{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 "" -1 256 "" 0 1 0 0 0 0 0 1 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 0 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 128 128 1 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "" 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 258 49 "High School Modules > Algebra by Gregory A. Moore" }}{PARA 3 "" 0 "" {TEXT 257 22 " Functions & Gra phs" }}{PARA 0 "" 0 "" {TEXT -1 132 "\nThe connection between function s and their graphs, the graphs of common functions, and the horizontal line test for function-hood.\n" }}{PARA 0 "" 0 "" {TEXT 259 153 "[Dir ections : Execute the Code Resource section first. Although there will be no output immediately, these definitions are used later in this wo rksheet.]" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 7 "0. Code" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "resta rt; with(plots): " }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 22 "1. Functio ns & Graphs " }}{PARA 0 "" 0 "" {TEXT -1 303 "\nFunctions can be expre ssed as graphs too. For each value of x, there is a value of f(x). If \+ we let y = f(x), we have a point (x,y). If we graph all of these point s, we have a \"picture\" of what the function looks like. The graph of a function tells us many thing about the relationship between x and y .\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> x^3 - 20*x ^2 + 100*x - 40; \nplot( f(x), x = -3..18, y = -120..120, thickness = \+ 2, color = green);" }}}{PARA 0 "" 0 "" {TEXT -1 219 "\nFunctions can t ake many different forms. Some of these functions are beyond the scope of your algebra class, but you can just back and enjoy the scenery fo r a moment. You don't have to be a chef to enjoy a great meal. " }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "f := x -> sqrt( abs(sin(x) \+ + cos(x) )): \nplot( f(x), x = 0..4*Pi, y = 0..1.3, thickness = 2, col or = gold);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "f := x -> e xp( x - 3*sqrt(x)*sin(x) ) - exp( x + 3*sqrt(x)*cos(x) ); \nplot( f(x) , x = 0..3, thickness = 2, color = magenta);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "f := x -> floor(1 + sin(8*x)) * x;\nplot( f(x), x = 0..10, thickness = 2, color = aquamarine, discont = true);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "f := x -> sin( exp(1/cos(x)) ); \nplot( f(x), x = 0..3, thickness = 1, color = green);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "f := x -> floor( 5*tan(x) ) ; \nplot( f(x), x = 0..2*Pi, y = -10..10, \n thickness = 3 , color = tan, discont = true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "f := x -> sqrt( x + abs(sin(x) - cos(x) )); \nplot( f(x), x = 0..10*Pi, y = 0..6, thickness = 2, color = blue);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 105 "f := x -> exp( x/3 ) + x*(1 + sin(x))*cos( x^1.5);\nplot( f(x), x = 0..10, thickness = 2, color = sienna);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "f := x -> abs(x) - abs(x-3) + abs(x-5) - abs(x-9);\nplot( f(x), x = -5..14, thickness = 2, color = orange, discont = true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 19 "2. Common Functions" }} {PARA 0 "" 0 "" {TEXT -1 88 "\nLets examine some of the garden variety functions we'll find sprouting up in the text.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "f := x -> 3*x - 4; plot( f(x), x = -3..3, \n \+ title=\"(non vertical) lines are functions too!\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "f := x -> x^2; plot( f(x ), x = -3..3, title=\"Classic Parabola\");" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 75 "f := x -> -x^2; plot( f(x), x = -3..3, title=\"Cla ssic Inverted Parabola\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "f := x -> 12-(x-3)^2; plot( f(x), x = -2..7, title=\"Another Para bola\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := x -> abs(x ); plot( f(x), x = -3..3, title=\"Classic Absolute Value\");" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "f := x -> sqrt(x); plot( f( x), x = -1..16, title=\"Squareroot\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "f := x -> x^3; plot( f(x), x = -3..3,y = -15..15, ti tle=\"Classic Cubic\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 " f := x -> x^3 - 4*x; plot( f(x), x = -3..3,y = -10..10, title=\"Curvy Cubic\");" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 110 "f := x -> 1/ x; plot( f(x), x = -3..3, y = -10..10, \n title= \"Reciprocal\", discont = true);" }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n " }}}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 21 "3. Vertical Line Test" }} {PARA 0 "" 0 "" {TEXT -1 90 "\nWhen looking at a graph it is fairly ea sy to see if its the graph of a function or not.\n\n" }{TEXT 256 43 "V ertical Line Test - (tests for a function)" }{TEXT -1 223 "\n\nIf EVER Y vertical line crosses a graph in at most one point, then the graph i s the graph of a function. Here are some examples of graphs which FAIL S this test. Thus none of these are functions. They are merely relatio ns.\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 220 "expr := 4*x^2 - 3*y^ 2 + y^3 = 1;\nplots[display](\n implicitplot( expr ,x=-2..2, y=-2. .4,color = blue, thickness = 2),\n plot( \{seq([[k/4,-2],[k/4,4]], k=-8..8)\}, x=-2..2, y= -2..4, \n color = gold));" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 192 "expr := y^2 = x + 1.5;\npl ots[display](\n implicitplot( expr ,x=-2..2, y=-2..2,color = blue, thickness = 2),\n plot( \{seq([[k/4,-2],[k/4,2]], k=-8..8)\}, x=- 2..2, y= -2..2, color = gold));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "expr := 3*(x)^2 + 9*(y)^4 = 8;\nplots[display](\n implicitp lot( expr ,x=-2..2, y=-1..1,\n color = blue, thickness = 2, numpoints = 1000),\n plot( \{seq([[k/4,-1],[k/4,1]], k=-8..8 )\}, x=-2..2, y= -1..1, color = gold));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 84 "\nHere are some exam ples which PASS the Vertical line test - thus they are functions." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 188 "f := x -> x^2;\nplots[displ ay](\n plot( f(x) ,x=-2..2, y=-2..4,color = green, thickness = 2), \n plot( \{seq([[k/4,-2],[k/4,4]], k=-8..8)\}, x=-2..2, y= -2..4, \+ \n color = tan));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 196 "f := x -> 2+2*sin(Pi*x);\nplots[display](\n plot( f(x) ,x=-2. .2, y=0..4,color = green, thickness = 2),\n plot( \{seq([[k/4,-2], [k/4,4]], k=-8..8)\}, x=-2..2, y= 0..4, \n color = tan));" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 206 "f := x -> 1 + abs(x+1)-ab s(x-1);\nplots[display](\n plot( f(x) ,x=-2..2, y=-2..4,color = gr een, thickness = 2),\n plot( \{seq([[k/4,-2],[k/4,4]], k=-8..8)\}, x=-2..2, y= -2..4, \n color = tan));" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 204 "f := x -> 1/x;\nplots[display](\n plot( f (x) ,x=-2..2, y=-5..5,color = green, thickness = 2, discont = true),\n plot( \{seq([[k/4,-5],[k/4,5]], k=-8..8)\}, x=-2..2, y= -5..5, \n color = tan));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT 260 36 " \n \251 2002 Waterloo Maple Inc " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{MARK "0 0" 12 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 } {PAGENUMBERS 0 1 2 33 1 1 }