3.1LinearFunctionsandRelations

Inmathematics,asetoforderedpairsiscalleda . Example: Thesetofallfirstcomponentsiscalledthe . Wecanalsocallthesevaluesthe values.Thesetofallsecondcomponentsiscalledthe . Wecanalsocallthesevaluesthe values.1.Statethedomainandrangeoftheexampleabove.(usesetnotation) Domain: Range:Arelationisa ifeachinputhasexactlyoneoutput.Inotherwords,eachx‐coordinateisrelatedtoexactlyoney‐coordinate. Istherelationintheaboveexampleafunction?Whyorwhynot?2.Statewhetherornottherelationisafunction.Whyorwhynot? Afunctioncanalsobedefinedbywritingaformula.Typicallyweuse notationtodefineafunction.Inthisnotation,thexrepresentsthe andtheyfrepresentsthe .Thenotation meanstoevaluatethefunctionwhen .3.Examinethefunction .Findthefollowing.

A offunctionsiswhentwofunctionscanbecombinedtoformanewfunction.The offwithgiswritten andisdefinedas: Inotherwords,theoutputofgbecomestheinputoff.4.If and ,findthefollowing: a.

f g[ ] x( ) b.

g f[ ] x( )

Didyoufindaandbtobethesame?Whatdoesthismean?

Questions5‐10:Let f x( ) = x2 + 4 and g x( ) = x3,determinethevalueofeachofthefollowing.

5. f −4( ) 6. g 9( ) 7.

f g[ ] −6( ) 8.

g f[ ] 2( )

9. f g 12( )( ) 10. g f −5( )( )

3.2GraphingFunctionsandRelations

Anotherwaythatwecandetermineifarelationisafunctionisbylookingatagraphoftherelation.Arelationisafunctionifandonlyifno lineintersectsthegraphmorethanonce.Questions1‐2:Graphtherelationandthenstatewhetherornottherelationisafunction.1. 2. Questions3‐4:Graphtherelationdefinedbyeachofthefollowingrulesoverthedomain .3. 4.

3.3GraphingLinearEquations

Linearequationscontainbothan anda .Thegraphofalinearequationisa .Therearetwoformsoflinearequationsthatwewilllookat: 1. ,whichisintheform . 2. ,whichisintheform . Slope‐InterceptForm: • Themrepresentsthe oftheline.Theslopeisthe ofthe

line.o Whentheslopeispositive,thegraphgoes fromlefttoright.o Whentheslopeisnegative,thegraphgoes fromlefttoright.o Whengraphing,theslopecanbethoughtofas over ,wherethe

numeratordeterminesthe displacementandthedenominatordeterminesthe displacementbetweenpointsontheline.

• Thebrepresentsthe oftheline.Inotherwords,it’swheretheline

crossesthe .Questions1‐2:Determinetheslopeandy‐interceptofthelinesbelow.

1. 2.

GraphinginSlope‐InterceptForm:• First,makesurethattheequationisintheform .• Plotthe first.Usethisasyourstartingpoint.• Usetheslopetoplotanotherpoint,usingtheideaofriseoverrun.• Drawalinethroughyourpointswitharrowsoneachend.

Questions3‐4:Graphtheequation.

3. 4.

GraphinginStandardForm:• Onewaytographfromstandardformistotransformtheequationintoslope‐interceptform.• Asecondwaytographinstandardformistofindthex­interceptandy­intercept.Plotthetwopoints

anddrawalinethroughthem.x‐intercept:Thex‐interceptisintheform .Sotofindthex‐intercept,youwillneedtosubstitute intotheequationfor andsolvefor .y‐intercept:They‐interceptisintheform .Sotofindthey‐intercept,youwillneedtosubstitute intotheequationfor andsolvefor .Youcanusethismethodtofindthexandy‐interceptsinslope‐interceptformaswell!Questions5‐6:Findthex‐interceptandy‐interceptforthefollowingequations.

5. 3x − 2y = 12 6. x5+y6= 1

Questions7‐8:Graphusingintercepts.7. 5x + 2y = −10 8. −6x + 3y = 12 9. Determinethevalueofksothatthegivenpointwillbeonthegraphofthegivenequation. −4x + 6k = 3y; 3,−10( ) VerticalandHorizontalLines:Theequationforaverticallineisintheform: .Theslopeofaverticallineis .Theequationforahorizontallineisintheform: .Theslopeofahorizontallineis .Questions10‐11:Graphtheequation.10. y = 5 11. x = −3

AbsoluteValueGraphs:Absolutevaluegraphsareintheshapeofa .Question12:Useatableofvaluestographthefunction.12. y = x Onyourgraphingcalculatorgraphthefollowingfunctions: Y1: y = x Y2: y = x + 3 Y3: y = x − 4 Whathappenedtothegraph?Nowgraphthefollowing, Y1: y = x2 Y2: y = x − 3( )2 Y3: y = x + 2( )2 Whathappenedtothegraph?Nowgraphthefollowing? Y1: y = x Y2: y = x − 5 + 3 Y3: y = x +1 − 4 Whathappenedtothegraph?13.Withoutyourcalculator,graphthefollowingfunctions. a. y = x −1 − 2 b. y = x + 2 + 3

3.4GraphingLinearInequalities

Tographalinearinequality,treattheinequalityasifitwereanequalsignandgraphtheline.Iftheinequalityis or ,drawtheline .Iftheinequalityis or ,drawtheline .Todeterminewheretoshade,testapoint!Chooseapointoneithersideofthelineandsubstituteitintotheinequality.Ifitmakesthestatementtrue,shadeonthatsideoftheline.Ifitmakesthestatementfalse,shadeontheothersideoftheline.Examples:a. b. Questions1‐6:Graphtheinequality.1. 2.

3. −2y < 7 4. −x +13y ≤ 2

5. 7 − 2x ≤ 4 6. y < x +1 + 3

3.5SlopeofaLine

Slopeisameasureofthesteepnessofaline.Whenlookingatagraph,wecandetermineslopebytakingriserun

,ortheverticaldisplacementdividedbythehorizontaldisplacementbetweentwopointsontheline.

Tofindtheslopeofalineusingitscoordinates,weuse theformula:

m =y2 − y1x2 − x1

Questions1‐3:Determinetheslopeofthelinethrougheachpairofpoints.

1. 4,−1( ), 2, 7( ) 2. 7,2( ), −5,2( ) 3.12,9⎛

⎝⎜⎞⎠⎟, 52,−1⎛

⎝⎜⎞⎠⎟

Questions4‐5:Determinetheslopeofthelineandgraphtheline.4. 5x + 4y = 20 5. −x + 4y = 12

Withslopeinterceptform,weplottedthe andthenusedthe tofindasecondpoint.WecanuseANYpointonthelineandtheslopetographusingthesamemethod.Questions6‐7:UsethegivenpointPandthegivenslopemtodeterminethecoordinatesofasecondpointQ.Then,drawthelinethroughPandQ.

6. P −2,5( ); m =12 7. P −3,7( ); m = −

52

Theslopecanalsobeusefulindeterminingmissingcoordinatesofpointsonaline.Questions8‐10:Determinethevalueofathatmakestheslopeofalinethroughthetwogivenpointsequaltothegivenvalueofm.

8. 3,6( ) and −1,2a( ) ;m =54

9. a,1( ) and −2,−4( ) ;m =32

10. a + 3,5( ) and 1,a − 2( ) ;m = 4

3.6FindingtheEquationofaLine

Whatarethetwothingsneededtowritetheequationofalineinslope‐interceptform? 1. 2. WriteanEquationGivenaSlopeandaPointontheLine:Example: Setupyourequationbysubstitutingintheslope. Whatcanyoudotofindthey‐intercept? Findthey­intercept! Whatistheequationoftheline?Questions1‐2:Determineanequationintheform ofthelinepassingthroughthegivenpointPwiththegivenslopem.

1. 2.

WriteanEquationGiven2PointsontheLine:Example: Findtheslope!

Setupyourequationbysubstitutingintheslope. Whatcanyoudotofindthey‐intercept? Findthey­intercept! Whatistheequationoftheline?

Questions3‐4:Determineanequationintheform ofthelinecontainingthetwogivenpoints.

3. 4.

ParallelandPerpendicularLines:Inyourgraphingcalculator,graphthefollowing:

Whatdoyounoticeaboutallthreegraphs?Whatisthesameabouteachequation?Whatisdifferentabouteachequation?So,tosumitup lineshavethe slope!Questions5‐6:Determinetheequationintheform ofthelinethroughthegivenpointsatisfyingthegivenequation.

5. ;parallelto 6. ;paralleltothelinethrough

and

Perpendicularlinesontheotherhandhave slopes!Questions7‐9:Findtheslopeofalineperpendiculartothegivenline.

7. 8. 9.

Questions10‐11:Determinetheequationintheform ofthelinethroughthegivenpointsatisfyingthegivenequation.10. perpendicularto 11. perpendiculartothelinethrough

and