{VERSION 5 0 "IBM INTEL NT" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 128 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 128 128 128 1 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 0 "" 0 "" {TEXT 257 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT 256 42 " Function Graphi ng - Shifting & Lifting" }}{PARA 0 "" 0 "" {TEXT -1 56 "\nExploration \+ of linear translation of common functions.\n" }}{PARA 0 "" 0 "" {TEXT 258 579 "[Directions : Execute the Code Resource section first. Althou gh there will be no output immediately, these definitions are used lat er in this worksheet.\n\nOn each of the animation plots : click once o n the graphic - you'll see a black frame appear around the graph. Then go to the Maple tool bar - and click the PLAY button (small triangle) to play the animation.\n\nNote : This work sheet has a great deal of \+ graphics which take up computer resources. If your computer runs out o f RAM, or you experience crashes, re-execute the CODE section before e ach of the subsequent sections.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 22 "restart; with(plots): " }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 53 "blip := (t, u, v) -> Heaviside(t-u) - Heaviside(t-v ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 491 "ShiftPlot := proc( f , left, right, a, b)\n local A, B, C, D1, D2;\n A := plot( f(x), x = left..right, color = green, linestyle = 2):\n B := plot( f(x-a )+b, x = (left+a)..(right+a), color = blue, thickness = 2):\n C := \+ plot( [[0,.05],[a,.05],[a,b]], color = gold, \n \+ thickness = 1, linestyle = 3):\n D1 := plottools[disk]([0,0], .10, color=red):\n D2 := plottools[disk]([a,b], .10, color=red):\n p lots[display](A,B,C,D1,D2, scaling=constrained);\nend proc:\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 802 "ShiftPlotSolid := proc( f, \+ left, right, a, b)\n local A,A2,B,B2, C, D1, D2, c1,c2, d;\n c1 \+ := COLOR(RGB, .8, .8, .6);\n c2 := COLOR(RGB, .6, .6, .5);\n d : = .01* f(right) ;\n A := plot( f(x), x = left..right, color = c1, \+ color = c1, filled = true):\n A2 := plot( f(x), x = left..right, c olor = c2, thickness = 2):\n B := plot( f(x-a)+b, x = (left+a)..(r ight+a), color = c1, filled = true):\n B2 := plot( f(x-a)+b, x = (l eft+a)..(right+a), color = c2, thickness = 2):\n C := plot( [[0,.0 5],[a,.05],[a,b]], color = gold, \n thickness = 1, linestyle = 3):\n D1 := plottools[disk]([0,0], d, color=red):\n D2 := plottools[disk]([a,b], d, color=red):\n plots[display]([D 1,D2, A,B2,A2,C], scaling=constrained);\nend proc:\n\nShiftPlotSolid(x ->x^2, -3, 5, 2, 2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1265 "ShiftAnimate := proc(f,left,right,bot,top,a,b)\n local v1,v2,ar1, ar2,A,B,C,wid,domain,range;\n\n v1 := [a, 0]: v2 := [a,b]: wid := m in(a,b)/12:\n domain := left..right; range := bot..top;\n ar1 := plo ts[arrow]( v1,\n shape = double_arrow, color = gold , difference = \+ true,\n width = wid, head_width = 1.8*wid, head_length = 1.2*wid): \n ar2 := plots[arrow]( v1, v2,\n shape = double_arrow, color = or ange , difference = true,\n width = wid, head_width = 1.8*wid, head _length = 1.8*wid):\n\n\n A:=animate( f(x) *blip(t, -.01,.2)\n + f(x- a*(5*t-1)) *blip(t, .2, .4)\n \+ + f(x-a) *blip(t, .4, .6)\n + (f(x-a) + b*(5*t-3)) *blip(t, .6, .8)\n + (f(x-a) + b) *bl ip(t, .8, 1.20)\n , x = domain, t=0..1, view=[domain,r ange],frames=36, \n color = blue, thickness = 2):\n B:=animat e( f(x) *blip(t, .01, .7)\n , x = domain, t=0.. 1, view=[domain,range],frames=36, \n color = gold, thick ness = 1):\n C:=animate( f(x-a) *blip(t, .41, 1.00)\n \+ , x = domain, t=0..1, view=[domain,range],frames=36, \n \+ color = green, thickness = 2):\nplots[display](A,B,C,ar1, ar2,sc aling=constrained);\n end proc:\n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 12 "1. Parabola s" }}{PARA 0 "" 0 "" {TEXT -1 103 "\nWhenever a function y = f(x) is r eplaced by (y-b) = f(x-a), of y = f(x-a) + b, the graph will shifted \+ " }{TEXT 259 1 "a" }{TEXT -1 25 " units right, and lifted " }{TEXT 260 1 "b" }{TEXT -1 152 " units up. In particular, when f(x) = x^2, th en (y-b) = (x-a)^2 will have the same shape, but moved accordingly. Le ts see an animation of this change.\n\n" }{TEXT 261 15 " Exampl e" }{TEXT -1 3 " : " }{XPPEDIT 18 0 "y - 4 = (x-2)^2" "6#/,&%\"yG\"\" \"\"\"%!\"\"*$,&%\"xGF&\"\"#F(F," }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> x^2;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 32 "ShiftPlotSolid( f, -3, 5, 2, 4);" }}}{PARA 0 "" 0 " " {TEXT -1 184 " \nHere is an animation which shows the shift. Click o n the graph to make the black border appear around it. Then go to the \+ Maple tool bar and hit the play button to play the animation." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ShiftAnimate(f,-3, 5, 0, 9, \+ 2, 4);\n\n " }}}{PARA 0 "" 0 "" {TEXT -1 10 "\nWhile (x-" }{TEXT 263 1 "a" }{TEXT -1 38 ") causes a shift of a units right, (x+" }{TEXT 264 1 "a" }{TEXT -1 9 ") = (x-(-" }{TEXT 265 1 "a" }{TEXT -1 21 ")) ca uses a shift of " }{TEXT 267 2 "-a" }{TEXT -1 29 " units right, which \+ is means " }{TEXT 266 1 "a" }{TEXT -1 90 " units left. It works the sa me way with y-b...where (y+b) means a shift of b units down.\n\n" } {TEXT 262 15 " Example" }{TEXT -1 3 " : " }{XPPEDIT 18 0 "y+12 \+ = (x+6)^2;" "6#/,&%\"yG\"\"\"\"#7F&*$,&%\"xGF&\"\"'F&\"\"#" }{TEXT -1 3 ". \n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ShiftPlotSolid( \+ f, -7,4, -6, -12);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 18 "2. Absolute V alues" }}{PARA 0 "" 0 "" {TEXT -1 58 "\nThe same principle works for t he absolute value function\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "f := x -> abs(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "Sh iftAnimate(f,-3, 8, 0, 9, 5, 1);\nShiftPlotSolid( f, -3, 8, 5, 1);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ShiftAnimate(f,-3, 8, -4, 6 , 1, -4);\nShiftPlotSolid( f, -3, 8, 1, -4);" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 8 "3. Cubes" }} {PARA 0 "" 0 "" {TEXT -1 51 "\nOf course this also applies to the cubi c function\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> x^ 3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "ShiftAnimate(f, -3, 3 , -6, 6, 4,12);\nShiftPlotSolid( f, -3, 3, 4, 12);" }}}{PARA 0 "" 0 " " {TEXT -1 97 "\nHere is another type of cubic function, but the princ iple of shifting and lifting still applies." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 92 "f := x -> x^3 - 2*x;\nShiftAnimate(f,-3, 3, -6, 6, \+ -4, 15);\nShiftPlotSolid( f, -3, 3, -3,15);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "4. Square Roots" }} {PARA 0 "" 0 "" {TEXT -1 138 "\nThe squareroot is limited in its domai n to [0, infinity), but it shifts in the same way, and its domain shif ts left or right accordingly\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "f := x -> sqrt(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "ShiftAnimate(f,0, 9, 0, 6, 5, 2);\nShiftPlotSolid( f,0, 8, 5, 2);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "ShiftAnimate( f, -4, 8, - 6, 6, -1, 3);\nShiftPlotSolid( f, -4, 8, -1, 3);" }}}{PARA 0 " " 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "5. Rec iprocals" }}{PARA 0 "" 0 "" {TEXT -1 342 "\nUnlike all of the function s above, the reciprocal function is slightly more difficult to graph. \+ This is because the basic form of the parabola, cubic, absolute value, and square root all pass through the point (0,0). On the other hand, \+ the reciprocal does not. It does pass through (1,1), and this point ex periences the same shift and lift.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> 1/x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "ShiftAnimate(f,-3, 6, -5, 6, 1, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "ShiftAnimate(f,-8, 8, -5, 4, -5, -1);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 15 "6. Ex ponentials" }}{PARA 0 "" 0 "" {TEXT -1 230 "\nLike the reciprocal func tion, the exponential function is slightly more difficult to graph bec ause it does not go through the origin. It does go through the point ( 0,1), and this point gets moved according to the same princples.\n" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f := x -> 2^x;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ShiftAnimate(f,-3, 6, -6, 6, 4, 2); \nShiftPlotSolid( f, -3, 6, 4, 2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "ShiftAnimate(f,-3, 5, -6, 6, 5, 2);\nShiftPlotSolid( \+ f, -3, 5, 5, 2);" }}}{EXCHG }{EXCHG }{EXCHG }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 268 36 "\n \251 2002 Waterloo Ma ple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 26 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }