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©2013MathematicsVisionProject|MVPInpartnershipwiththeUtahStateOfficeofEducation
LicensedundertheCreativeCommonsAttribution‐NonCommercial‐ShareAlike3.0Unportedlicense.
Math1CPtoMath2HonorsSummerBridge
QuadraticFunctionsModule9H
Adaptedfrom
TheMathematicsVisionProject:ScottHendrickson,JoleighHoney,BarbaraKuehl,
TravisLemon,JanetSutoriuswww.mathematicsvisionproject.org
InpartnershipwiththeUtahStateOfficeofEducation
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SDUHSDMath1CPtoMath2HSummerBridge
Module9HOverview
Belowarethestandards,concepts,andvocabularyfromtheCOMPLETEMODULEinIntegratedMath1Honors
PrerequisiteConcepts&Skills:
Graphingfunctions Identifyingdomain,range,andintervalsofincreaseanddecrease Usingpatternstofindrecursiveandexplicitrules Usingdifferenceswithinatableofvaluestodeterminethetypeoffunctionthatcanbeusedtorepresenta
dataset Comparingfeaturesoffunctions Comparinglinearandexponentialfunctions Findingmultiplerepresentationsforacontext
SummaryoftheConcepts&SkillsinModule9H:
Usepatternstodeterminethetypeoffunction Usemultiplerepresentationstocomparelinear,quadratic,andexponentialfunctions Determinedomain,range,maximum,andminimumvaluesforquadratics Determineaverageratesofchange
ContentStandardsandStandardsofMathematicalPracticeCovered:
ContentStandards:F.BF.1,A.SSE.1,A.CED.2,F‐BF,F‐LE,F.LE.2,F.LE.3 StandardsofMathematicalPractice:
1. Makesenseofproblems&persevereinsolvingthem2. Attendtoprecision3. Reasonabstractly&quantitatively4. Constructviablearguments&critiquethereasoningofothers5. Modelwithmathematics6. Useappropriatetoolsstrategically7. Lookfor&makeuseofstructure8. Lookfor&expressregularityinrepeatedreasoning
Module9HVocabulary:
Quadraticfunctions Maximumvalue Minimumvalue Averagerateofchange Binomial Perimeter Area Domain Range First&seconddifferences Parabola Explicitequation Recursiveequation Increasing Decreasing Intervalnotation Continuous Discrete Acceleration Initialspeed Instantaneousspeed Averagespeed Atrest Exponentialfunction
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SDUHSDMath1CPtoMath2HSummerBridge
ConceptsUsedintheNextModule: Determineifacontextrepresentsalinear,exponential,orquadraticfunction Determinethemaximumorminimumvalue Findmultiplerepresentationsfromacontext Determinethetransformationsbeingappliedtoafunction Identifyfeaturesoffunctions(domain,range,intervalsofincrease/decrease)
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SDUHSDMath1CPtoMath2HSummerBridge
Module9H–QuadraticFunctions9.1HSolidificationofquadraticfunctionsbeginsasquadraticpatternsareexaminedinmultiplerepresentationsandcontrastedwithlinearrelationshipsandFocusspecificallyonthenatureofchangebetweenvaluesinaquadraticbeinglinearandfocusonmaximum/minimumpointaswellasdomainandrangeforquadratics(F.BF.1,A.SSE.1,A.CED.2,F.LE,F.IF.9)WarmUp:SomethingtoTalkAbout–ADevelopUnderstandingTaskClassroomTask:RabbitRun–ASolidifyUnderstandingTaskReady,Set,GoHomework:QuadraticFunctions9.1H9.2HFocusonmaximum/minimumpointaswellasdomainandrangeforquadratics(F.BF.1,A.SSE.1,A.CED.2,F.LE)ClassroomTask:Scott’sMachoMarch–ASolidifyUnderstandingTaskReady,Set,GoHomework:QuadraticFunctions9.2H9.3HExaminingquadraticfunctionsonvarioussizedintervalstodetermineaverageratesofchange(F.BF.1,A.SSE.1,A.CED.2)ClassroomTask:LookoutBelow–ASolidifyUnderstandingTaskReady,Set,GoHomework:QuadraticFunctions9.3H
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SDUHSDMath1CPtoMath2HSummerBridge
© 2013
www.flickr.com/photos/robwallace/ 9.1HWarmUp–SomethingtoTalkAbout
ADevelopUnderstandingTaskCellphonesoftenindicatethestrengthofthephone’ssignalwithaseriesofbars.Thelogobelowshowshowthismightlookforvariouslevelsofservice.
Figure1 Figure2 Figure3 Figure4
1. Assumingthepatterncontinues,drawthenextfigureinthesequence.2. HowmanyblockswillbeintheFigure10logo?3. Examinethesequenceoffiguresandfindaruleorformulaforthenumberoftilesinanyfigurenumber.
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SDUHSDMath1CPtoMath2HSummerBridge
©2013www.flickr.com
/photos/bishi/2314705514
9.1HRabbitRunASolidifyUnderstandingTaskMishahasanewrabbitthatshenamed“Wascal”.ShewantstobuildWascalapensothattherabbithasspacetomovearoundsafely.Mishahaspurchaseda72footrolloffencingtobuildarectangularpen.1. IfMishausesthewholerolloffencing,whataresomeofthepossibledimensionsofthepen?2. IfMishawantsapenwiththelargestpossiblearea,whatdimensionsshouldsheuseforthepen?Justifyyour
answer.3. Writeamodelfortheareaoftherectangularpenintermsofthelengthofoneside.Includebothanequation
andagraph.
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SDUHSDMath1CPtoMath2HSummerBridge
4. Whatkindoffunctionisthis?Why?5. HowdoesthisfunctioncomparetothesecondtypeofblockIlogosinIRule?
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SDUHSDMath1CPtoMath2HSummerBridge
Name: QuadraticFunctions 9.1HReady,Set,GoReadyTopic:AddingandmultiplyingbinomialsSimplifythefollowingexpressions.Forthepartbproblems,multiplyusingthegivenareamodel.1a. 6 1 10 b. 6 1 10
2a. 8 3 3 4 b. 8 3 3 4
3a. 5 2 7 13 b. 5 2 7 13
4a. 12 3 4 3 b. 12 3 4 3
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SDUHSDMath1CPtoMath2HSummerBridge
5. 5 5
6. Comparethestyleyouranswersin#1–4(parta)toyouranswersin#1–4(partb).Lookforapatterninthe
answers.Howaretheydifferent?7. Theanswerto#5isadifferent“shape”thantheotherpartbanswers,eventhoughyouwerestillmultiplying.
Explainhowitisdifferentfromtheotherproducts.Trytoexplainwhyitisdifferent.Find2examplesofmultiplyingbinomialsthatwouldhaveasimilarsolutionas#5.
8. Tryaddingthetwobinomialsin#5. 5 5 Doesthisanswerlook
differentthanthoseinparta?Explain.
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SDUHSDMath1CPtoMath2HSummerBridge
SetTopic:ComparingareaandperimeterCalculatetheperimeterandtheareaofthefiguresbelow.Youranswerswillcontainavariable.9.
a. Perimeter: b. Area:
10.
a. Perimeter: b. Area:
11.
a. Perimeter: b. Area:
12.
a. Perimeter: b. Area:
13.Comparetheperimetertotheareaineachofproblems9‐12.Inwhatwayarethenumbersandunitsinthe
perimetersandareasdifferent?
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Topic:Recognizinglinear,exponential,andquadraticequations.14. Ineachsetof3functions,onewillbelinearandonewillbeexponential.Oneofthethreewillbeanew
categoryoffunction.Statethetypeoffunctionrepresented(linear,exponential,ornewfunction).Listthecharacteristicsineachtableand/orgraphthathelpedyoutoidentifythelinearandtheexponentialfunctions.Forthegraph,placeyouraxessothatyoucanshowall5points.Identifyyourscale.Findanexplicitandrecursiveequationforeach.
a.
2 17
1 12
0 7
1 2
2 3
Typeandcharacteristics?Explicitequation: Recursiveequation:
b.
2
1
0 1
1 5
2 25
Typeandcharacteristics?Explicitequation: Recursiveequation:
c.
2 9
1 6
0 5
1 6
2 9
Typeandcharacteristics?Explicitequation: Recursiveequation:
GoTopic:GreatestCommonFactor(GCF)FindtheGCFofthegivennumbers.16.15 and25 17. 12 and32 18. 17 and51
⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
19.6 ,18 , 12 20. 49 and36 21. 11 ,33 ,and3
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© 2013 ww.flickr.com/photos/perspective//ElvertBarnes
9.2HScott’sMachoMarchASolidifyUnderstandingTaskPart1:Afterlookinginthemirrorandfeelingflabby,Scottdecidedthathereallyneedstogetinshape.Hejoinedagymandaddedpush‐upstohisdailyexerciseroutine.Hestartedkeepingtrackofthenumberofpush‐upshecompletedeachdayinthebargraphbelow,withdayoneshowinghecompletedthreepush‐ups.Afterfourdays,Scottwascertainhecancontinuethispatternofincreasingthenumberofpush‐upsforatleastafewmonths.
1. Modelthenumberofpush‐upsScottwillcompleteonanygivenday.Includebothexplicitandrecursive
equations.Scott’sgymissponsoringa“MachoMarch”promotionduringthemonthofMarch.Thegoalof“MachoMarch”istoraisemoneyforcharitybydoingpush‐ups.Scotthasdecidedtoparticipateandhassponsorsthatwilldonatemoneytothecharityifhecandoatotalofatleast500push‐ups.Asabonus,theywilldonateanadditional$10forevery100push‐upshecandobeyondhisgoalof500push‐ups.2. Estimatethetotalnumberofpush‐upsthatScottwilldoinamonthifhecontinuestoincreasethenumberof
push‐upshedoeseachdayinthepatternshownabove.3. Howmanytotalpush‐upswillScotthavedoneafteronefullweek?
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4. Modelthetotalnumberofpush‐upsthatScotthascompletedonanygivendayduring“MachoMarch”.Includebothrecursiveandexplicitequations.
5. WillScottmeethisgoalandearnthedonationforthecharity?Willhegetabonus?Ifso,howmuch?Explain.Part2:MultiplyingBinomials&UsingPatternstoDetermineRecursiveEquations:Thefollowingproblemsarefactorizationsofnumericalexpressionscalledquadratics.Giventhefactors,multiplythebinomialsusingtheareamodeltofindthequadraticexpression.Simplifytheexpressionsbycombiningliketermsandwritingthetermsindescendingorderofpowers(example: .Thendistributethecoefficient,ifoneexists.1. 5 7 2. 3. 2 9 4
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SDUHSDMath1CPtoMath2HSummerBridge
4. 3 1 4 5. 2 3 5 1 6. 2 5 7 3 1
Usefirstandseconddifferencestoidentifythepatterninthetablesaslinear,quadratic,orneither.Writetherecursiveequationforthepatternsthatarelinearorquadratic.7.
3 232 171 110 51 12 3 13
a. Pattern:b. Recursiveequation:
8.
3 42 01 20 21 02 43 10
a. Pattern:b. Recursiveequation:
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SDUHSDMath1CPtoMath2HSummerBridge
Name: QuadraticFunctions 9.2HReady,Set,Go!ReadyTopic:ApplyingtheslopeformulaCalculatetheslopeofthelinebetweenthegivenpoints.Useyouranswertoindicatewhichlineisthesteepest.1. 3, 7 5, 17 2. 12, 37 4, 3 3. 11, 24 21, 40 4. 55, 75 15, 40 Steepestline: SetTopic:InvestigatingperimetersandareasAdamandhisbrotherareresponsibleforfeedingtheirhorses.Inthespringandsummer,thehorsesgrazeinanunfencedpasture.Thebrothershaveerectedaportablefencetocorralthehorsesinagrazingarea.Eachdaythehorseseatallofthegrassinsidethefence.Thentheboysmoveittoanewareawherethegrassislongandgreen.Theportablefenceconsistsof16separatepiecesoffencingeach10feetlong.Thebrothershavealwaysarrangedthefenceinalongrectanglewithonelengthoffenceoneachendand7piecesoneachsidemakingthegrazingarea700sq.ft.Adamhaslearnedinhismathclassthatarectanglecanhavethesameperimeterbutdifferentareas.Heisbeginningtowonderifhecanmakehisdailyjobeasierbyrearrangingthefencesothatthehorseshaveabiggergrazingarea.Hebeginsbymakingatableofvalues.Helistsallofthepossibleareasofarectanglewithaperimeterof160ft.,whilekeepinginmindthatheisrestrictedbythelengthsofhisfencingunits.Herealizesthatarectanglethatisorientedhorizontallyinthepasturewillcoveradifferentsectionofgrassthanonethatisorientedvertically.Soheisconsideringthetworectanglesasdifferentinhistable.Usethisinformationtoanswerquestions5–9onthenextpage.
5. FillinAdam’stablewithallofthearrangementsforthefence. Lengthin
“fencing”unitsWidthin
“fencing”unitsLengthinft. Widthinft. Perimeter Area
1unit 7units 10ft 70ft 160ft 700 a. 2units 160ftb. 3units 160ftc. 4units 160ftd. 5units 160fte. 6units 160ftf. 7units 160ft
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6. DiscussAdam’sfindings.Explainhowyouwouldrearrangethesectionsoftheporta‐fencesothatAdamwillbeabletodolesswork.Explainwhat“lesswork”meansforAdamandhisbrother.
7. MakeagraphofAdam’s investigation. Let lengthbe the
independentvariableandareabethedependentvariable.Labelandscaletheaxes.
8. Whatistheshapeofyourgraph? 9. Explainwhatmakesthisfunctionquadratic.
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GoTopic:Comparinglinearandexponentialratesofchange.Indicatewhichfunctionischangingfaster.10.
11.
12.
13.
14.
15.
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SDUHSDMath1CPtoMath2HSummerBridge
16a. Examinethegraphattheleftfrom0to1.Whichgraphdoyouthinkisgrowingfaster?
b. Nowlookatthegraphfrom2to3.Whichgraph
isgrowingfasterinthisinterval?
Topic:Linear,Exponential,&QuadraticSequencesDetermineifeachsequenceislinear,exponential,orquadratic.Explainhowyoucandeterminethetypeofpatternbaseduponthetable.Thenfindtherecursiveandexplicitequationsforeachpattern.17.
1 32 93 274 815 243
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
18.
1 62 43 24 05 2
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
19.1 42 73 124 195 28
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
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20. 1 52 83 114 145 17
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
21.
1 22 83 184 325 50
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
22.1 12 43 164 645 256
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
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© 2013
www.flickr.com/photos/melissawall 9.3HLookOutBelow!
ASolidifyUnderstandingTaskWhathappenswhenyoudropaball?Itfallstotheground.Thatquestionsoundsassillyas“Whydidthechickencrosstheroad?”Seriously,ittookscientistsuntilthesixteenthandseventeenthcenturiestofullyunderstandthephysicsandmathematicsoffallingbodies.Wenowknowthatgravityactsontheobjectthatisfallinginawaythatcausesittoaccelerateasitfalls.Thatmeansthatifthereisnoairresistance,itfallsfasterandfaster,coveringmoredistanceineachsecondasitfalls.Ifyoucouldslowtheprocessdownsothatyoucouldseethepositionoftheobjectasitfalls,itwouldlooksomethinglikethepicturebelow.
Tobemoreprecise,objectsfallataconstantrateofaccelerationonearthofabout32feetpersecondpersecond.Thesimplestcaseoccurswhentheobjectstartsfromrest,thatis,whenitsspeediszerowhen 0.Inthiscase,theobject’sinstantaneousspeedafter1secondis32feetpersecond;after2seconds,itsinstantaneousspeedis2 32 64feetpersecond;andsoon.Otherplanetsandmoonseachhaveadifferentrateofacceleration,butthebasicprincipalremainsthesame.Iftheaccelerationonaparticularplanetisg,thentheobject’sinstantaneousspeedafter1secondisgunitspersecond;after2seconds,itsinstantaneousspeedis2 unitspersecond;andsoon.Inthistask,wewillexplorethemathematicsoffallingobjects,butbeforewestartthinkingaboutfallingobjectsweneedtobeginwithalittleworkontherelationshipbetweenspeed,time,anddistance.
Part1: AveragespeedanddistancetraveledConsideracarthatistravelingatasteadyrateof30feetpersecond.Attime 0,thedriverofthecarstartstoincreasehisspeed(accelerate)inordertopassaslowmovingvehicle.Thespeedincreasesataconstantratesothat20secondslater,thecaristravelingatarateof40feetpersecond.1. Graphthecar’sspeedasafunctionoftimeforthis20‐secondtimeinterval.
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2. Calculatetheaveragespeedofthecarforthis20‐secondtimeinterval.3. Findthetotaldistancethecartravelsduringthis20‐secondtimeinterval.4. Explainhowtouseareatofindthetotaldistancethecartravelsduringthis20‐secondinterval.Thisproblemillustratesanimportantprinciple:Ifanobjectistravelingwithconstantacceleration,thenitsaveragespeedoveranytimeintervalistheaverageofitsbeginningspeedanditsfinalspeedduringthattimeinterval.Let’sapplythisideatoapennythatisdropped(initialspeedis0when 0)fromthetopoftheEmpireStateBuilding.Besuretoreferencethesecondparagraphatthestartofthetaskforinformationabouttheconstantrateofacceleration.5. Whatwillitsspeedbeafter1second?6. Graphthepenny’sspeedasafunctionoftimein1secondintervals.
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SDUHSDMath1CPtoMath2HSummerBridge
7. Whatistheaveragespeedofthepennyinthe1‐secondinterval?8. Whatisthetotaldistancethatthepennyfellinthe1‐secondinterval?9. FindtheheightoftheEmpireStateBuilding.Howlongwillittakeforthepennytohittheground?10.CompletethetablebelowforthespeedsanddistancestraveledofapennydroppedfromthetopoftheEmpire
StateBuilding.
Time InstantaneousSpeed AverageSpeed DistanceTraveled
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SDUHSDMath1CPtoMath2HSummerBridge
Part2: Falling,Falling,FallingWhentheastronautswenttothemoon,theyperformedGalileo’sexperimenttotesttheideathatanytwoobjects,nomattertheirmass,willfallatthesamerateifthereisnoairresistance.Becausethemoondoesn’thaveairresistance,wearegoingtopretendlikewe’retheastronautsdroppingmoonrocksandthinkingaboutwhathappens.Onthesurfaceofthemoontheconstantaccelerationincreasesthespeedofafallingobjectby6feetpersecondpersecond.Thatis,ifanobjectisdroppednearthesurfaceofthemoon(e.g.,itsinitialspeedis0when
0),thentheobject’sinstantaneousspeedafter1secondis6feetpersecond,after2seconds,itsinstantaneousspeedis12feetpersecond,andsoon.11.Usingthisinformation,completethetablebelowforthespeedsanddistancestraveledofanobjectthatis
droppedfromaheightof200feetabovethesurfaceofthemoonasafunctionoftheelapsedtime(inseconds)sinceitwasdropped.
Time InstantaneousSpeed AverageSpeed DistanceTraveled
12.Howdidyoufindthedistancethattheobjecthastraveled?13.Writeanequationforthedistancetheobjecthasfallenasafunctionofelapsedtimet.14.Approximatelyhowlongwillittakefortheobjecttohitthesurfaceofthemoon?15.Writeanequationfortheheightoftheobjectabovethesurfaceofthemoonasafunctionofelapsedtimet.
Reminder:theobjectisdroppedfromaheightof200feetabovethesurfaceofthemoon.
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SDUHSDMath1CPtoMath2HSummerBridge
16.Supposetheobjectwasnotdropped,butwasthrowndownwardfromaheightof250feetabovethesurfaceofthemoonwithaninitialspeedof10feetpersecond.Rewriteyourequationfortheheightoftheobjectabovethesurfaceofthemoonasafunctionofelapsedtimettotakeintoaccountthisinitialspeed.
17.Howisyourworkonthesefallingobjectsproblemsrelatedtoyourworkwiththerabbitrunsintheprevious
task?18.Whyarethe“distancefallen”and“heightabovetheground”functionsquadratic?
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SDUHSDMath1CPtoMath2HSummerBridge
Name: QuadraticFunctions 9.3HReady,Set,Go!ReadyTopic: EvaluatingexponentialfunctionsFindtheindicatedvalueofthefunctionforeachvalueofx. , , , , ,
1. 3 2. 3.
SetTheSearsTowerinChicagois1730feettall.Ifapennywereletgofromthetopofthetower,thepositionabovethegrounds(t)ofthepennyatanygiventimetwouldbe 16 1730.4. Fillinthemissingpositionsinthechartbelow.Thenaddtogetthedistancefallen.
Distancefromgrounda. 0sec
b. 1sec
c. 2sec
d. 3sec
e. 4sec
f. 5sec
g. 6sec
h. 7sec
i. 8sec
j. 9sec
k. 10sec
5. Howfarabove thegroundisthepennywhen7secondshavepassed?
6. Howfarhasitfallenwhen7secondshavepassed? 7. Hasthepennyhitthegroundat10seconds?Justify
youranswer.
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SDUHSDMath1CPtoMath2HSummerBridge
Theaveragerateofchangeofanobjectisgivenbytheformula ,whereristherateofchange,disthedistancetraveled,andtisthetimeittooktotravelthegivendistance.Weoftenusesomeformofthisformulawhenwearetryingtocalculatehowlongatripmaytake.8. Ifourdestinationis225milesawayandwecanaverage75mph,thenweshouldarrivein3hours.
3 Inthiscaseyouwouldberearrangingtheformulasothat .However,ifyourmother
findsoutthatthetriponlytook2½hours,shewillbeupset.Usetherateformulatoexplainwhy.9. Howistheslopeformula liketheformulaforrate?
Forthefollowingquestions,referbacktotheSear’sTowerproblem(questions6–9).10. Findtheaveragerateofchangeforthepennyontheinterval[0,1]seconds.11. Findtheaveragerateofchangeforthepennyontheinterval[6,7]seconds.12. Explainwhythepenny’saveragespeedisdifferentfrom0to1secondthanbetweenthe6thand7thseconds.13. Whatistheaveragespeedofthepennyfrom[0,10]seconds?14. Whatistheaveragespeedofthepennyfrom[9,10]seconds?15. Findthefirstdifferencesonthetablewhereyourecordedthepositionofthepennyateachsecond.Whatdo
thesedifferencestellyou?16. Takethedifferenceofthefirstdifferences.Thisiscalledthe2nddifference.Didyouranswersurpriseyou?
Whatdoyouthinkthismeans?
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SDUHSDMath1CPtoMath2HSummerBridge
GoTopic: Evaluatingfunctions17.Find 9 giventhat 10. 18.Find 3 giventhat 2 4.
19.Find 1 giventhat 5 3 9. 20.Find giventhat .
22. Find 2 giventhat 7 11 .Topic:Linear,Exponential,orQuadraticSequencesDetermineifeachsequenceislinear,exponential,orquadratic.Explainhowyoucandeterminethetypeofpatternbaseduponthetable.Thenfindtherecursiveandexplicitequationsforeachpattern.23.
1 02 33 84 155 24
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
24.
1 62 123 244 485 96
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
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SDUHSDMath1CPtoMath2HSummerBridge
25. 1 32 123 274 485 75
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation:
26.
1 32 73 114 155 19
TypeofPattern:Howdidyoudeterminethetypeofpatternbasedonthetable?RecursiveEquation:ExplicitEquation: