Lesson 1 Definition Math1节定义数学Word文件下载.docx

Lesson 1 Definition Math1节定义数学Word文件下载.docx

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3.Describeproblemsituations（linear,quadraticandexponential）byusing

tabular,graphicalandsymbolicrepresentations.

10thgrade

2.Describeandcomparecharacteristicsofthefollowingfamiliesof

functions:

squareroot,cubic,absolutevalueandbasictrigonometric

functions;

e.g.,generalshape,possiblenumberofroots,domainand

range.

6.a）Studentswilllearnhowtographapiecewisefunctionbasedoneachpiecewithaspecificdomain.

b）Studentswillbeabletowriteapiecewisefunctionwithproperdomainbasedonagraph.

c）Studentswillbeabletousepiecewisefunctionstomodelandanswerquestionsgivenareal-lifesituation.

d）Studentswillbeabletographapiecewisefunctionontheircalculator.

7.Calculator

8.a）Definitionsaregivenfirst.Applicationsarelater.

b）Groupsofthree

9.Informal,Reflective,Formative,Self-guidedinvestigation

Lesson1:

IntroductiontoPiecewiseFunctions

Goal:

Tofamiliarizethestudentswiththebasicideasinvolvedwithpiecewisefunctions.

Definition–

Thefollowinggraphiscalledapiecewisefunctionbecausethefunctionisdefinedbytwoormoredifferentequationsappliedtodifferentpartsofthefunction’sdomain.

Noticethatitappearstobecomposedofthreesegments,eachadifferentlinearfunctionoveraparticulardomain.Pleasenoteafilledcircleincludesthatpoint,whileanopencircledoesnotincludethatpoint.

1.Whatisthedomainforthefirst（left）segment?

TheRange?

2.Whatisthedomainforthesecond（middle）segment?

3.Whatisthedomainforthethird（right）segment?

4.Howmanyequationsdoyouthinkyouwouldhavetousetowriteruleforthefollowingpiecewisefunction?

Noticethatitappearstobecomposedoftworays,eachadifferentlinearfunctionoveraparticulardomain.

5.Whatisthedomainforthefirst（left）ray?

6.Whatisthedomainforthesecond（right）ray?

Lesson2a:

TablesandGraphsofPiecewiseFunctions–

Givenapiecewisefunction,studentswilllearnhowtomakeatableofvaluesandsketchthegraphonthespecifieddomain.

Bothofthefollowingnotationscanbeusedtodescribeapiecewisefunctionoverthefunction’sdomain:

f（x）=

orf（x）=

7.

Completethefollowingtableofvaluesforthepiecewisefunctionoverthegivendomain.

x

f（x）

-5

-3

1

1.7

1.9

2

2.2

4

6

8.Graphtheorderedpairsfromyourtabletohandsketchthegraphofthepiecewisefunction.

9.Howmanypiecesdoesyourgraphhave?

Why?

10.Arethepiecesraysorsegments?

11.Arealltheendpointssoliddotsoropendotsorsomeofeach?

12.Wereallthesexvaluesnecessarytographthispiecewisefunction,orcouldthishavebeengraphedusinglesspoints?

13.Whichxvalueswere“critical”toincludeinordertosketchthegraphofthispiecewisefunction?

14.Canyougeneralizewhichx-valuesareessentialtoinputintoyourtabletomakeahandsketchedgraphofapiecewiselinearfunction?

15.Nowgraphthispiecewisefunction:

f（x）=

bycompletingatableofvaluesforthepiecewisefunctionoverthegivendomain.

16.

Whydidyouchoosethexvaluesyouplacedintothetable?

17.Graphtheorderedpairsfromyourtabletohandsketchthegraphofthepiecewisefunction.

18.Howmanypiecesdoesyourgraphhave?

19.Arethepiecesraysorsegments?

20.Arealltheendpointsfilledcirclesoropencirclesorsomeofeach?

21.Wasitnecessarytoevaluatebothpiecesofthefunctionforthex-value1?

Whyorwhynot?

22.Whichxvalueswere“critical”toincludeinordertographthispiecewisefunction?

Explain.

Lesson2b:

UsingHandheldGraphingTechnologytoGraphPiecewiseFunctions

Inthislesson,wewilllearnhowtographapiecewisefunctionontheTI-nspire.

Graphthepiecewisefunction

23.Enteryourfunctiononthecalculator.

a.Turnthecalculatoron.

b.PressHome;

2:

AddGraphs&

Geometry

c.Withthecurseronthefunctionbar,pressandholdctrlwhilepressingthemultiplicationkey（x）.

Youwillseethefollowingonyourscreen:

d.Highlightthepiecewisefunction（highlighted）.

e.Pressenter.

f.Typethenumberofpiecestoyourpiecewisefunction.

g.Pressenter.

h.Usethenavigationarrowstomovethecursortothecellstotypetheequationsanddomains.

i.Pressenter.Yourgraphwilllooklikethis:

j.Ifyoucannotseepartsofthegraph,youcanremovethefunctionbarbypressingandholdingctrlwhilepressinggtotogglebackandforthbetweenshowingthefunctionbarandnotshowingthefunctionbar.Youmayalsochangethewindow.

24.Whataretheadvantages/disadvantagestographingapiecewisefunctionusinghandheldgraphingtechnology?

Lesson3:

Writingpiecewisefunctionsgivenagraph.

Studentswilllearntowritearuleforeachpieceofthegraphspecifyingtheproperdomain.

25.

Let’srevisitthefirstpiecewisegraphfromLesson1.Canyouidentifytheequationsofthelinesthatcontaineachsegment?

a.Leftsegment（blue）equation=

b.Middle（red）equation=

c.Right（green）equation=

26.Next,listthedomainofeachsegment.

a.Leftsegment（blue）domain=

b.Middle（red）domain=

c.Right（green）domain=

27.Now,putthedomaintogetherwiththeequationstowritethepiecewisefunctionforthegraph.

Lesson4:

ModelingwithPiecewiseFunctions.

Studentswillbeabletomodelareallifesituationwithapiecewisefunction.

28.Taxes:

Giventhetableattheright,writethepiecewisefunctionthatdescribesthefederaltaxpaidbasedonthetaxableincome（x）forapersonwithfilingstatussingle.Hint,thispiecewisefunctionshouldhave6pieces.

*Thischartwasobtainedfrom:

Thewordbetweenisambiguous,butthetaxtableisnot.Thisshouldread“AtLeast_____butlessthan_____”notjusttheword“between”.

29.Graphthispiecewisefunctionusingyourhandheldgraphingtechnology.

30.Youcompletedyourtaxformsandfoundthatyourtaxableincomewas$80,450.Yourtaxadvisorisadamantthatyoudepositatleast$1650intoanIRAor403btoreduceyourtaxableincome.Youthinksheisjusttryingtomakemoneyoffofyouliketypicalfinancial“advisors”.Shesaystheamountyouwillbesavingintaxesisalmostdoubleyourdeposit.Isshecorrect?

Howmuchwillshesaveyouintaxes?

Explainandshowyourwork.

31.TextingPlans:

TextingfromT-mobilecosts$.15pertextwithnoplan.Inaddition,theyofferthreeothertextingplans,showntotheright,thatincludeacertainnumberoftextswithadditionaltextsovercosting$.15pertext.

Decidewhichplansarelinearpiecewisefunctionsandwhicharenotpiecewise（circleyouranswer）.

a.Noplan–Piecewise?

YesorNo

b.400textplan–Piecewise?

c.1000textplan–Piecewise?

d.Unlimitedtextplan–Piecewise?

32.IfyousaidthatNoplanandUnlimitedplanwereNOTpiecewise...youarewise.Now,writethefunctionrulesforeachwherexisthenumberoftextsandf（x）isthetotalmonthlycost.

a.Noplan–

b.400textplan–

c.1000textplan–

d.Unlimitedtextplan–

33.Mydaughterusesapproximately1200textspermonth.Howmuchwouldthiscostmeundereachplan?

a.Noplan

b.400textplan

c.1000textplan

d.Unlimitedtextplan

34.Iutilizeabout90textspermonth.WhichplanshouldIpurchaseformyphone?

35.Givetheintervalnumberoftextsthatwouldmakeeachoftheseplansthebestonetopurchase（thisinformationwouldbegoodtogivetooursalespeoplewhentheyareadvisingcustomersonwhichplantopurchase）.

36.Extension（s）:

Researchtofindapiecewisefunctionthatwillfitoneofthefollowingreallifeapplications:

a.HeartRatebefore/during/afterexercise.

b.Repairservicecharge:

（$150forhousecallupto1hour,then$25per15minutesthereafter）.

c.AutoInsuranceratesbasedonage（between16-23yrold,23-35yrolds,ect.）

d.other

Lesson5:

OtherPiecewiseFunctions（StepFunctions）

Understandingtheproperties&

characteristicsofpiecewiseconstantfunctions（orstepfunctions）.

Astepfunctionisapiecewiseconstantfunction.Inotherwords,eachpieceisafunctionwhosevaluesdonotvaryandarethusconstant.Thisissometimescalledastaircasefunction.

37.Sketchthegraphofthefunctionbelow.Whatvaluesofxwillmakethisfunctiontrue?

Inotherwords,whereisthisfunctiondefined,orwhatisitsdomain?

Whyistherangeno