Linear Functions And Geometry Study...
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NAME:___________________________ Math __________, Period ____________ Mr. Rogove Date:__________ Linear Functions and Geometry Study Guide 1 Linear Functions And Geometry Study Guide FUNCTIONS Functions are rules that assign each input exactly one output. We have described functions in four different ways: Verbally/Written Description I have $500 in my bank account now, and deposit $75 per week. Equation = 75 + 500 = 75 + 500 Table Weeks (x) Money (y) 0 500 1 575 2 650 3 725 5 875 Graph A linear function is a special kind of function where the function rule is specifically a linear equation in the form = + . Characteristics of Linear Functions: • The rate of change of a linear function stays constant. • When the slope of a linear function is negative, the function is decreasing. When the slope of linear function is positive, the function is increasing. • Linear functions graph as straight lines. • Linear functions describe proportional relationships. REMINDER: Some functions don’t involve numbers at all. Example: Input is car model (i.e. Accord), and output is car manufacturer (i.e Honda). When we talk about functions, we say that the output is a function of the input. Example: The money in my bank account is a function of the number of weeks I’ve saved.
Transcript of Linear Functions And Geometry Study...
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide1
Linear Functions And Geometry
Study Guide
FUNCTIONS Functionsarerulesthatassigneachinputexactlyoneoutput.Wehavedescribedfunctionsinfourdifferentways:Verbally/WrittenDescriptionIhave$500inmybankaccountnow,anddeposit$75perweek.
Equation
𝑦 = 75𝑥 + 500 𝐨𝐫 𝑓 𝑥 = 75𝑥 + 500
Table
Weeks(x)
Money(y)
0 5001 5752 6503 7255 875
Graph
Alinearfunctionisaspecialkindoffunctionwherethefunctionruleisspecificallyalinearequationintheform𝑦 = 𝑚𝑥 + 𝑏.CharacteristicsofLinearFunctions:
• Therateofchangeofalinearfunctionstaysconstant.
• Whentheslopeofalinearfunctionisnegative,thefunctionisdecreasing.
Whentheslopeoflinearfunctionispositive,thefunctionisincreasing.
• Linearfunctionsgraphasstraightlines.
• Linearfunctionsdescribeproportionalrelationships.
REMINDER:Somefunctionsdon’tinvolvenumbersatall.Example:Inputiscarmodel(i.e.Accord),andoutputiscarmanufacturer(i.eHonda).Whenwetalkaboutfunctions,wesaythattheoutputisafunctionoftheinput.Example:ThemoneyinmybankaccountisafunctionofthenumberofweeksI’vesaved.
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide2
WHAT DOES A FUNCTION LOOK LIKE?
**Inatableofvalues,thereareNOx-values(inputvalues)repeated**Onagraph,itmeansthataverticallinewillonlypassthroughthefunctionONCE.Discretev.ContinuousFunctionsAdiscretefunctionisafunctionthatonlyhasaspecificsetofinputs(suchasintegers).Example:Aboxofcookiescosts$3.00.Youcan’tbuyafractionalboxofcookies.Acontinuousfunctionisafunctionthatcouldincluderationalnumberinputvalues.Example:Apoundofgrapesis$3.00.Youcanbuy3.5poundsofgrapes.
GEOMETRY (Volume of 3D shapes) RemembertheseformulasCylinder
𝑉 = 𝜋𝑟! ℎ
𝑉 = 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡
Cone
𝑉 =13𝜋𝑟! ℎ
𝑉 = !
!𝑎𝑟𝑒𝑎 𝑜𝑓 𝑏𝑎𝑠𝑒 ×
ℎ𝑒𝑖𝑔ℎ𝑡
Sphere
𝑉 =43𝜋𝑟!
Height
Areaofbase(𝜋𝑟^2)
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide3
PROBLEM SET
Pleasecompleteallproblemsandsubmitasyoutakethisassessment1.Rachelishiringaplumbertore-pipeherkitchenandbathroom.Oneplumbingcompany,LeakProofPlumbingCompany,ischarginga$500formaterialsplus$120perhour.Anothercompany,DripFreeSince2003Inc.,doesnotchargeformaterial,buttheirhourlyrateis$165.Athirdcompany,CleanYerPipes.com,submitsabidtodotheworkfor$2300nomatterhowlongittakes.Nocompanyprovidesanestimateofhowmanyhoursitwilltake.a.Writelinearequationsthatmodelthechargesforeachofthethreecompanies.Leakproof:𝑦 = 500+ 120𝑥DripFreeSince2003:𝑦 = 165𝑥CleanYerPipes.com:𝑦 = 2300b.Ifittakes8hoursfortheworktobecompleted,whichcompanywillprovidethebestvalue?Leakproof:𝑦 = 500+ 120 8 = 500+ 960 = $1460DripFreeSince2003:𝑦 = 165 8 = $1320CleanYerPipes.com:$2300DripFreeisthecheapest…c.ForwhattimeintervalisLeakProofPlumbingCompanythecheapestalternative?DripFreeischeapestfrom0hourstoapproximately11hoursand7minutes.LeakProofischeapestifyourjobwilltakelongerthan11hours,7minutes,butlessthan15hours.d.AtwhatpointdoesitbecomemosteconomicaltohireCleanYerPipes.com?CleanYerPipes.comischeapestifthejobwilltakelongerthan15hours.
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide4
2.LIGHTBULBS.Incandescentlightbulbshavebeenusedformanydecadestoprovidelightinhomesandbusinesses.Theycostabout$1.50each,andtheylastabout1,000hours.NewerCFLlightbulbshavebeenintroducedthataremoreexpensive—costing$10.00each,buttheylastabout8,000hours.a.AfterhowmanyhourswillitpaytogettheCFLbulbs?(thinkaboutusinglinearfunctionstohelpyouanswer).Ifyouusetheincandescentbulbsfor6,000hours,you’llspend$9.00(becauseyou’llneed6bulbsat$1.50each).Ifyouusetheincandescentbulbsfor6,001hours,you’llneeda7thbulbandthetotalcostforthebulbswillriseto$10.50,makingitmoreexpensivethanthe$10.00CFLbulb.b.Tomakethingsmoreinteresting,whatifItoldyouthattheincandescentbulbsaremuchmoreinefficient,andtheyadd$.01125perhourtoyourelectricbill.CFLbulbswillonlyadd$0.00345toyourelectricbillforeachhourtheyareinuse.Howwillthisaffectyourdeterminationofwhenit’smoreeconomicaltobuyincandescentbulbsv.CFLbulbs?Explainyouranswerusingwordsandequations.Now,wecanuseequations…Theequationfortheincandescentbulbisasfollows:𝐶 = 0.01125ℎ + 1.50whereCisthecostindollars,andhisthenumberofhoursthebulbsareinuse(note,thatweneedtoaddanother$1.50forevery1000hoursused—thiswillgraphlikeapiecewisefunction.)TheequationiftheCFLbulb:𝐶 = 0.00345ℎ + 10.00againwhereCisthecostindollarsandhisthenumberofhoursthebulbsareinuse.(noteweneedtoaddanother$10.000forevery8000hoursused—thiswillalsographasapiecewisefunction.Ifwesettheaboveequationsequaltooneanotherwefindthatatabout1,090hours,thecostwouldbeequal,butwewouldhavehadtobuyasecondincandescentbulb,solet’sassumewebuy2incandescentbulbs,makingthefirstequation:𝐶 = 0.01125ℎ + 3.00.Usingthisnumber,wewouldfindthatthebreakevenpointwouldbeabout897hours…so,armedwiththisinformation,wecanfigureoutthatafter1,001hours,you’dhavetobuyanotherincandescentbulbandatTHATpoint,you’dneedanewbulbandthatwouldmaketheincandescentbulbsmoreexpensive.Anotherwaytolookatitasfollows:ifyouusedeachfor1000hours:Incandescentwouldcost0.01125 × 1000+ 1.50 = $12.75.theCFLwouldbe0.00345×1000+ 10.00 = $13.40.Atthatpoint,you’dhavetospendanother$1.50ontheincandescentbulb.
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide5
c.EvenmoreexpensivethanCFLbulbsarenewerLEDlightbulbs.LEDlightbulbscostabout$20eachbuttheylastforanamazing25,000hours,andtheycostalmostnothingtouseperhour(thecostis$0.00135)AfterhowmanyhoursofusewilltheLEDbulbsbethemosteconomical?Justifyyouranswer.WecanusethesamelogictoseewhentheCFLbulbsbecomemoreexpensive….buttheshortanswer(withoutusinganyequations)isthatassoonasyoubuythesecondCFLbulb(after8,000hours)you’llbespendingmoresimplybecauseatthatpoint,thecostofthebulbswillbeequal,andthecostofkeepingthelightsonismuchcheaperfortheLEDbulbs…soafter8001hours,theCFLbulbswillbemoreexpensive.d.Ifthetypicalhouseholdhastheirlightson4hoursaday,howmanyYEARSwouldyouexpecteachtypeofbulbtolast?Incandescent:4hoursadayfor1000hoursmeansthebulbwouldlast250days.
250365 = 0.685 𝑦𝑒𝑎𝑟𝑠
CFL:4hoursadayfor8000hoursmeanthebulbwouldlast2000days.2000365 = 5.48 𝑦𝑒𝑎𝑟𝑠
LED:4hoursadayfor25,000hoursmeansthebulbswouldlast6250days.6250365 = 17.12 𝑦𝑒𝑎𝑟𝑠
Basedonthisusage,ifyourparentsinstalledanLEDbulbwhenyouwereborn,you’dbeabletousethemtoshinelightonyourcollegeapplications.3.POWERBALL.Therecentpowerballlotterywasworth$1.6Billion.Winnerscandecidetotakealumpsumpayoutortheycanget30annualpayments.Iftheydecidetotakealumpsum,winnerscouldexpectapproximately$992,000,000.Ofcourse,thewinnerswouldn’tgetALLthatmoney.Therearetaxestopay-thefederalgovernmentwillshave39.6%fromthetotal.a.HowmuchisthelumpsumpaymentAFTERtaxesaretakenout?Thiswould60.4%of$992,000,000or$599,168,000.b.HowmuchiseachannualinstallmentAFTERtaxesaretakenout?Thiswouldbe60.4%of1,600,000,000dividedby30.
0.604 1,600,000,00030 = $32,213,333.33
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide6
c.Ifyoutakethelumpsumpayment,howmuchmoney(indollars)doyouhavetoearneachyearforthelumpsumtoturnouttobeabetterinvestmentthantheannualinstallments?Explainhowyoufiguredthisout.Ifyoutakethelumpsum,you’llendupwith$599,168,000.Ifyoutaketheinstallments,you’lleventuallyget$966,400,000.Thismeansthatoverthecourseof30years,you’llhavetoinvestyourmoneyinsomethingthatwillgetyouanaverageof!"",!"",!!!!!"",!"#,!!!
!"inorderforthelumpsumtobemorelucrative.Thisequates
to$12,241,066.67eachyearfor30yearsthatyouneedtomakeinorderforthelumpsumtobeworthasmuchmoneyastheannualinstallmentsfor30years.This$12MillionPlusequatestoareturnonyourinvestmentofalittlemorethan2%eachyear.Thisdoesn’ttakeintoaccounttheconceptofcompoundinginterest.Historically,it’seasytomakea2%returnonyourinvestmentandthisiswhymanypeopletakethelumpsum.d.Whatwouldyoudo?Explainwhy.I’ddefinitelytakethelumpsum.Seeaboveforrationale.4.Assumetheearthisperfectlyroundandthattheequatorisagoodmeasureoftheabsolutelargestcircumference.Iftheequatoris24,900,whatisthevolumeoftheearth?(expressyouranswerintermsofpi).𝑉!"!!"! =
!!𝜋𝑟!Thecircumferenceof24,900
canhelpusfindtheradius…because𝐶 =2𝜋𝑟…solet’ssolveforr.
24900 = 2𝜋𝑟
12450𝜋 = 𝑟
So,armedwiththisinfo,wecanfindthevolumeintermsofpi…
𝑉 =43𝜋
12450𝜋
!
=43 1929781125000
𝜋!
=2573041500000
𝜋! 𝑐𝑢𝑏𝑖𝑐 𝑚𝑖𝑙𝑒𝑠
NAME:___________________________ Math__________,Period____________
Mr.Rogove Date:__________
LinearFunctionsandGeometryStudyGuide7
5a.Manyicecreamconesare1.5inchesindiameteratthetop,andwouldstandabout4inchestall.Howmuchicecreamwouldbeabletofitinsidethecone(assumethaticecreamdoesnotpileontopofthecone,butisleveledatthetopofthecone).
𝑉!"#$ =13𝜋𝑟
!ℎ
=13𝜋
34
!
4
=34𝜋 𝑐𝑢𝑏𝑖𝑐 𝑖𝑛𝑐ℎ𝑒𝑠
5b.Usingtheinformationfromabove,let’ssayinsteadofcones,theymade“icecreamcylinders”foryoutocarryyourdeliciousiceddairytreat.Whatarepossibledimensionsforacylinderthatwilltwiceasmuchicecreamastheconeabove?Thecylinderwouldneedtohaveavolumeof1.5𝜋cubicinches.Ifthediameterofthecylindricalconewere2inches,andtheradiuswere1inch,theheightwouldbe1.5inchesinordertogettwiceasmuchicecream.
