[PDF] Graph Transformations Only after f has done





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Graph Transformations

Only after f has done its job do you add d to get the new function f(x) + d. 67. ') g. Page 2. Because 



math1414-transformations-of-functions.pdf

function f (x) = x. 2 . Step 2: Now that we know the library function we will be using we need a set of points from the graph of f (x) = x. 2 to work with.



AP Calculus AB Sample Student Responses and Scoring

e ? ? ? ? ? ?. ?. = ?. = The slope of the line tangent to the graph of f at. 3. 2 x e ?. ?. 2?. 2 e ?. The absolute minimum value of f on 0. 2 x.



Posltlve +

(E) x = 2 is a vertical asymptote of the graph of f. The graph of a differentiable function f is shown above. ... 1(oJ: I ~ (01) ~ (I/O) ~ 5(1)=0. 7.



1.7 Transformations

Suppose we wanted to graph the function defined by the formula g(x) = f(x)+2. Let's take a minute to remind ourselves of what g is doing. We start with an input 



AP Calculus Exam Prep Assign13KEY

A) Find the maximum area of a rectangle that has two vertices on the x-axis and two vertices on the graph of f. Justify your answer. A = lw = 2x. ( )e x2. dA dx.



Functions and Their Graphs Jackie Nicholas Janet Hunter Jacqui

Sketch the graph of f(x)=3x ? x2 and find If y = x2 + 2x and x = (z ? 2)2 find y when z = 3. b. Given L(x)=2x + 1 and M(x) = x2 ? x



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Write a rule for g described by the transformations of the graph of f. 7. f(x) = x²; vertical stretch by a factor of 2 and a reflection in the x-axis followed 



AP Calculus AB Sample Student Responses and Scoring

7. 6. I. 5. I. A. I. 3. I l J. . J. -1 I. . -s-4 -3 -2 Ao 1 2 3 4 5 6 t 11_2. ._ I. -3. I. I. X. Graph of g. 3. The graph of the continuous function g



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Unauthorized copying or reuse of any part of this page is illegal. 2. If ƒ (x) = x³ ? x² + x ? 1 then ƒ'(2) = (A) 10. (B) 9. (C) 7 f'= 3x²-21.

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(

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