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.