforfuturegrowththatprojectsanincreaseinnetincomeof0.5millionperyear,whilethemanagementofComputafestdevelopsaplanaimedatincreasingitsnetincomeby15%eachyear.
a. Expresstheprojectednetincomesinthesetwobusinessplansasrecursiveformulas.
b. Writeanexplicitequationforthenetincomeasafunctionoftimeforeachcompany’sbusinessplan.
c. Compareyouranswersinaandb.Howarethetworepresentationssimilar?Howdotheydiffer?Whatrelationshipsarehighlightedineachrepresentation?
d. Explainwhyifbothcompaniesareabletomeettheirnetincomegrowthgoals,thenetincomeofComputafestwilleventuallybelargerthanthatofCalcu‐rama.InwhatyearwillthenetincomeofComputafestbelargerthanthatofCalcu‐rama?
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GoTopic:Writingexplicitequationsforlinearandexponentialmodels.Writetheexplicitequationforthetablesandgraphsbelow.12. 13.
1 4 81
0 2 3 27
1 10 2 9
2 50 1 3
14.
15.
16.
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Topic:Solvingsystemsthroughgraphing.Findthesolutiontothesystemsofequationsbygraphing.
17. 3 4
18.2 23 15
19.4
2 1 0
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© 2012 www.flickr.com/photos/teddylam
bec
4.4MakingMyPointASolidifyUnderstandingTaskZacandSionewereworkingonpredictingthenumberofquiltblocksinthispattern:
Whentheycomparedtheirresults,theyhadaninterestingdiscussion:Zac: Igot 6 1becauseInoticedthat6blockswereaddedeachtimesothepatternmusthavestartedwith
1blockat 0.Sione: Igot 6 1 7becauseInoticedthatat 1therewere7blocksandat 2therewere13,soI
usedmytabletoseethatIcouldgetthenumberofblocksbytakingonelessthanthen,multiplyingby6(becausethereare6newblocksineachfigure)andthenadding7becausethat’showmanyblocksinthefirstfigure.Here’smytable:
1 2 3 n7 13 19 6 1 7
1. WhatdoyouthinkaboutthestrategiesthatZacandSioneused?Areeitherofthemcorrect?Whyorwhynot?
Useasmanyrepresentationsasyoucantosupportyouranswer.
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ThenextproblemZacandSioneworkedonwastowritetheequationofthelineshownonthegraphbelow.
Whentheywerefinished,hereistheconversationtheyhadabouthowtheygottheirequations:Sione: Itwashardformetotellwherethegraphcrossedtheyaxis,soIfoundtwopointsthatIcouldreadeasily,
1, 3 and 9, 7 .Ifiguredoutthattheslopewas andmadeatableandcheckeditagainstthegraph.Here’smytable:
1 3 5 7 9
3 4 5 6 7 1 3
Iwassurprisedtonoticethatthepatternwastostartwiththen,subtract1,multiplybytheslopeandthensubtract3.Igottheequation 1 3.
Zac: Hey—IthinkIdidsomethingsimilar,butIusedthepoints, 7, 6 and 9, 7 .
Iendedupwiththeequation: 9 7.Oneofusmustbewrongbecauseyourssaysthatyousubtract1tothenandminesaysthatyousubtract9.Howcanwebothberight?
2. Whatdoyousay?Cantheybothberight?
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Zac: Myequationmademewonderiftherewassomethingspecialaboutthepoint 9, 7 sinceitseemedtoappearinmyequation 9 7whenIlookedatthenumberpattern.NowI’mnoticing
somethinginteresting—thesamethingseemstohappenwithyourequation, 9 2andthepoints 9, 2 .
3. DescribethepatternthatZacisnoticing.4. FindanotherpointonthelinegivenaboveandwritetheequationthatwouldcomefromZac’spattern.5. Whatwouldthepatternlooklikewiththepoint , ifyouknewthattheslopeofthelinewasm?6. Couldyouusethispatterntowritetheequationofanylinearfunction?Whyorwhynot?
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ZacandSionewentbacktoworkonanextensionofthequiltproblemtheywereworkingonbefore.Nowtheyhavethispattern:
Zac: Thisoneworksalotlikethelastquiltpatterntome.Theonlydifferenceisthatthepatternisdoubling,soI
knewitwasexponential.Ithoughtthatitstartswith7blocksanddoubles,sotheequationmustbe7 2 .
Sione: Idon’tknowaboutthat.Iagreethatitisanexponentialfunction—justlookatthatgrowthpattern.But,I
madethistable:
1 2 3
7 14 28 7 2
Iusedthenumbersinthetableandgotthisequation: 7 2 .Thisseemsjustlikeallthatstuffweweredoingwiththelines,butIthinkthatthegraphsofthesetwoequationswouldbedifferent.Thereissomethingdefinitelywronghere.
7. WhatisdifferentaboutthethinkingthatZacandSioneusedtocometodifferentequations?8. Howaretheirresultssimilartotheirresultsonthelinearquiltpatternabove?Howaretheydifferent?
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Zac: Iknow!Let’strydoingthesamethingwithyourexponentialfunctionasthelinearfunction.Whatifwetookthepoint 1, 7 andwrotetheequationthisway:
2 7
SeewhatIdid?Ididthesubtract1thingwiththexandthenaddedonthe7fromthey‐valueofthepoint.I’llbetthisisareallygoodshortcuttrick.
9. Graphthethreeexponentialfunctionstoverifytheirequivalenceortoshowthattheyarenotequivalent.
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Name: LinearandExponentialFunctions 4.4Ready,Set,Go!ReadyTopic:Writingequationsoflines.Writetheequationofalineinslope‐interceptform: ,usingthegiveninformation.1. 7, 4 2. , 3 3. 16, Writetheequationofthelineinpoint‐slopeform: ,usingthegiveninformation.4. 9, 0, 7 5. , 6, 1 6. 5, 4, 11 7. 2, 5 3, 10 8. 0, 9 3, 0 9. 4, 8 3, 1
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SetTopic:GraphinglinearandexponentialfunctionsMakeagraphofthefunctionbasedonthefollowinginformation.Addyouraxes.Chooseanappropriatescaleandlabelyourgraph.Thenwritetheequationofthefunction.10. Thebeginningvalueofthefunctionis5andits
valueis3unitssmallerateachstage.Equation:
11. Thebeginningvalueis16anditsvalueis smallerateachstage.
Equation:
12. Thebeginningvalueis1anditsvalueis10timesasbigateachstage.Equation:
13. Thebeginningvalueis 8anditsvalueis2unitslargerateachstage.
Equation:
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GoTopic:Recognizinglinearandexponentialfunctions.Foreachrepresentationofafunction,decideifthefunctionislinear,exponential,orneither.14.3 4 3
15.ideofasquare Areaofasquare
1inch 1in22inches 4in23inches 9in24inches 16in2
16.
Topic:Slope‐interceptformRewritetheequationsinslope‐interceptform.17.2 10 6 12
18. 13 8 14
19. 11 7 2
20. 3 2 9 12
21. 2 10 25
22. 1 3
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4.5Warm‐UpThefourformsofalinearequationBelowarethe4formsofthesamelinearequation.Completethetable:
Slope‐Intercept Point‐Slope Standard RecursiveFormRateofChange x‐intercept y‐intercept
3 2 3 5 13 3 2 0 21 3
7 8 5 4 280 7
1
3 6 1 2 3 90 3
1
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©2012 www.flickr.com/photos/cannongod
4.5EfficiencyExpertsASolidifyUnderstandingTaskPart1:InvestigatingLinearFormsInourworksofar,wehaveworkedwithlinearandexponentialequationsinmanyforms.Someoftheformsofequationsandtheirnamesare:
Name Equation
SlopeInterceptForm
wheremistheslopeandbisthey‐intercept1
PointSlopeForm
wheremistheslopeand , arethecoordinatesofapointontheline
3 4
StandardForm
2 2
RecursiveForm1
Givenaninitialvalue andD=constantdifferenceinconsecutiveterms
0 1, 1
1. Createatableofvaluesforeachequationabove.Verifythatthefourequationsareequivalent.
SlopeInterceptForm: PointSlopeForm
StandardForm: RecursiveForm
2. Explainhowyouknowthatthefourequationsarelinear.
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ContextInvestigation:Whichformbesttellsthestory?Youhavebeenappointedasamathematicsefficiencyexpert.Yourjobistocomparethesefourformsofequationsfordifferentusesanddecidewhichformismostefficientandeffectiveforeachuse.Determinewhichequationwillbethemostefficientandeffectiveforeachscenario.3. Inhisjobsellingvacuums,Joemakes$500eachmonthplus$20foreachvacuumhesells.
StandardForm: Slope‐InterceptForm:20 500 20 500
4. TheTreeHuggerGranolaCompanymakestrailmixwithcandiesandnuts.Thecostofcandiesforatrailmixis
$2perpoundandthecostofthenutsis$1.50perpound.Thetotalcostofabatchoftrailmixis$540.
StandardForm: SlopeInterceptForm:2 1.5 540 360
5. GrandmaBillingsisworkingonaquiltwithblocksinthefollowingpattern.
StandardForm: RecursiveForm:6 1 1 7, 1 6
6. Apoolisfillingataconstantrateof5gallonsperminute.After13minutes,thereare500gallonsinthepool.
PointSlopeForm: StandardForm:5 13 500 5 435
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EquationInvestigation:Determinewhichofthefourformsoflinearequationstousetowriteeachequation.Thenwritetheequationusingthechosenform.7. Writetheequationofthelinewithaslopeof 2throughthepoint 2, 5 .8. Writetheequationofthelinethroughthepoints 1, 2 and 4, 1 .9. Writetheequationofthearithmeticsequencethatstartswith 7andeachtermdecreasesby3.GraphInvestigation:Graphthefollowingequations.Thendescribehoweachrepresentationofthelinearfunctionassistsincompletingthegraph.10. 5
Howdoestheslopeinterceptformassistincompletingthegraph?
11. 3 5 15
Howdoesthestandardformassistincompletingthegraph?
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12. 4 6 5
Howdoesthepointslopeformassistincompletingthegraph?
13. 0 2, 1 5
Howdoestherecursiveformassistincompletingthegraph?
EfficiencyUsingtheFourFormsofLinearEquations:14.Fillintheresourcepageattheendofthistaskwithyourfindingsforlinearfunctions.
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Part2:InvestigatingExponentialFormsDuringthecourseoftheyear,wehavealsoworkedwithformsofexponentialequations,withafewmorecomingbeforetheendofthemodule.Theformsofexponentialequationsthatwehaveseensofar:
Name Equation
ExplicitForm 10 3
RecursiveForm1 ⋅
Givenaninitialvalue r=constantratiobetweenconsecutiveterms
1 10, 1 ⋅ 3
ContextInvestigation:Whichformtellsthestorybest?Determinewhichequationwillbethemostefficientandeffectiveforeachscenario.15.GrandmaBillingshasstartedpiecingherquilttogetherandhascreatedthefollowinggrowthpattern:
Whichequationbestmodelsthenumberofsquaresineachblock?ExplicitForm: RecursiveForm:
7 2 1 7, 1 ⋅ 2
16.ThepopulationoftheresorttownofJavaHotSpringsin2003wasestimatedtobe35,000peoplewithan
annualrateofincreaseofabout2.4%.WhichequationbestmodelsthenumberofpeopleinJavaHotSprings,witht=thenumberofyearsfrom2003?
ExplicitForm: RecursiveForm:
35,000 1.024 0 35,000, 1 ⋅ 1.024
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GraphInvestigation:Graphthefollowingequations.Thendescribehoweachrepresentationofthelinearfunctionassistsincompletingthegraph.17. 2 1.8
Howdoestheexplicitformassistincompletingthegraph?
18. 0 5, 1 ⋅ 0.6
Howdoestherecursiveformassistincompletingthegraph?
EfficiencyUsingtheTwoFormsofExponentialEquations:19.Fillintheresourcepageattheendofthistaskwithyourfindingsforexponentialfunctions.
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4.5ResourcePage
Whendoyouusethisformfor…
FormofLinearEquation
Situationsincontext(i.e.storyproblems)
WritingEquations Graphing
SlopeInterceptForm
PointSlopeForm
StandardForm
RecursiveForm1 , 1
Whendoyouusethisformfor…
FormofExponentialEquation
Situationsincontext(i.e.storyproblems)
WritingEquations
Graphing
ExplicitForm⋅
RecursiveForm1 , 1 ⋅
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SDUHSDMath1CollegePrep
Name: LinearandExponentialFunctions 4.5Ready,Go!ReadyTopic:SimpleinterestWhenapersonborrowsmoney,thelenderusuallycharges“rent”onthemoney.This“rent”iscalledinterest.Simpleinterestisapercent“r”oftheoriginalamountborrowed“P”multipliedbythetime“t”,usuallyinyears.Theformulaforcalculatingtheinterestis .Calculatethesimpleinterestowedonthefollowingloans.1. $1000 11% 2 2. $6500 12.5% 5 3. $20,000 8.5% 6 4. $700 20% 6 GoTopic:Solvingmulti‐stepequationsSolvethefollowingequations5. 12 6 4 5 2 3 1
6. 5 2 4 3 5 19
7. 7 3 4 2 6 2 3 17
8. 2 1 6 3
9. Whatdoesitmeanwhenyouhavesolvedanequation?10.Explainhowalinearequationcanhavemorethanonesolution.
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Topic:GeometricmeansForeachgeometricsequencebelow,findthemissingtermsinthesequence.11.
x 1 2 3 4 5
y 10 0.625
12.
x 1 2 3 4 5
y g
13.
x 1 2 3 4 5
y 3 243
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4.6Warm‐UpInterestJuanitaborrowed$1,000andagreedtopay15%simpleinterestfor5years.Juanitadidnothavetomakeanypaymentsuntiltheendofthe5years,butthenshehadtopaybacktheamountborrowed“P”plusalloftheinterest“i”forthe5years“t.”BelowisachartthatshowshowmuchmoneyJuanitaowedthelenderattheendofeachyearoftheloan.
EndofYear
Interestowedfortheyear
Totalamountofmoneyowedtothelendertopaybacktheloan.
1 $1000 0.15 $150 A=Principal+interest=$1150
2 $1000 0.15 $150 $1300
3 $1000 0.15 $150 $1450
4 $1000 0.15 $150 $1600
5 $1000 0.15 $150 $1750
1. a. Lookforthepatternyouseeinthechartabovefortheamount(A)owedtothelender.Writeafunction
thatbestdescribesAwithrespecttotime(inyears). b. Whattypeoffunctiondoesthefunctionfrompartarepresents?2. Considerifthelendercharged15%oftheamountowedinsteadof15%oftheamountoftheoriginalloan.
Calculatetheinterestowedeachyearifthelenderrequired15%oftheamountowedattheendofeachyear.Notethattheinterestowedattheendofthefirstyearwouldstillbe$150.Completethetablebelow.EndofYear
StartingAmountfortheYear Interest
TotalAmountattheEndoftheYear
1 $1000 $1000 0.15 $150 $1150
2 $1150 $1150 0.15 $1150
3
4
5
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SDUHSDMath1CollegePrep
© 2012 www.flickr.com/photos/civisi
4.6UpaLittle,DownaLittleASolidifyUnderstandingTaskOneofthemostcommonapplicationsofexponentialgrowthiscompoundinterest.Forexample,MamaBigbucksputs$20,000inabanksavingsaccountthatpays3%interestcompoundedannually.“Compoundedannually”meansthatattheendofthefirstyear,thebankpaysMama3%of$20,000,sotheyadd$600totheaccount.Mamaleavesheroriginalmoney($20,000)andtheinterest($600)intheaccountforayear.Attheendofthesecondyearthebankwillpayinterestontheentireamount,$20,600.Sincethebankispayinginterestonapreviousinterestamount,thisiscalled“compoundinterest”.1. ModeltheamountofmoneyinMamaBigbucks’bankaccountaftertyears.
#ofYears Money
Equation:
2. Isthefunctionlinearofexponential?Supportyouranswerwithfeaturesfromthetableandgraph.3. UseyourmodeltofindtheamountofmoneythatMamahasinheraccountafter20years.
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Aformulathatisoftenusedforcalculatingtheamountofmoneyinanaccountthatiscompoundedannuallyis:1
Where:A=amountofmoneyintheaccountaftertyearsP=principal,theoriginalamountoftheinvestmentr=theannualinterestratet=thetimeinyears4. ApplythisformulatoMama’sbankaccountandcomparetheresulttothemodelthatyoucreated.5. Basedupontheworkthatyoudidincreatingyourmodel,explainthe 1 partoftheformula.Anothercommonapplicationofexponentialfunctionsisdepreciation.Whenthevalueofsomethingyoubuygoesdownacertainpercenteachyear,itiscalleddepreciation.Forexample,MamaBigbucksbuysacarfor$20,000anditdepreciatesatarateof3%peryear.Attheendofthefirstyear,thecarloses3%ofitsoriginalvalue,soitisnowworth$19,400.6. ModelthevalueofMama’scaraftertyears.
#ofYears Money
Equation:
7. Isthefunctionlinearofexponential?Supportyouranswerwithfeaturesfromthetableandgraph.
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8. UseyourmodeltofindhowmanyyearswillittakeforMama’scartobeworthlessthan$500?9. HowisthesituationofMama’scarsimilartoMama’sbankaccount?10.Whatdifferencesdoyouseeinthetwosituations?11.ConsideryourmodelforthevalueofMama’scaranddevelopageneralformulafordepreciation.
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Name: LinearandExponentialFunctions 4.6Set,Go!SetTopic:Evaluateusingtheformulasforsimpleinterestorcompoundinterest.Giventheformulaforsimpleinterest: ,calculatethesimpleinterestpaid.(Remember,i=interest,P=theprincipal,r=theinterestrateperyearasadecimal,t=timeinyears)1. Findthesimpleinterestyouwillpayona5yearloanof$7,000at11%peryear.2. Howmuchinterestwillyoupayin2yearsonaloanof$1500at4.5%peryear?Use tocompletethetable.Allinterestratesareannual.
i = P r t
3. $11,275 12% 3years
4. $1,428 $5,100 4%
5. $93.75 $1,250 6months
6. $54 8% 9monthsGiventheformulaforcompoundinterest: 1 ,writeacompoundinterestfunctiontomodeleachsituation.Thencalculatethebalanceafterthegivennumberofyears.(Remember:A=thebalanceaftertyears,P=theprincipal,t=thetimeinyears,r=theannualinterestrateexpressedasadecimal)7. $22,000investedatarateof3.5%compoundedannuallyfor6years.8. $4300investedatarateof2.8%compoundedannuallyfor15years.
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9. Supposethatwhenyouare15yearsold,amagicgeniegivesyouthechoiceofinvesting$10,000atarateof7%or$5,000atarateof12%.Eitherchoicewillbecompoundedannually.Themoneywillbeyourswhenyouare65yearsold.Whichinvestmentwouldbethebest?Justifyyouranswer.
GoTopic:UsingorderofoperationswhenevaluatingexpressionsEvaluatetheexpressionsforthegivenvaluesofthevariables.10. 6 10;when 7and 3 11. ;when 2,and 612. 1 ;when 5,and 3 13. 9 4 ;when 4,and 514. ;when 7,and 2 15. 7 2 ;when 2,and 4
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4.7Warm‐UpWritingalinearequationinslope‐interceptformWritetheequationofthelineinslope‐interceptformgiventhefollowinginformation.(PandQarepointsontheline)1. 0 6, 1 2. 4, : 5, 8 3. 14 2 9 0 4. : 17, 4 , : 5, 26 5. 9 6 6. 11, 8 , : 1, 8
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©2012 www.flickr.com/photos/bfurlong
4.7XMarkstheSpotAPracticeUnderstandingTaskTablePuzzles1. Usethetablestofindthemissingvaluesofx:
a. .
2 4.4
10 10
8.6
4 0.2
1.2
b.
10 10
3 6
5
0
10
c. Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwo
tablesabove?
d.
5 243
81
3
2 9
e.
5 32
8
1
2
f. Whatequationscouldbewritten,intermsofxonly,foreachoftherowsthataremissingthexinthetwo
tablesabove?
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2. Whatstrategydidyouusetofindthesolutionstoequations(fromquestion1c)generatedbythetablesthatcontainedlinearfunctions?
3. Whatstrategydidyouusetofindthesolutionstoequations(fromquestion1f)generatedbythetablesthat
containedexponentialfunctions?GraphPuzzles4. Thegraphof 3isgivenbelow.Usethegraphtosolvetheequationsforxandlabelthesolutions.
a. 5 3 __________ LabelthesolutionwithanAonthegraphb. 3 1 __________ LabelthesolutionwithaBonthegraphc. 0.5 3 1 __________ LabelthesolutionwithaConthegraph
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5. Thegraphof 3 isgivenbelow.Usethegraphtosolvetheequationforxandlabelthesolutions.
a. 3 __________ LabelthesolutionwithanAonthegraphb. 3 9 __________ LabelthesolutionwithaBonthegraphc. 3√3 3 __________ LabelthesolutionwithaConthegraphd. 1 3 __________ LabelthesolutionwithaDonthegraphe. 6 3 __________ LabelthesolutionwithanEonthegraph
6. Howdoesthegraphhelptofindsolutionsforx?EquationPuzzles:Solveeachequationforx:7. 5 125 8. 7 6 9 9. 10 10,000
10.2.5 0.9 1.3 11.6
12. 16
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Name: LinearandExponentialFunctions 4.7Ready,Set,Go!ReadyTopic:Discretevs.continuous1. Giveacontextualexamplethatcanbemodeledbyadiscretefunction.2. Giveacontextualexamplethatcanbemodeledbyacontinuousfunction.Topic:Arithmeticandgeometricmeans3. Thefirstand5thtermsofasequencearegiven.Fillinthemissingnumbersforanarithmeticsequence.Then
fillinthenumbersforageometricsequence.
Arithmetic 6250 10Geometric 6250 10
Topic:Comparingratesofchange4. Comparetherateofchangeinthepairoffunctionsinthe
graphbyidentifyingtheintervalwhereitappearsthatischangingfasterandtheintervalwhereitappears
that ischangingfaster.Verifyyourconclusionsbymakingatableofvaluesforeachfunctionandexploringtheratesofchangeinyourtables.
Topic:Determiningifasequencesislinear,exponential,orneither.5. Identifythefollowingsequencesaslinear,exponential,orneither.
a. 23, 6, 11, 28, … b. 49, 36, 25, 16, … c. 5125, 1025, 205, 41,…d. 2, 6, 24, 120,… e. 0.12, 0.36, 1.08, 3.24, … f. 21, 24.5, 28, 31.5, …
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SetTopic:ComparinglinearandexponentialfunctionsusingmultiplerepresentationsDescribethedefiningcharacteristicsofeachtypeoffunctionbyfillinginthecellsofeachtableascompletelyaspossible.
6 5 6 5
6. Typeofgrowth
7. Whatkindofsequencecorrespondstoeachmodel?
8. Makeatableofvalues
x y
x y
9. Findtherateofchange
10.Grapheachequation.Comparethegraphs.Whatisthesame?Whatisdifferent?
11.Findthey‐interceptforeachfunction.
12.Writetherecursiveformofeachequation.
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Therewere2girlsinmygrandmother’sfamily,mymotherandmyaunt.Theyeachhad3daughters.Mytwosisters,3cousins,andIeachhad3daughters.Eachoneofour3daughtershavehad3daughters.13. Ifthepatternofeachgirlhaving3daughterscontinuesfor2moregenerations(mymomandauntbeingthe1st
generation,Iwanttoknowaboutthe5thgeneration),howmanydaughterswillbebornthen? 14.Writetheexplicitequationforthispattern. 15.Createatableandagraphdescribingthispattern.Isthissituationdiscreteorcontinuous?
GoTopic:Solvingmulti‐stepequationsSolvethefollowingequations.16.5 3 2 6
17. 6 12 10 2 3 6
18.13 12
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Recallthefollowingformulas: Simpleinterest Compoundinterest Usingtheformulasforsimpleinterestorcompoundinterest,calculatethefollowing.19.Thesimpleinterestonaloanof$12,000ataninterestrateof17%for6years.20.Thesimpleinterestonaloanof$20,000ataninterestrateof11%for5years.21.Theamountowedonaloanof$20,000,at11%,compoundedannuallyfor5years.22.Comparetheinterestpaidin#26totheinterestpaidin#27.Whichkindofinterestdoyouwantifyouhaveto
takeoutaloan?23.Theamountinyoursavingsaccountattheendof30years,ifyoubeganwith$2500andearnedaninterestrate
of7%compoundedannually.
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Module4ReviewHomeworkCharacteristicsofFunctions:Foreachofthefunctionsfindthefollowinginformation.1.
x‐intercept:____________________y‐intercept:____________________Rateofchangebetween 1and 2:____
2.
x‐intercept:_____________________y‐intercept:_____________________Rateofchangebetween 1and 2:____
3. Discussandcomparethefunctionsbyanalyzingtheratesofchange,intercepts,andwhereonefunctionis
greaterorlessthantheother.: rateofchange:
x‐intercept:y‐intercept:
: rateofchange:
x‐intercept:y‐intercept:
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LinearandExponentialModels:4. Writeanexplicitformulatomodelthenumberofdotsperday.Day1 Day2 Day3
Modelthefunctionusingthetableprovided: Useyourtabletocreateagraphofthefunction:
Day NumberofDots
5. Sherryhasahugedollcollectionof80dolls.Hermomtellsherthatsheneedstogetridof5peryeartogetit
downtoadecentnumberbeforeleavingforcollege.Writeanexplicitformulatomodelthenumberofdollsperyear.Ifsheis12,howmanywillshehaveleftwhensheis18?
6. YouboughtaToyotaCorollain2004for$12,500.Thecar’svaluedepreciatesby7%ayear.Howmuchisthe
carworthnow?Howmuchisitworthin2020?
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7. Thepopulationofalargecityincreasesbyarateof3%ayear.Whenthe2000censuswastaken,thepopulationwas1.2million.a. Writeamodelforthispopulationgrowth.
b. Whatshouldthepopulationbenow?Whatistheprojectedpopulationfor2020?
8. BankPlans:Supposeyouworkedmowinglawnsallsummerandearned$50.Twosavingsinstitutions,LinearLuckandExponentialExperimentwantyoutoletthem“holdontoyourmoney”forawhile.
LinearLuck: Thissavingsplanwilladd$100toyourbalanceforeverymonththatyouleaveyourmoneyin
theaccount.ExponentialExperiment: Thissavingsplanwillmultiplyyourbalanceby2everymonththatyouleaveyour
moneyintheiraccount.Analyzetheplans:Writetheexplicitfunctionforeachaccount,anddecidewhichaccountisbestatwhattimes.