[PDF] Graph Transformations

Graph Transformations

effect function become points of new graph f(x) + d (a, b) 7 (a, b + d) shift up by d f(x) d (a, b) 7 The graph of 2g(3x) is obtained from the graph of g(x) by shrinking the horizontal 

ap15_calculus_ab_q2pdf - College Board

x = − + + Let R and S be the two regions enclosed by the graphs of f and g shown in the figure 2 0 0 997427 1 006919 2  

AP Calculus AB Sample Student Responses and Scoring

PDF

Calc Sem 2 Unit 1 PT and Answer Keypdf

e the shaded region bounded by the graph of y = In x and the line y = x - 2, as shown above (a) Find the area Let f and g be the functions given by f(x) = et and g(x) = ln x een r

AP Calculus Exam Prep Assign13KEY

graph of f (x) intersects both axes 4) If f (x) = 4 x 1 and g(x) = 2x, then the solution set of f (g(x)) 

Chapter 25-HW

the graph of y = f(x) to graph the function g(x) = f(x + 3) y = f(x) shift 3 units left oa Eco 1 LEHET

Graphing Standard Function & Transformations

a picture of the graph of g(x) = x2 1 It is obtained from the graph of f(x) = x 2 by shifting it down 1 

chapt 2 and 7 ANSpdf

h diagram represents a one-to-one function? ID : { x1 x 5 R 0x2 4 / 3 5) The functions f and g are defined by f(x) = x² and g(x) = 2x respectively Find h(x) if h(x) = f(2x)g(–2x)

EXAM QUESTIONS - MadAsMaths

gram above shows the graph of the function f , defined as ( ) 1 4, , 2 1 f x x x Question 19 (***) The functions f and g 

[PDF] graphe de markov sureté de fonctionnement

[PDF] graphiques anorexie 2015 france

[PDF] graphisme et psychomotricité

[PDF] graphisme maternelle a imprimer gratuit

[PDF] graphisme maternelle gs

[PDF] graphisme maternelle moyenne section

[PDF] graphisme maternelle petite section

[PDF] graphomotricité exercice

[PDF] graphomotricité psychomotricité

[PDF] gravity and grace

[PDF] gray anatomie pour les étudiants 2e édition pdf

[PDF] gray anatomy for students pdf download

[PDF] gray's anatomie pdf francais

[PDF] gray's anatomy for students pdf

[PDF] gray's anatomy pdf français

GraphTransform ations

Therearemanyt imeswheny ou'llknowv erywellwhatthegr aphofa particularfunctionlookslike,and you'llwanttoknowwh atthegraphofa verysimila rfunctionlookslike.I nthischapter,we'lldiscu sssomewaysto drawgraphsin thesecircumstances. Supposeyouknowwhatt hegraphof afunctionf(x)loo kslike.Suppo se d2Rissomen umberthati sgreaterthan0,an dyouareaskedto grapht he functionf(x)+d.The grapho fthenewfunctionisea sytodescr ibe:just takeevery pointinthegraphoff(x),andmo veit upadistanceofd.Tha t is,if(a,b)isapointinthegraphoff(x),then (a,b+d)isapointinthe graphoff(x)+d. Asan expla nationforwhat'swrittenabove:If(a,b)isapointinthegraph off(x),the nthatmeansf(a)=b.Hen ce,f(a)+d=b+d,whi chistosay that(a,b+d)isapointinthegraphoff(x)+d. Thechart onthenextpagedescri besh owtouseth egraphoff(x)tocreate thegraphof somes imilarf unctions.Throughoutthe chart,d>0,c>1,and (a,b)isapointinthegraphoff(x). Noticethatallof the"newfunct ions"inth echart di↵erfrom f(x)bysome algebraicmanipulationthat happensafter fplaysitspartas afunction. For example,firstyouputxintothefuncti on,thenf(x)is whatco mesout.Th e functionhasdoneitsjob.O nlyafterfhasdoneit sjobdoyouadddtoget thene wfunctionf(x)+d. 67
(9') g Becauseallofthealgebr aictrans format ionsoccur afterthefunctiondoes itsjob,all ofthechan gestopoint sinth esecondcolumn ofthechartoccur intheseco ndcoordina te.Thus,all thechangesinthegraphsoccurinth e verticalmeasurementsoft hegraph.

NewHowpoin tsingraphoff(x)visuale↵ect

functionbecomepointsof newgraph f(x)+d(a,b)7!(a,b+d)shiftupbyd f(x)d(a,b)7!(a,bd)shiftdownbyd cf(x)(a,b)7!(a,cb)stretchverticallybyc 1 c f(x)(a,b)7!(a, 1 c b)shrinkverticallyby 1 c f(x)(a,b)7!(a,b)flipoverthe x-axis

Examples.

•Thegraphof f(x)=x 2 isagra pht hatweknowhowtod raw.It's drawnonpage5 9. Wecanu sethisg raphth atweknowandt hechartaboveto drawf(x)+2, f(x)2,2f(x), 1 2 f(x),andf(x).Orto writethe previousfiv efunctions withoutthenameofth efunctionf,th esearethefiv efunctionsx 2 +2,x 2 2, 2x 2 x 2 2 ,andx 2 .These graphs aredrawnonthenextpa ge. 68
69
c3\ a 2. S!V-X zx- c1'l 4LLS z

Urv\Of'

Z_ N Wecou ldalsomakesimpleal gebraic adjustmentstof(x)beforethefunc - tionfgetsac hancetodo itsjob.Forexample,f(x+d)isthefunctionwhere youfirs tadddtoan umbe rx,an donlyaftert hatdoyoufeedanu mberinto thefunc tionf. Thechar tbelowissimilar tothechartonpage 68.Thedi ↵erenceinthe chartbelowis thatthealgebrai cmani pulationsocc urbeforeyo ufeedanum- berinto f,an dthusallo fthechangesoccu rinthefir stcoordi natesofpoi nts inthe graph.Allof thevisualchange sa↵ectthe horizontal measurementsof thegraph. Inth echartbelow, justasintheprevi ouschart,d>0,c>1,and( a,b)is apo intinthegraph off(x).

NewHowpoi ntsingraphoff(x)visuale↵ect

functionbecomepointsof newgraph f(x+d)(a,b)7!(ad,b)shiftleftbyd f(xd)(a,b)7!(a+d,b)shiftrightby d f(cx)(a,b)7!( 1 c a,b)shrinkhorizontallyby 1 c f( 1 c x)(a,b)7!(ca,b)stretchhorizontallyby c f(x)(a,b)7!(a,b)flipoverthe y-axis Oneimporta ntpointofcautiontokeepinmind isthatmostofthevisual horizontalchangesdescribedinthec hartabovearetheexa ctoppositeofthe e↵ectthatm ostpeopleant icipateafte rhavingseenthecharto npage68.To 70
getanide aforwh ythat'strue let'swork throughon eexample.We'llsee whythefirs trowofthe previouscha rtistrue, thatis we'llseew hythegraph off(x+d)isthegraphoff(x)shiftedleftbyd: Supposethatd>0.If( a,b)isapointthatiscontainedinthegraphof f(x),then f(a)=b.Hen ce,f((ad)+d)=f(a)=b,whi chistosaythat (ad,b)isapointinthegraphoff(x+d).The visualchangeb etweenthe

Examples.

•Beginningwiththegraphf(x)=x 2 ,we canuset hechar tonthe previouspagetodrawtheg raphsoff(x+2),f(x2),f(2x),f( 1 2 x),and f(x).We couldalternatively writethesef unctionsas(x+2) 2 ,(x2) 2 (2x) 2 x 2 2 ,and(x) 2 .The graphs ofthesefunctionsaredra wnonth enext page.

Noticeonthenextp ageth atthegrap hof(x)

2 isthe sameastheg raph ofouroriginal functionx 2 .Tha t'sbecausewhenyouflip thegraphofx 2 overthey-axis,you'l lgetthesamegraphthatyoustartedwith .Th atx 2 and (x) 2 havethesameg raphmeanst hattheyare thesamefunction.W eknow thisasw ellf romtheiralgebra:becaus e(1) 2 =1,weknowthat(x) 2 =x 2 71
72
c3\ L1x_' ''.ii (2.) etc. b7Z zx z) çx 2 (xz) -2 2 (x_2)2 2

Transformationsbeforeandaftertheoriginalfunct ion

Aslo ngasthereis onlyon etypeofoperati oninvo lved"insidet hefuncti on" "outsideof thefunction"- eithermultiplication oraddition- youcan apply therule sfromthetw ochartsonpage68 and70to transformthegraphofa function.

Examples.

•Let'slookat thefunction2f(x+3).There isonlyonekindof operationinsideoftheparent heses,andthatoperat ionisa ddition - yo uare adding3. Thereisonlyon ekindo foperationout sideofth eparentheses,andthat operationismultiplication-y oua remultiplyingby2,andyouaremultiplying by1. Sotofind thegraphof 2f(x+3),takethegraphoff(x),shif tittothe leftbyadist anceof 3,s tretchverticallybyafact orof2, andthe nflipover thex-axis. (Therearethreetra nsformatio nsthatyouhavet operforminthisproblem: shiftleft,str etch,andflip .Youhavetodoallthr ee,butth eorderinwhich youdot hemisn 'timportant.Y ou'llgetthesam eanswereitherway.) 73
73
7 4(x) -LI.-'

4I(x43

2tk3) -4 2.Lt. o'le'( .z-axS 73
7 4(x) -LI.-'

4I(x43

2tk3) -4 2.Lt. o'le'( .z-axS 73
7 4(x) -LI.-'

4I(x43

2tk3) -4 2.Lt. o'le'( .z-axS 73
7 4(x) -LI.-'

4I(x43

2tk3) -4 2.Lt. o'le'( .z-axS •Thegraph of2g(3x)isobtainedfromthegraphofg(x)byshrinking thehorizon talcoordinateby 1 3 ,an dstretching theverticalcoordinateby2. (You'dgetthes ameanswerhe reify ourevers edtheorderofthetransfor- mationsandstretchedve rticallyb y2beforeshrinkinghorizontall yby 1 3 .The orderisn't important.) 74
7: - (x) 4, 7c' 'I II 'I' -I -4-I -t N

Exercises

For#1-10, supposethatf(x)=x

8 .Mat cheachofthenu mberedfunctio ns ontheleft withtheletter edfunctionon theright thatitequals.

1.)f(x)+2 A.)( x)

8

2.)3f(x)B.)

1 3 x 8

3.)f(x)C.)x

8 2

4.)f(x2)D.)x

8 +2 5.) 1 3 f(x)E.)( x 3 8

6.)f(3x)F.) x

8

7.)f(x)2G.)(x2)

8

8.)f(x)H.) (3x)

8

9.)f(x+2)I.) 3x

8

10.)f(

x 3 )J.) (x+2) 8

For#11and# 12,suppos eg(x)=

1 x .Mat cheachofthenu mberedfunctio ns ontheleft withtheletter edfunctionon theright thatitequals.

11.)4g(3x7)+2A. )

6 2x+5 3

12.)6g(2x+5)3B.)

4 3x7 +2 75
Giventhegraph off(x)above,matchthefollowingfourfunctionswith theirgraphs.

13.)f(x)+214.) f(x)215.)f(x+2)16. )f(x2)

76
theirgraphs.

13.)f(x)+214.)f(x)215.)f(x+2)16.)f(x2)

60
3 -II~ g c-f f-Lit theirgraphs.

13.)f(x)+214.)f(x)215.)f(x+2)16.)f(x2)

60
3 (~f -t III I i-LI- 3 c-f I.' g (Lf~ 3.- -IIII~ 8 f-tm 7i~. (_(7L 3 II t' 8 (L~L

Exercises

For#1-10,supposef(x)=x

8 .Matcheachofthenumberedfunctionson

1.)f(x)+2.A.)(x)

8

2.)3f(x)B.)

1 3 x 8

3.)f(x)C.)x

8 2

4.)f(x2)D.)x

8 +2 5.) 1 3 f(x)E.)( x 3 8

6.)f(3x)F.)x

8

7.)f(x)2G.)(x2)

8

8.)f(x)H.)(3x)

8

9.)f(x+2)I.)3x

8

10.)f(

quotesdbs_dbs19.pdfusesText_25