Algebra 2 Unit 3: Linear Functions Algebra 2 Unit 3: Linear … · 2018. 11. 29. · Algebra 2 Unit 3: Linear Functions Ms. Talhami 4 DIRECT VARIATION COMMON CORE ALGEBRA II HOMEWORK

Algebra2Unit3:LinearFunctions

Ms.Talhami 1


Name_________________


Ms.Talhami 2

DIRECT VARIATION COMMONCOREALGEBRAII

Webeginour linearunitby lookingatthesimplest linearrelationshipthatcanexistbetweentwovariables,namelythatofdirectvariation.Wesaythattwovariablesaredirectlyrelatedorproportionaltooneanotherifthefollowingrelationshipholds.

Exercise#1:Ineachofthefollowing,xandyaredirectlyrelated.Solveforthemissingvalue.

(a) 15 when 5

? when 9

y x

y x

= =

= =

(b) 6 when 4

? when 10

y x

y x

= − =

= = −

(c) 12 when 16

? when 24

y x

y x

= =

= =

Exercise#2:Thedistanceapersoncantravelvariesdirectlywiththetimetheyhavebeentravelingifgoingataconstantspeed.IfPhoenixtraveled78milesin1.5hourswhilegoingataconstantspeed,howfarwillhetravelin2hoursatthesamespeed?Exercise#3:Jennaworksajobwhereherpayvariesdirectlywiththenumberofhoursshehasworked.Inoneweek, sheworked35hoursandmade$274.75.Howmanyhourswouldsheneed towork inorder toearn$337.55?

PROPORTIONALORDIRECTRELATIONSHIPS

Twovariables,xandy,haveadirect(proportional)relationshipifforeveryorderedpair wehave:

Stated succinctly,ywill always be a constantmultiple of x. The value ofk is known as the constant ofvariation.


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Wewill now examine the graph of a direct relationship and seewhy it is indeed the simplest of all linearfunctions.

Exercise#4:Twovariables,xandy,varydirectly.When 6x = then 4y = .Thepointisshownplottedbelow.

(a) Findthey-valuesforeachofthefollowingx-values.Ploteachpointandconnect.

3x = 6x = − (b)Whatistheconstantofvariationinthisproblem?Whatdoes

itrepresentonthisline?(c)Writetheequationofthelineyouplottedin(a).Directrelationshipsoftenexistbetweentwovariableswhosevaluesarezerosimultaneously.Exercise#3:Themilesdrivenbyacar,d,variesdirectlywiththenumberofgallons,g,ofgasolineused.Abagailisabletodrive 336d = mileson 8g = gallonsofgasolineinherhybridvehicle.

(a) Calculate the constant of variation for the

relationship . Includeproperunits inyour

answer.

(b)Give a linear equation that represents therelationship betweend andg. Express youranswerasanequationsolvedford.

(c) HowfarcanAbagaildriveon gallonsofgas?

(d)HowmanygallonsofgaswillAbagailneed inordertodrive483miles?

y

x


Ms.Talhami 4

DIRECT VARIATION COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Ineachofthefollowing,thevariablepairgivenareproportionaltooneanother.Findthemissingvalue. (a) 8 when 16

? when 18

b a

b a

= =

= =

(b) 10 when 14

? when 21

y x

y x

= =

= =

(c) 2 when 6

? when 15

w u

w u

= − =

= = −

2. Inthefollowingexercises,thetwovariablesgivenvarydirectlywithoneanother. Solveforthemissing

value. (a) 12 when 8

? when 6

p q

p q

= =

= =

(b) 21 when 9

? when 6

y x

y x

= =

= = −

(c) 5 when 2

? when 8

z w

z w

= − =

= =

3. Ifxandyvarydirectlyand 16 when 12y x= = ,thenwhichofthefollowingequationscorrectlyrepresents

therelationshipbetweenxandy?

(1) 34

y x= (3) 192xy =

(2) 28y x+ = (4) 43

y x=


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APPLICATIONS

4. ThedistanceMax’sbikemoves isdirectlyproportional tohowmanyrotationshisbike’scrankshafthasmade.IfMax’sbikemoves25feetaftertworotations,howmanyfeetwillthebikemoveafter15rotations?

5. Forhisworkout,theincreaseinJacob’sheartrateisdirectlyproportionaltotheamountoftimehehasspent

workingout. Ifhisheartbeathasincreasedby8beatsperminuteafter20minutesofworkingout,howmuchwillhisheartbeathaveincreasedafter30minutesofworkingout?

6. Whenaphotographisenlargedorshrunken,itswidthandlengthstayproportionaltotheoriginalwidthand

length.Rojasisenlargingapicturewhoseoriginalwidthwas3inchesandwhoseoriginallengthwas5inches.Ifitsnewlengthistobe8inches,whatistheexactvalueofitsnewwidthininches?

7. Forasetamountoftime,thedistanceKirkcanrunisdirectlyrelatedtohisaveragespeed.IfKirkcanrun3

miles inwhile runningat6milesperhour,howfarcanherun in thesameamountof time ifhisspeedincreasesto10milesperhour?

REASONING8. Twovariablesareproportionaliftheycanbewrittenat y kx= ,wherekissomeconstant.Thisleadstothe

fact that when 0x = then 0y = as well. Is the temperature measured in Celsius proportional to thetemperaturemeasuredinFahrenheit?Explain.


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AVERAGE RATE OF CHANGE COMMONCOREALGEBRAII

Whenwemodelusingfunctions,weareveryofteninterestedintheratethattheoutputischangingcomparedtotherateoftheinput.Exercise#1:Thefunction ( )f x isshowngraphedtotheright.(a) Evaluateeachofthefollowingbasedonthegraph: (i) ( )0f (ii) ( )4f (iii) ( )7f (iv) ( )13f (b) Find the change in the function, fΔ , over each of the

following domain intervals. Find this both by subtractionandshowthisonthegraph.

(i)0 4x≤ ≤ (ii) 4 7x≤ ≤ (iii)7 13x≤ ≤ (c) Whycan'tyousimplycomparethechangesinffrompart(b)todetermineoverwhichintervalthefunction

changingthefastest?(d) Calculatetheaveragerateofchangeforthefunctionovereachoftheintervalsanddeterminewhichinterval

hasthegreatestrate. (i)0 4x≤ ≤ (ii) 4 7x≤ ≤ (iii)7 13x≤ ≤ (e) Usingastraightedge,drawinthelineswhoseslopesyoufoundinpart(d)byconnectingthepointsshown

onthegraph.Theaveragerateofchangegivesameasurementofwhatpropertyoftheline?

y

x


Ms.Talhami 7

Theaveragerateofchangeisanexceptionallyimportantconceptinmathematicsbecauseitgivesusawaytoquantifyhowfastafunctionchangesonaverageoveracertaindomaininterval.Althoughweuseditsformulainthelastexercise,westateitformallyhere:Exercise#2:Considerthetwofunctions ( ) 5 7f x x= + and ( ) 22 1g x x= + .(a) Calculatetheaveragerateofchangeforbothfunctionsoverthefollowingintervals.Doyourworkcarefully

andshowthecalculationsthatleadtoyouranswers.

(i) 2 3x− ≤ ≤ (ii)1 5x≤ ≤ (b) Theaveragerateofchangeforfwasthesameforboth(i)and(ii)butwasnotthesameforg.Whyisthat?Exercise#3:Thetablebelowrepresentsalinearfunction.Fillinthemissingentries.

AVERAGERATEOFCHANGE

Forafunctionoverthedomaininterval ,thefunction'saveragerateofchangeiscalculatedby:

x 1 5 11 45

y -5 1 22


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AVERAGE RATE OF CHANGE COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Forthefunction ( )g x giveninthetablebelow,calculatetheaveragerateofchangeforeachofthefollowing

intervals. (a) 3 1x− ≤ ≤ − (b) 1 6x− ≤ ≤ (c) 3 9x− ≤ ≤ (d)Explainhowyoucantellfromtheanswersin(a)through(c)thatthis isnotatablethatrepresentsa

linearfunction.2. Considerthesimplequadraticfunction ( ) 2f x x= .Calculatetheaveragerateofchangeofthisfunctionover

thefollowingintervals: (a)0 2x≤ ≤ (b) 2 4x≤ ≤ (c) 4 6x≤ ≤ (d)Clearlytheaveragerateofchangeisgettinglargeratxgetslarger.

Howisthisreflectedinthegraphoffshownsketchedtotheright?

x 4 6 9 8 13 12 5


Ms.Talhami 9

3. Whichhasagreateraveragerateofchangeovertheinterval 2 4x− ≤ ≤ ,thefunction ( ) 16 3g x x= − orthe

function ( ) 22f x x= ?Providejustificationforyouranswer.APPLICATIONS4. Anobjecttravelssuchthatitsdistance,d,awayfromitsstartingpointisshownasafunctionoftime,t,in

seconds,inthegraphbelow. (a) Whatistheaveragerateofchangeofdoverthe

interval 5 7t≤ ≤ ? Includeproperunits inyouranswer.

(b) Theaveragerateofchangeofdistanceovertime

(what you found in part (a)) is known as theaveragespeedofanobject.Istheaveragespeedofthisobjectgreaterontheinterval0 5t≤ ≤ or11 14t≤ ≤ ?Justify.

REASONING5. Whatmakestheaveragerateofchangeofalinearfunctiondifferentfromthatofanyotherfunction?What

isthespecialnamethatwegivetotheaveragerateofchangeofalinearfunction?

Time(seconds)

Distan

ce(fee

t)


Ms.Talhami 10

FORMS OF A LINE COMMONCOREALGEBRAII

Linear functions come ina varietyof forms. The two shownbelowhavebeen introduced inCommonCoreAlgebraIandCommonCoreGeometry.Exercise#1:Considerthelinearfunction ( ) 3 5f x x= + .

Exercise#2:Consideralinewhoseslopeis5andwhichpassesthroughthepoint ( )2, 8− .

Exercise#3:Whichofthefollowingrepresentsanequationforthelinethatisparallelto 3 72

y x= − andwhich

passesthroughthepoint ( )6, 8− ?

(1) ( )28 63

y x− = − + (3) ( )38 62

y x+ = −

(2) ( )38 62

y x− = + (4) ( )28 63

y x+ = − −

TWOCOMMONFORMSOFALINE

Slope-Intercept: Point-Slope:

wheremistheslope(oraveragerateofchange)ofthelineand representsonepointontheline.

(a) Determine they-interceptof this functionbyevaluating .

(b) Find its average rate of change over theinterval .

(a) Write theequationof this line inpoint-slopeform, .

(b)Write the equation of this line in slope-interceptform, .


Ms.Talhami 11

Exercise#4:Alinepassesthroughthepoints ( ) ( )5, 2 and 20, 4− .

Exercise#5:Thegraphofalinearfunctionisshownbelow.(a)Writetheequationofthislinein y mx b= + form.(b)Whatmustbetheslopeofalineperpendiculartothe

oneshown?(c)Drawalineperpendiculartotheoneshownthat

passesthroughthepoint ( )1, 3 .(d)Writetheequationofthelineyoujustdrewinpoint-

slopeform.

(a) Determinetheslopeofthislineinsimplestrationalform.

(b)Writeanequationofthislineinpoint-slopeform.

(c) Writeanequationforthislineinslope-interceptform.

(d) Forwhatx-valuewillthislinepassthroughay-valueof12?

y

x

(e)Doesthelinethatyoudrewcontainthepoint?Justify.


Ms.Talhami 12

FORMS OF A LINE COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Whichofthefollowinglinesisperpendicularto 5 73

y x= − andhasay-interceptof4?

(1) 5 43

y x= + (3) 345

y x= −

(2) 3 45

y x= − + (4) 3 45

y x= +

2. Whichofthefollowinglinespassesthroughthepoint ( )4, 8− − ?

(1) ( )8 3 4y x+ = + (3) ( )8 3 4y x+ = −

(2) ( )8 3 4y x− = − (4) ( )8 3 4y x− = +

3. Whichofthefollowingequationscoulddescribethegraphofthelinearfunctionshownbelow?

(1) 2 43

y x= − (3) 2 43

y x= − −

(2) 2 43

y x= + (4) 2 43

y x= − +

4. Foralinewhoseslopeis 3− andwhichpassesthroughthepoint ( )5, 2− :5.Foralinewhoseslopeis0.8 andwhichpassesthroughthepoint ( )3,1− :

y

x

(a) Writetheequationofthislineinpoint-slopeform, .


(a) Writetheequationofthislineinpoint-slopeform, .



Ms.Talhami 13

REASONING6. Thetwopoints ( ) ( )3, 6 and 6, 0− areplottedonthegridbelow.

(a)Find an equation, in y mx b= + form, for the linepassing through these two points. Use of the grid isoptional.

(b)Doesthepoint ( )30, 16− lieonthisline?Justify.7. Alinearfunctionisgraphedbelowalongwiththepoint ( )3,1 .

(a) Drawalineparalleltotheoneshownthatpassesthroughthepoint ( )3,1 .

(b)Writeanequationforthelineyoujustdrewinpoint-slope

form.

(c) Betweenwhattwoconsecutiveintegersdoesthey-interceptofthelineyoudrewfall?

(d)Determinetheexactvalueofthey-interceptofthelineyoudrew.

y

x

y

x


Ms.Talhami 14

LINEAR MODELING COMMONCOREALGEBRAII

InCommonCoreAlgebraI,youusedlinearfunctionstomodelanyprocessthathadaconstantrateatwhichonevariablechangeswithrespecttotheother,oraconstantslope.Inthislessonwewillreviewmanyofthefacetsofthistypeofmodeling.Exercise#1:DiawasdrivingawayfromNewYorkCityataconstantspeedof58milesperhour.Hestarted45milesaway.

InExercise#1, it isclear fromthecontextwhatboththeslopeandthey-interceptof this linearmodelare.Althoughthisisoftenthecasewhenconstructingalinearmodel,sometimestheslopeandapointareknown,inwhichcase,thepointslopeformofthealineismoreappropriate.Exercise#2:Edelynistryingtomodelhercell-phoneplan.Sheknowsthatithasafixedcost,permonth,alongwitha$0.15chargepercallshemakes.Inherlastmonth’sbill,shewascharged$12.80formaking52calls.

(a) Write a linear function that gives Dia’sdistance,D, fromNewYorkCityasafunctionofthenumberofhours,h,hehasbeendriving.

(b) IfDia’sdestinationis270milesawayfromNewYork City, algebraically determine to thenearesttenthofanhourhowlongitwilltakeDiatoreachhisdestination.

(a) Createalinearmodel,inpoint-slopeform,forthe amount Edelyn must pay, P, per monthgiventhenumberofphonecallsshemakes,c.

(b)How much is Edelyn’s fixed cost? In otherwords,howmuchwould shehave topay formakingzerophonecalls?


Ms.Talhami 15

Manytimeslinearmodelshavebeenconstructedandweareaskedonlytoworkwiththesemodels.Modelsintherealworldcanbemessyanditisoftenconvenienttouseourgraphingcalculatorstoplotandinvestigatetheirbehavior.Exercise#3:Afactoryproduceswidgets(genericobjectsofnoparticularuse).Thecost,C,indollarstoproducewwidgetsisgivenbytheequation 0.18 20.64C w= + .Eachwidgetsellsfor26cents.Thus,therevenuegained,R,fromsellingthesewidgetsisgivenby 0.26R w= .

(a) Use your graphing calculator to sketch andlabel each of these linear functions for theinterval . Besuretolabelyoury-axiswithitsscale.

(b)Use your calculator’s INTERSECT command todetermine the number of widgets, w, thatmustbeproducedfortherevenuetoequalthecost.

(c) If profit is defined as the revenueminus thecost,createanequationintermsofwfortheprofit,P.

w

Dollars

(e)Whatistheminimumnumberofwidgetsthatmustbesoldinorderfortheprofittoreachatleast$40?Illustratethisonyourgraph.

(d)Usingyourgraphingcalculator,sketchagraphof the profit over the interval .UseaTABLEonyourcalculatortodetermineanappropriateWINDOW forviewing. Label thexandyinterceptsofthislineonthegraph.

w

Dollars


Ms.Talhami 16

LINEAR MODELING COMMONCOREALGEBRAIIHOMEWORK

APPLICATIONS1. Whichofthefollowingwouldmodelthedistance,D,adriverisfromChicagoiftheyareheadingtowards

thecityat58milesperhourandstarted256milesaway?

(1) 256 58D t= + (3) 58 256D t= +

(2) 256 58D t= − (4) 58 256D t= − 2. Thecost,C,ofproducingx-bikesisgivenby 22 132C x= + .Therevenuegainedfromsellingx-bikesisgiven

by 350R x= .Iftheprofit,P,isdefinedas P R C= − ,thenwhichofthefollowingisanequationforPintermsofx?

(1) 328 132P x= − (3) 328 132P x= + (2) 372 132P x= + (4) 372 132P x= −

3. Theaveragetemperatureoftheplanetisexpectedtoriseatanaveragerateof0.04degreesCelsiusperyearduetoglobalwarming.Theaveragetemperatureintheyear2000was14.71degreesCelsius.TheaverageCelsiustemperature,C,isgivenby 14.71 0.04C x= + ,wherexrepresentsthenumberofyearssince2000.

(a) What will be the average temperature intheyear2100?

(b) Algebraically determine the number ofyears,x,itwilltakeforthetemperature,C,toreach20degreesCelsius.Roundtothenearestyear.

(c) Sketch a graph of the average yearlytemperature below for the interval

. Be sure to label your y-axisscaleaswellastwopointsontheline(they-interceptandoneadditionalpoint).

(d)What does this model project to be theaverageglobaltemperaturein2200?

x

C(Celsius)


Ms.Talhami 17

4. FabioisdrivingwestawayfromAlbanyandtowardsBuffaloalongInterstate90ataconstantrateofspeedof62milesperhour.Afterdrivingfor1.5hours,Fabiois221milesfromAlbany.

5. AparticularrockettakingofffromtheEarth’ssurfaceusesfuelataconstantrateof12.5gallonsperminute.

Therocketinitiallycontains225gallonsoffuel.

(a) Writea linearmodelforthedistance,D,thatFabioisawayfromAlbanyasafunctionofthenumberofhours,h,thathehasbeendriving.Write your model in point-slope form,

.

(b) Rewrite this model in slope-intercept form,.

(c) How far was Fabio from Albany when hestartedhistrip?

(d) IfthetotaldistancefromAlbanytoBuffalois290 miles, determine how long it takes forFabiotoreachBuffalo.Roundyouranswertothenearesttenthofanhour.

(a) Determinealinearmodel,in form,fortheamountoffuel,y,asafunctionofthenumber of minutes, x, that the rocket hasburned.

(b) Belowisageneralsketchofwhatthegraphofyour model should look like. Using yourcalculator,determinethexandyinterceptsofthis model and label them on the graph atpointsAandBrespectively.

(c)Therocketmuststillcontain50gallonsoffuelwhen it hits the stratosphere. What is themaximumnumber ofminutes the rocket cantaketohitthestratosphere?Showthispointonyourgraphbyalsographingthehorizontalline and showing the intersectionpoint. x

y

A

B


Ms.Talhami 18

INVERSES OF LINEAR FUNCTIONS COMMONCOREALGEBRAII

Recall that functions have inverses that are also functions if they are one-to-one. With the exception ofhorizontallines,alllinearfunctionsareone-to-oneandthushaveinversesthatarealsofunctions.Inthislessonwewillinvestigatetheseinversesandhowtofindtheirequations.

Exercise#1:Onthegridbelowthelinearfunction 2 4y x= − isgraphedalongwiththeline y x= .

(a) How can you quickly tell that 2 4y x= − is a one-to-onefunction?

(b)Graphtheinverseof 2 4y x= − onthesamegrid.Recallthat

thisiseasilydonebyswitchingthexandycoordinatesoftheoriginalline.

(c)Whatcanbesaidaboutthegraphsof 2 4y x= − anditsinverse

withrespecttotheline y x= ?

Aswecanseefrompart(e) inExercise#1, inversesof linearfunctionsincludetheinverseoperationsoftheoriginalfunctionbutinreverseorder.Thisgivesrisetoasimplemethodoffindingtheequationofanyinverse.Simplyswitchthexandyvariablesintheoriginalequationandsolvefory.

Exercise#2:Whichofthefollowingrepresentstheequationoftheinverseof 5 20y x= − ?

(1) 1 205

y x= − + (3) 1 45

y x= −

(2) 1 205

y x= − (4) 1 45

y x= +

y

x

(d)Findtheequationoftheinversein form.

(e) Find theequationof the inverse in

form.


Ms.Talhami 19

Althoughthisisasimpleenoughprocedure,certainproblemscanleadtocommonerrorswhensolvingfory.Careshouldbetakenwitheachalgebraicstep.

Exercise#3:Whichofthefollowingrepresentstheinverseofthelinearfunction 2 83

y x= + ?

(1) 3 82

y x= − (3) 3 82

y x= − +

(2) 3 122

y x= − (4) 3 122

y x= − +

Exercise#4:Whatisthey-interceptoftheinverseof 3 95

y x= − ?

(1) 15y = (3) 9y =

(2) 19

y = (4) 53

y = −

Sometimesweareaskedtoworkwithlinearfunctionsintheirpoint-slopeform.Themethodoffindingtheinverseandplottingit,though,donotchangejustbecausethelinearequationiswritteninadifferentform.

Exercise#5:Whichofthefollowingwouldbeanequationfortheinverseof ( )6 4 2y x+ = − ?

(1) ( )12 64

y x− = + (3) ( )6 4 2y x− = − +

(2) ( )12 64

y x− = − + (4) ( )2 4 6y x+ = − −

Exercise#6: Whichofthefollowingpoints liesonthegraphofthe inverseof ( )8 5 2y x− = + ?Explainyourchoice.

(1) ( )8, 2− (3) ( )10, 40−

(2) ( )8, 2− (4) ( )2, 8− Exercise#7:Whichofthefollowinglinearfunctionswouldnothaveaninversethatisalsoafunction?Explainhowyoumadeyourchoice. (1) y x= (3) 2y = (2) 2y x= (4) 5 1y x= −


Ms.Talhami 20

INVERSES OF LINEAR FUNCTIONS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. Thegraphofafunctionanditsinversearealwayssymmetricacrosswhichofthefollowinglines?

(1) 0y = (3) y x=

(2) 0x = (4) 1y =

2. Whichofthefollowingrepresentstheinverseofthelinearfunction 3 24y x= − ?

(1) 1 83

y x= + (3) 1 243

y x= − +

(2) 1 83

y x= − − (4) 1 13 24

y x= −

3. Ifthey-interceptofalinearfunctionis8,thenweknowwhichofthefollowingaboutitsinverse?

(1)Itsy-interceptis 8− . (3)Itsy-interceptis 18.

(2)Itsx-interceptis8. (4)Itsx-interceptis 8− .

4. Ifbothwereplotted,whichofthefollowinglinearfunctionswouldbeparalleltoitsinverse?Explainyourthinking.

(1) 2y x= (3) 5 1y x= −

(2) 2 43

y x= − (4) 6y x= +

5.Whichofthefollowingrepresentstheequationoftheinverseof 4 243

y x= + ?

(1) 4 243

y x= − − (3) 3 184

y x= −

(2) 3 184

y x= − + (4) 4 243

y x= −

6.Whichofthefollowingpointsliesontheinverseof ( )2 4 1y x+ = − ?

(1) ( )2, 1− (3)1 ,12

⎛ ⎞⎜ ⎟⎝ ⎠

(2) ( )1, 2− (4) ( )2,1−


Ms.Talhami 21

7. Alinearfunctionisgraphedbelow.Answerthefollowingquestionsbasedonthisgraph. (a)Writetheequationofthislinearfunctionin y mx b= + form. (b)Sketchagraphoftheinverseofthisfunctiononthesamegrid. (c)Writetheequationoftheinversein y mx b= + form. (d)Whatistheintersectionpointofthislinewithitsinverse?APPLICATIONS

8. Acartravelingataconstantspeedof58milesperhourhasadistanceofy-milesfromPoughkeepsie,NY,givenbytheequation 58 24y x= + ,wherexrepresentsthetimeinhoursthatthecarhasbeentraveling.

(c) Giveaphysicalinterpretationoftheansweryoufoundinpart(b).Considerwhattheinputandoutput

oftheinverserepresentinordertoanswerthisquestion.REASONING

9. Giventhegenerallinearfunction y mx b= + ,findanequationforitsinverseintermsofmandb.

(a) Findtheequationoftheinverseofthislinear

functionin form.

(b) Evaluatethefunctionyoufoundinpart(a)foraninputof .

y

x


Ms.Talhami 22

PIECEWISE LINEAR FUNCTIONS COMMONCOREALGEBRAII

Functions expressed algebraically can sometimes bemore complicated and involvedifferent equations fordifferentportionsoftheirdomains.Theseareknownaspiecewisefunctions(theycomeinpieces).Ifallofthepiecesarelinear,thentheyareknownaspiecewiselinearfunctions.

Exercise#1:Considerthepiecewiselinearfunctiongivenbytheformula ( )3 3 0

1 4 0 42

x xf x

x x

− − ≤ <⎧⎪= ⎨ + ≤ ≤⎪⎩

.

(a) Createatableofvaluesbelowandgraphthefunction.(b) Statetherangeoffusingintervalnotation.Notonlyshouldwebeabletographpiecewisefunctionswhenwearegiventheirequations,butweshouldalsobeabletotranslatethegraphsofthesefunctionsintoequations.Exercise#2:Thefunction ( )f x isshowngraphedbelow.Writeapiecewiselinearformulaforthefunction.Besuretospecifyboththeformulasandthedomainintervalsoverwhichtheyapply.

y

x

x 0 1 2 3 4

y

x


Ms.Talhami 23

Piecewiseequationscanbechallengingalgebraically.Sometimesinformationthatwefindfromthemcanbemisleadingorincorrect.Exercise #3: Consider the piecewise linear function

( ) 12

5 22 2x x

g xx x− <⎧

= ⎨ + ≥⎩.

(e) Howcanyouresolvethefactthatthealgebraseemstocontradictyourgraphicalevidenceofx-intercepts?

Exercise #4: For the piecewise linear function ( ) 2 10 05 1 0x x

f xx x

− + ≤⎧= ⎨ − >⎩

, find all solutions to the equation

( ) 1f x = algebraically.

(a) Determine the y-intercept of this functionalgebraically. Why can a function have onlyoney-intercept?

(b) Findthex-interceptsofeach individual linearequation.

(c) Graphthepiecewiselinearfunctionbelow.

(d)Whydoesyourgraphcontradict theanswersyoufoundinpart(b)?

y

x


Ms.Talhami 24

PIECEWISE LINEAR FUNCTIONS COMMONCOREALGEBRAIIHOMEWORK

FLUENCY

1. For ( )13

5 3 28 2 37 3

x xf x x x

x x

− < −⎧⎪= + − ≤ <⎨⎪ + ≥⎩

answerthefollowingquestions.

(a) Evaluateeachofthefollowingbycarefullyapplyingthecorrectformula: (i) ( )2f (ii) ( )4f − (iii) ( )3f (iv) ( )0f (b) Thethreelinearequationshavey-interceptsof 3, 8 and 7− respectively.Yet,afunctioncanhaveonly

oney-intercept.Whichoftheseisthey-interceptofthisfunction?Explainhowyoumadeyourchoice. (c) Calculatetheaveragerateofchangeoffovertheinterval 3 9x− ≤ ≤ .Showthecalculationsthatleadto

youranswer.

2. Determinetherangeofthefunction ( ) 32 64 2 29 2 x

x xg x

x < ≤

+ − ≤ ≤⎧= ⎨− +⎩

graphically.

y

x


Ms.Talhami 25

3. Determineapiecewiselinearequationforthefunction ( )f x shownbelow.Besuretospecifynotonlytheequations,butalsothedomainintervalsoverwhichtheyapply.

REASONING

4. Stepfunctionsarepiecewisefunctionsthatareconstants(horizontallines)overeachpartoftheirdomains.Graphthefollowingstepfunction.

( )

2 0 33 3 57 5 105 10 12

xx

f xxx

− ≤ <⎧⎪ ≤ <⎪= ⎨ ≤ <⎪⎪ ≤ ≤⎩

5. Find all x-intercepts of the function ( ) 12

2 8 5 14 1 1

4 10 1 4

x xg x x x

x x

+ − ≤ < −⎧⎪= − − − ≤ <⎨⎪− + ≤ ≤⎩

algebraically. Justify your work by

showingyouralgebra.Besuretocheckyouranswersversusthedomainintervalstomakesureeachsolutionisvalid.

y

x

y

x


Ms.Talhami 26

SYSTEMS OF LINEAR EQUATIONS COMMONCOREALGEBRAIISystemsofequations,ormorethanoneequation,arisefrequentlyinmathematics.Tosolveasystemmeanstofindallsetsofvaluesthatsimultaneouslymakeallequationstrue.Ofspecialimportancearesystemsoflinearequations. You have solved them in your last two Common Core math courses, but we will add to theircomplexityinthislesson.Exercise#1:Solvethefollowingsystemofequationsby:(a)substitutionand(b)byelimination.(a)3 2 92 7x yx y+ = −+ = −

(b)3 2 92 7x yx y+ = −+ = −

You should be very familiar with solving two-by-two systems of linear equations (two equations and twounknowns).Inthislesson,wewillextendthemethodofeliminationtolinearsystemsofthreeequationsandthreeunknowns.TheselinearsystemsserveasthebasisforafieldofmathknownasLinearAlgebra.Exercise#2:Considerthethree-by-threesystemoflinearequationsshownbelow.Eachequationisnumberedinthisfirstexercisetohelpkeeptrackofourmanipulations.

2 156 3 354 4 14

x y zx y zx y z

+ + =− − =

− + − = −

(1)

(2)

(3)

(a) Theadditionpropertyofequalityallowsustoaddtwoequationstogethertoproduceathirdvalid equation. Create a system by addingequations (1)and (2)and (1)and (3).Why isthisaneffectivestrategyinthiscase?

(b)Usethisnewtwo-by-twosystemtosolvethethree-by-three.


Ms.Talhami 27

Just as with two by two systems, sometimes three-by-three systems need to be manipulated by themultiplicationpropertyofequalitybeforewecaneliminateanyvariables.Exercise#3: Consider the systemof equations shownbelow.Answer the followingquestions basedon thesystem.4 3 62 4 2 385 7 19

x y zx y zx y z

+ − = −− + + =

− − = −

Exercise#4:Solvethesystemofequationsshownbelow.Showeachstepinyoursolutionprocess.

4 2 3 235 3 37

2 4 27

x y zx y zx y z

− + =+ − = −

− + + =

(a) Which variable will be easiest to eliminate?Why? Use the multiplicative property ofequalityandeliminationtoreducethissystemtoatwo-by-twosystem.

(b) Solvethetwo-by-twosystemfrom(a)andfindthefinalsolutiontothethree-by-threesystem.


Ms.Talhami 28

SYSTEMS OF LINEAR EQUATIONS COMMONCOREALGEBRAIIHOMEWORKFLUENCY1. Thesumoftwonumbersis5andthelargerdifferenceofthetwonumbersis39.Findthetwonumbersby

settingupasystemoftwoequationswithtwounknownsandsolvingalgebraically.2. Algebraically,findtheintersectionpointsofthetwolineswhoseequationsareshownbelow.

4 3 136 8

x yy x

+ = −= −

3. Showthat 10, 4, and 7x y z= = = isasolutiontothesystembelowwithoutsolvingthesystemformally.

2 254 5 12 8 32

x y zx y zx y z

+ + =− − =

− − + =

4. Inthefollowingsystem,thevalueoftheconstantcisunknown,butitisknownthat 8x = − and 4y = are

thexandyvaluesthatsolvethissystem.Determinethevalueofc.Showhowyouarrivedatyouranswer.

5 2 3 811

2 35

x y zx y zx y cz

− + + =− + = −− + =


Ms.Talhami 29

5. Solvethefollowingsystemofequations.Carefullyshowhowyouarrivedatyouranswers.

4 2 212 2 13

3 2 5 70

x y zx y zx y z

+ − =− − + =

− + =

6. Algebraicallysolvethefollowingsystemofequations.Therearetwovariablesthatcanbereadilyeliminated,

butyouranswerswillbethesamenomatterwhichyoueliminatefirst.

2 5 353 4 31

3 2 2 23

x y zx y zx y z

+ − = −− + =

− + + = −

7. Algebraically solve the following systemof equations. This systemwill takemoremanipulationbecause

therearenovariableswithcoefficientsequalto1.

2 3 2 334 5 3 546 2 8 50

x y zx y zx y z

+ − =+ + =

− − − = −

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Algebra 2 Unit 3: Linear Functions Algebra 2 Unit 3: Linear … · 2018. 11. 29. · Algebra 2 Unit 3: Linear Functions Ms. Talhami 4 DIRECT VARIATION COMMON CORE ALGEBRA II HOMEWORK