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