3.1 Linear Functions and Relations
Transcript of 3.1 Linear Functions and Relations
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.
3.5SlopeofaLine
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