ࡱ> ')&7  bjbjUU .:7|7| %llllxxxxTA2Y&Y&Y&lAnAnAnAnAnAnA$C ?EAlY&$lY&Y&Y&A.AA...Y&VllA.Y&lA..q5r>Tl? @ȑx(F??A0A?E7,VE?.Transformation of Functions: Part 2 If we start with the function y = f(x) We can apply several transformations to this function to produce a new equation in terms of f(x) outside of the function  y = a f (k(x - d)) + c  inside function where x and y are the independent and dependent variables respectively. f( ) is some function. k is a horizontal compression d is a horizontal shift a is a vertical expansion c is a vertical shift Exercise Given a starting function f(x), express the second function in terms of f(x) and determine the values for k, d, a and c. a)  EMBED Equation.3  (  EMBED Equation.3  b)  EMBED Equation.3  (  EMBED Equation.3  Some Hints to Help Understand Transformations There are a few tips to help you remember what the constants a, c, k, d do to the graph of the function. Multipliers such as a and k expand and compress Numbers that add or subtract such as c and d shift the function. Constants inside the function deal with horizontal transformations. Constants outside the function deal with vertical transformations. For constants inside the function; THINK OPPOSITE! Large values of k actually compress the function rather than expand. When a number is added to x, it actually shifts the function left not right. Order of Transformations The order for which transformations are carried out is critical to producing the correct graph for a given function. Procedure: 1. Deal with transformations inside the function first but perform them in the opposite order of operations; ie SAMDEB. 2. Deal with the transformations outside of the function and perform them in the order of operations; ie BEDMAS. For a function expressed in the form, y = a f (k(x - d)) + c We should perform transformations dealing with the constants k, d, a, c in that order. Activity Graph the parent functions on the left by using the table of values, then graph the functions on the right using a set of transformations. Show any rough work on the back of this page.  a)  EMBED Equation.3   EMBED Equation.3  xy -2-1012  b)  EMBED Equation.3   EMBED Equation.3  xy 0149 c)  EMBED Equation.3   EMBED Equation.3  xy -2-1012 x y x y y x y x y x y x %CDNOZ[nopqrsuvijI H* jEHUjxc~M 5CJUV\aJ jvEHUjY~M 5CJUV\aJ j,EHUjY~M 5CJUV\aJ j jEHUjY~M 5CJUV\aJ jUjCJUmHnHuCJ>*3$%CDNOIbV$ & Fa$$  a$$ a$$a$ VWij'j/ 0 $^a$$ & Fa$$p^pa$$h^ha$$a$0 I J E 4 N O X Y    d$Q[&#$/If d rI J E N O X     , - . / q r       = > @ C D W ʾvp 5CJ\j5CJ EHU\jb~M CJUVaJj 5CJ EHU\jc~M CJUVaJjv 5CJ EHU\jwb~M CJUVaJjK 5CJ EHU\jc~M CJUVaJj5CJ U\j5CJU\mHnHu>* 5CJ \ 5>*CJ \) wwwwwwww d$Q[&#$/Ift$$Ifl0 xx 6Q[064 la wqqqqqqqqqq dt$$Ifl0 xx 6Q[064 la d$Q[&#$/If  ! $ % ' ( ) + , - / 0 1 qqqqt$$Ifl0h 6Q[064 la d$Q[&#$/If d 1 3 4 5 6 7 8 9 : ; < = > wqqqqqqqqqq dt$$Ifl0h 6Q[064 la d$Q[&#$/If W X Y Z ˿׹ 5CJH*\ 5CJ\j5CJ EHU\jd~M CJUVaJ 5CJ \j5CJ U\jr5CJ EHU\jc~M CJUVaJ/ wwwwt$$Ifl0 L 6Q[064 la d$Q[&#$/If wwqqqqqqo dt$$Ifl0 L 6Q[064 la d$Q[&#$/If  1h/ =!"#$%nLPNG  IHDR,,\ѩbKGD#2 cmPPJCmp0712`IDATx^흍:E/V>baӻNq ae^zaKX,a@ :KX,a@ :KX,a@Rd2?) @L5H3\Z!1+НB`~4QvC YiP~.~^~7wxn.W?|:n.~~o`+0.1ΚM1+w2a^ Y4&7|lpζzr!p}DXTgV9E4YCL+ N +rTnhݰ;n"B4*f㊂`Y7L[}Y B*,@]XTgXQ Aк!0|ԘEXTgunjV_o+ZX叵}7,ך:pmo|pa1Geݐ lm0F3,P,KX4*7к!pxЅ :kϰg=0U,0(֗Ŭy3+7kSкaǽe|5הpTygOd f:w+X~)4$siZa : nhxRcBa R[X7:J>I)Џ¢rC]ϭmq)Z7p:t@HYe,JE:Q|«߈k+`}tĄ%,@Hu [7n\{t /gݿ8KXU'͓/as?٬nqc)XӤj)P[Z_SkUC_qiku~hrڃ*[ZĪ*_ӚN~_ƦWӆ~7`[KUR /zs X=mPU4/^?XSuB=ƍp>yZ5eעq٦U&:YkphHG+fЦ;Qպe-f=cUM֏;Ho=?Oʵ \]YOoXR/l'.3CLC4ו+ʽhfԕzb57|P?t[{[s/gI3_A=Ҡ _Z_PϠk;"Rmu~mOt*,a@+îo 4Y_">f`j֯ @57oxɜo9eglz*{  Y+bV8.a YV\n8X~k/;l ⾖噌xWXTg 8g?(|V9U :zuiV'v+r,HrCoXr׹hӹp|f3 K2{ aeiV>v+Xq笵 P YVVnxmuSRs:?}X7+>a R[X7:ojV2f uGrcI;Șr)Z]K rA֗Ŭ=FoƷp+ArHFcV+2a Y4*7ܰr?ҥZ!0|EXTgun*^i]+rTnx{脹a37Guސ`ݰNtxQQKf9+@ KX,+*7n\?Rca RGX=7vuʾĬ51g[l,^aiԘ[haܦp>u ?c`uwB!/,a@ :KX,a@ :KX,a@ : ciIENDB`,DdhJ  C A? "2|^ÛueΥ)jDw`!b|^ÛueΥ)r@@8 |0xcdd``dd``baV d,FYzP1n:&&6! KA?H110| UXRY7&,e`abM-VK-WMcpX cF\_UBkGy 3@FhVpDwCasdBܞ'&? P 27)?!BDdm%4v16h0y{qĤ\Y\d.P"CDHg!t?2eyJDd|hJ  C A? "2I$;9Ȕ,5b8+pw`!I$;9Ȕ,5b8+` @|Nxcdd``dd``baV d,FYzP1n:&.! KA?H1 \ ߁qC0&dT20 KXB2sSRs~.b~i,@u@@ڈkc'HI% ׄY|S8?׀0ƕ('= ]_bAFUb' W&00zNBeׇd.XPŰ",` =q ~e%4v)lи`` #RpeqIj.C5E.B ~fNπ_DdL|J  C A? "2f$ǿӜw`!f$ǿӜ` 0cxcdd``dd``aaV d,FYzP1n:lB@?b 10sC0&dT200pE1At 2Ba`<b 30~fU@QUTQ{ъ՜kN 1 $ L &OA $37X/\!(?71arO VpZO0MbF\OtVTOQM ~Z%} Bܤ\E > ᰏH0f.I 7wM=Ѩ&80@``#RpeqIj.E.fB ~b>t&vDd0|J  C A? "2Q};2=w`!Q};2= `H0zxڕR=K`m,UH",&tq$VT0Vi"uu'.;&1<T8mbGmE(h>>J\874x(+ǮӰ]S(Z"[kʇw^\|`B/5+/$ts'sz_x2^[,D>|h^1sj-#g{SnPiHH ͖Fq> *fJhd4Y ex,+Dd@J  C A? "2k?oؠ#+5i w`!ak?oؠ#+5: /xcdd``ed``baV d,FYzP1n:&6! KA?H1:l ǀqC0&dT20 KXB2sSRs~.b?m:&?0b鬨0@Jt@AI9 W> ᰏH0f.I 7wM=Ѩ&80@``#RpeqIj.t= @ ]` @YO`h  !"#$%>(+,-/.021345768:9;<=BC?@ADEFHIJKLMNOPQRSTUVWXYZ[\]^_`abcdefghiRoot Entry7 F hґ*Data WordDocument6.:ObjectPool9 hґ_1300126176FOle CompObjfObjInfo  #&),/258;=>?@ABCEFGHIJKLMNOPQRSTUVWXYZ[\^ FMicrosoft Equation 3.0 DS Equation Equation.39q.H! f(x)=x 2 FMicrosoft Equation 3.0 DS EqEquation Native J_1300126132 FOle CompObj fuation Equation.39q?m y=3(x"5) 2 "4 FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo Equation Native  [_1300126165 F$$Ole  CompObj fObjInfoEquation Native F_1300128632"Fpnpn* f(x)= x  FMicrosoft Equation 3.0 DS Equation Equation.39q;5l y="5 2x"6  +4Ole CompObjfObjInfoEquation Native W_1300128726FpߵpߵOle CompObjfObjInfo FMicrosoft Equation 3.0 DS Equation Equation.39q)hl f(x)=|x| FMicrosoft Equation 3.0 DS EqEquation Native E_1300128375'F`)`)Ole CompObj fuation Equation.39q)   y=|2x"8| FMicrosoft Equation 3.0 DS Equation Equation.39qObjInfo!!Equation Native "E_1300128734,$F` ` Ole $CompObj#%%fObjInfo&'Equation Native (F_1300128402)FPUPU*q% f(x)= x  FMicrosoft Equation 3.0 DS Equation Equation.39q7l y=3 2x+2  "5Ole *CompObj(*+fObjInfo+-Equation Native .S_13001287421.FPPOle 0CompObj-/1fObjInfo03 FMicrosoft Equation 3.0 DS Equation Equation.39q.0r f(x)=1x FMicrosoft Equation 3.0 DS EqEquation Native 4J_13001289993F@đ@đOle 6CompObj247fuation Equation.39q.$l y=42x"8Oh+'0 , H T ` lx$Transformation of Functions: Part 2ObjInfo59Equation Native :J1TableGESummaryInformation(8<soDd|J  C A? "2`zB5Ta8Gw`!`zB5Ta8 `F0sxڕR=KA|]QDⴰPbB**1rFH VV)m [[+KHQٽ; 7f"D X^1) !}p.ȥ#]Ngf%ZO<,KD2 &ԧM+Z;w!)o+x2r{P 4mY*vOboE5t.D[ISCwO`Ӳ]l3(܏;;2k<iRs^t Qy>Ucu*ynٿ}/Ye["8Ոwnέ}E{ܾ^:^${To J^0bp Y>Py?DdlJ   C A ? "2֔z:TiMBtV$ }w`!u֔z:TiMBtV$ VL Cxcdd``f 2 ĜL0##0KQ* W􀙁MRcgbR v@=P5< %!@5 @_L ĺEvX@ V;35v! @Hfnj_jBP~nb@|"Z?LF]F\=X@:+*!'&?>c<Ƌ`{ `5|;4 ?@penR~C#WG!3dm%4.v0o8^+KRsA2u(2t5|b@#3X?XTjyADd8lJ   C A ? " 2lN%, +& `p221)W2lԡRYݏ`thranJohnforohnohnNormalrOwnerr2neMicrosoft Word 9.0u@H'@4A@jz@jzl՜.+,D՜.+,P  hp|  o  DocumentSummaryInformation8DCompObj]j$Transformation of Functions: Part 2 Title 8@ _PID_HLINKSAl$dhCC:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gifdhCC:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gifdhCC:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gifdhCC:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gifdh!CC:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gifdh%CC:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif  FMicrosoft Word Document MSWordDocWord.Document.89q i8@8 NormalCJ_HaJmH sH tH B@B Heading 1$ d@& 5>*CJ \<A@< Default Paragraph Font.>@. Title$a$ 5CJ \:J@: Subtitle  r 5>*CJ \8B@8 Body Text  r 5CJ \ !$ '&#" !$'    2$%CDNOIbVWij'j/0IJE4NOXY ! $ % ' ( ) + , - / 0 1 3 4 5 6 7 8 9 : ; < = > 000000000000000 0 0 0 0 0 00000000000000000000000 0 0 0 0 000000000000000000 00r0000r0r0r00000000000000000r0r0r0r0r0r00000r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r0r00000000000000000000000r0r0r0r0r0000000000000000000000000I W V0 1  Znpu,.q   C W Y ::::::::::l'(,b$L "2w@'b(  HB  C DHB  C DHB  C DNB  S DNB @ S DNB  S DXh 'v.Z, #  #"    c A  C:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif[image]"`'-Z,f  S  6$]& f  S  O,H$v.% Xh 'v.Z, #  #"   c A  C:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif[image]"`'-Z,f  S  6$]& f  S  O,H$v.% Xh 'v.Z, #  #"    c A  C:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif[image]"`'-Z,f  S   6$]&  f  S   O,H$v.%  Xh 'v.Z, #  #"    c A  C:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif[image]"`'-Z,f  S   6$]&  f  S   O,H$v.%  Xh 'v.Z,  #  #"  ! c A  C:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif[image]"`'-Z, f " S " 6$]& f # S # O,H$v.% Xh 'v.Z, $#  #"  % c A  C:\Program Files\TI Education\TI InterActive!\TIIimagefile5023.gif[image]"`'-Z,f & S & 6$]& f ' S ' O,H$v.% B S  ?> ? dpt$ d t$  ptTt T t t @t$4@(t4(BtBt4c(tD4tKNKMTWZACGHijhj/0IJ8:>?NOXY $ % ' ) + - / 1 3 @ C EFJKLMNNcdJKSTZqsu@AFG$Yhjo894578=>NO/q   ! " C Z 33333333333333333333333333333333333333KNaFFJNSTZqu@AFG$Yhjo78=>NOcj/q  < = @ @ C Z JohnbC:\Documents and Settings\Owner\Application Data\Microsoft\Word\AutoRecovery save of Document1.asdJohn5D:\Teaching\mcr3u_TransformationsofFunctionsPart2.docJohn~C:\Documents and Settings\Owner\Application Data\Microsoft\Word\AutoRecovery save of mcr3u_TransformationsofFunctionsPart2.asdJohn5D:\Teaching\mcr3u_TransformationsofFunctionsPart2.docJohn~C:\Documents and Settings\Owner\Application Data\Microsoft\Word\AutoRecovery save of mcr3u_TransformationsofFunctionsPart2.asdJohn5D:\Teaching\mcr3u_TransformationsofFunctionsPart2.docJohnID:\Teaching\Lourdes Winter 2009\mcr3u_TransformationsofFunctionsPart2.docOwnerlC:\Users\Owner\AppData\Roaming\Microsoft\Word\AutoRecovery save of mcr3u_TransformationsofFunctionsPart2.asdOwnerIC:\Teaching\Lourdes Winter 2009\mcr3u_TransformationsofFunctionsPart2.docOwner5C:\Teaching\mcr3u_TransformationsofFunctionsPart2.docAI;B$jkukT=5l ^`o()^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.h ^`OJQJo(h pp^p`OJQJo(oh @ @ ^@ `OJQJo(h ^`OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h PP^P`OJQJo(oh   ^ `OJQJo(^`.^`.pLp^p`L.@ @ ^@ `.^`.L^`L.^`.^`.PLP^P`L.h ^`OJQJo(h pp^p`OJQJo(oh @ @ ^@ `OJQJo(h ^`OJQJo(h ^`OJQJo(oh ^`OJQJo(h ^`OJQJo(h PP^P`OJQJo(oh   ^ `OJQJo(5l;BkukA         v<                 v<         ! $ % ' ( ) + , - / 0 1 3 4 5 @| | 41| |  `@UnknownG:Ax Times New Roman5Symbol3& :Cx Arial;Wingdings?5 :Cx Courier New"hnu&nu& f l!0d / 2Q#Transformation of Functions: Part 2JohnOwner