// Numbas version: finer_feedback_settings
{"name": "Transformaciones de funciones", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Transformaciones de funciones", "functions": {}, "rulesets": {"std": ["all", "fractionNumbers"]}, "preamble": {"css": "table#values th {\n  background: none;\n  text-align: center;\n}", "js": "function dragpoint_board() {\n  var scope = question.scope;\n  \n // var a = scope.variables.a.value;\n//  var c = scope.variables.c.value;\n\n  var yo0 = scope.variables.yo0.value;\n  var yo1 = scope.variables.yo1.value;\n  var yo2 = scope.variables.yo2.value;\n  var yo3 = scope.variables.yo3.value;  \n  var yo4 = scope.variables.yo4.value;\n    \n  var maxx = scope.variables.maxx.value;\n  var maxy = scope.variables.maxy.value;\n  \n  var div = Numbas.extensions.jsxgraph.makeBoard('500px','500px',{boundingBox:[-maxx,maxy,maxx,-maxy],grid:true});\n  $(question.display.html).find('#dragpoint').append(div);\n  \n  var board = div.board;\n  \n  \n   //create stationary points\n    \n  var op0 = board.create('point',[-2,yo0],{name:'',fixed:true,size:2,color:'black'});\n    var op1 = board.create('point',[-1,yo1],{name:'',fixed:true,size:2,color:'black'});\n    var op2 = board.create('point',[0,yo2],{name:'',fixed:true,size:2,color:'black'});\n    var op3 = board.create('point',[1,yo3],{name:'',fixed:true,size:2,color:'black'});\n    var op4 = board.create('point',[2,yo4],{name:'',fixed:true,size:2,color:'black'});\n    \n    \n    //create draggable points\n    //why are there are a cloine under each one?\n    var np0 = board.create('point',[-2,yo0],{name:'A',size:2,snapSizeX: 0.25,snapSizeY: 0.25,snapToGrid: true});\n    var np1 = board.create('point',[-1,yo1],{name:'B',size:2,snapSizeX: 0.25,snapSizeY: 0.25,snapToGrid: true});\n    var np2 = board.create('point',[0,yo2],{name:'C',size:2,snapSizeX: 0.25,snapSizeY: 0.25,snapToGrid: true});\n    var np3 = board.create('point',[1,yo3],{name:'D',size:2,snapSizeX: 0.25,snapSizeY: 0.25,snapToGrid: true});\n    var np4 = board.create('point',[2,yo4],{name:'E',size:2,snapSizeX: 0.25,snapSizeY: 0.25,snapToGrid: true});\n  \n  \n    \n  \n  //shorthand to evaluate a mathematical expression to a number\n  function evaluate(expression) {\n    try {\n      var val = Numbas.jme.evaluate(expression,question.scope);\n      return Numbas.jme.unwrapValue(val);\n    }\n    catch(e) {\n      // if there's an error, return no number\n      return NaN;\n    }\n  }\n  \n  // set up points array\n  var num_points = 5;\n  var points = [np0, np1, np2, np3, np4];\n  \n  \n  // this function sets up the i^th point\n  function make_point(i) {\n    \n    // calculate initial coordinates\n  //  var x = i-(num_points-1)/2;\n    \n    // create an invisible vertical line for the point to slide along\n   // var line = board.create('line',[[x,0],[x,1]],{visible: false});\n    \n    // create the point\n  //  var point = points[i] = board.create(\n   //   'point',\n   //   [i-(num_points-1)/2,0],\n   //   {\n    //    name:'',\n   //     size:2,\n    //    snapSizeY: 0.25,  // the point will snap to y-coordinates which are multiples of 0.1\n    //    snapSizeX: 0.25,\n   //     snapToGrid: true\n    //  }\n   // );\n    \n   var point = points[i];\n    \n    var x=point[0];\n    var y=point[1];\n    \n    // the contents of the input box for this point\n    var xstudentAnswer = question.parts[0].gaps[2*i].display.studentAnswer;\n    var ystudentAnswer = question.parts[0].gaps[2*i+1].display.studentAnswer;\n    \n    // watch the student's input and reposition the point when it changes.    \n    ko.computed(function() {\n      x = evaluate(xstudentAnswer());\n      y = evaluate(ystudentAnswer());\n      if(!(isNaN(x)) && !(isNaN(y)) && board.mode!=board.BOARD_MODE_DRAG) {\n        point.moveTo([x,y],100);\n      }\n    });\n    \n    // when the student drags the point, update the gapfill input\n    point.on('drag',function(){\n      var x = Numbas.math.niceNumber(point.X());\n      var y = Numbas.math.niceNumber(point.Y());\n      xstudentAnswer(x);\n      ystudentAnswer(y);\n    });\n    \n  }\n  \n  // create each point\n  for(var i=0;i<num_points;i++) {\n    make_point(i);\n  }\n  \n  // create the parabola\n  var parabool = board.create('spline',[np0, np1, np2, np3, np4],{strokeWidth:3,strokeColor:'red'});\n  var statparabool = board.create('spline',[op0, op1, op2, op3, op4],{strokeWidth:3,strokeColor:'black'});\n\n}\n\nquestion.signals.on('HTMLAttached',function() {\n  dragpoint_board();\n}); "}, "variables": {"hsh": {"name": "hsh", "definition": "if(selector[1]=1,random(-3..3 except 0),0)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>horizontal shift</p>"}, "yn": {"name": "yn", "definition": "map(vsc*y+vsh,y,yo)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>new y values after the transformation</p>"}, "xo": {"name": "xo", "definition": "list(-2..2)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>original x values</p>"}, "hsc": {"name": "hsc", "definition": "if(selector[3]=1,random(-2,-1,-0.5,0.5,2),1)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "xn": {"name": "xn", "definition": "map((x-hsh)/hsc,x,xo)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>new transformed x values</p>"}, "yo3": {"name": "yo3", "definition": "yo[3]", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "vsh": {"name": "vsh", "definition": "if(selector[0]=1,random(-3..3#0.5 except 0),0)\n", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>vertical shift</p>"}, "yo1": {"name": "yo1", "definition": "yo[1]", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "maxx": {"name": "maxx", "definition": "max(map(abs(a),a,xn)+5)+1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "yo": {"name": "yo", "definition": "repeat(random(-5..5),5)", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>the (random) original y values which relate to the x values</p>"}, "maxy": {"name": "maxy", "definition": "max(map(abs(a),a,yn)+5)+1", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "yo0": {"name": "yo0", "definition": "yo[0]", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "yo2": {"name": "yo2", "definition": "yo[2]", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "yo4": {"name": "yo4", "definition": "yo[4]", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "selector": {"name": "selector", "definition": "shuffle([0,0,1,1])", "templateType": "anything", "group": "Ungrouped variables", "description": "<p>order is ['vsh','hsh','vsc','hsc'] 1 is on 0 is off</p>"}, "recip": {"name": "recip", "definition": "1/hsc", "templateType": "anything", "group": "Ungrouped variables", "description": ""}, "vsc": {"name": "vsc", "definition": "if(selector[2]=1,random(-2,-1,-0.5,0.5,2),1)", "templateType": "anything", "group": "Ungrouped variables", "description": ""}}, "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Horizontal and vertical shifts and scales of a random cubic spline</p>"}, "parts": [{"showFeedbackIcon": true, "type": "gapfill", "gaps": [{"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "xn[0]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "xn[0]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "yn[0]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "yn[0]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "xn[1]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "xn[1]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "yn[1]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "yn[1]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "xn[2]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "xn[2]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "yn[2]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "yn[2]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "xn[3]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "xn[3]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "yn[3]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "yn[3]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "xn[4]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "xn[4]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}, {"showFeedbackIcon": true, "variableReplacements": [], "notationStyles": ["plain", "en", "si-en"], "minValue": "yn[4]", "scripts": {}, "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "maxValue": "yn[4]", "mustBeReducedPC": 0, "type": "numberentry", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "allowFractions": true, "marks": 1, "mustBeReduced": false}], "showCorrectAnswer": true, "steps": [{"showFeedbackIcon": true, "type": "information", "showCorrectAnswer": true, "marks": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "variableReplacements": [], "prompt": "<p>Vamos a suponer que $x$ es reemplazada en la funci&oacute;n, para establecer el orden en que las operaciones se realizan en la funci&oacute;n&nbsp;$y=\\simplify[fractionNumbers,all]{{vsc}f({hsc}x+{hsh})+{vsh}}$. Cuando evaluamos la funci&oacute;n con&nbsp;$x$:</p>\n<p><span data-jme-visible=\"hsc&lt;&gt;1\">Multiplicar $x$ por $\\var{hsc}$&nbsp;</span>&nbsp;</p>\n<p><span data-jme-visible=\"hsh&lt;&gt;0\">Operar con $\\var{hsh}$</span></p>\n<p>Lo anterior nos da $f$</p>\n<p><span data-jme-visible=\"vsc&lt;&gt;1\">Multiplicar $f$ por $\\var{vsc}$</span>&nbsp;</p>\n<p><span data-jme-visible=\"vsh&lt;&gt;0\">Operar el resultador anterior con $\\var{vsh}$</span>.</p>\n<p><span data-jme-visible=\"hsc&lt;&gt;1 &amp;&amp;hsh=0\">The $\\simplify[fractionNumbers,all]{f({hsc}x)}$ part of the equation means that to get the same $y$ value as the original graph the new $x$ value will have to be <span data-jme-visible=\"hsc=0.5\">$2$ times</span> <span data-jme-visible=\"hsc=-0.5\">$-2$ times</span> <span data-jme-visible=\"hsc=-1\">the negative of</span> <span data-jme-visible=\"hsc=2\">half of</span> <span data-jme-visible=\"hsc=-2\">the negative half of</span> what it was&nbsp;before&nbsp;(so that when you&nbsp;multiply the&nbsp;new $x$ value by $\\var{hsc}$ you get the old one).</span></p>\n<p><span data-jme-visible=\"hsc&lt;&gt;1 &amp;&amp;hsh=0\">Since the $x$ value is displayed in the horizontal&nbsp;direction, this means we&nbsp;stretch or&nbsp;scale horizontally&nbsp;by a factor of $\\simplify[fractionNumbers,all]{{recip}}$<span data-jme-visible=\"abs(hsc)&lt;&gt;1\">, or in other words, compress horizontally by a factor of $\\simplify[fractionNumbers,all]{{hsc}}$</span>.</span></p>\n<p><span data-jme-visible=\"hsh&lt;&gt;0 &amp;&amp; hsc=1\">The $\\simplify[fractionNumbers,all]{f(x+{hsh})}$ means that to get the same $y$ value as the original graph the new $x$ value will have to be $\\var{abs(hsh)}$ unit<span data-jme-visible=\"abs(hsh)&lt;&gt;1\">s</span>&nbsp;<span data-jme-visible=\"hsh&gt;0\">less</span> <span data-jme-visible=\"hsh&lt;0\">greater</span> than the original $x$ value (so that when you <span data-jme-visible=\"hsh&gt;0\">add $\\var{hsh}$ to</span>&nbsp;<span data-jme-visible=\"hsh&lt;0\">subtract $\\var{abs(hsh)}$ from</span>&nbsp;the new $x$ value you get the old one).</span></p>\n<p><span data-jme-visible=\"hsh&lt;&gt;0 &amp;&amp; hsc=1\">Since the $x$ value is displayed in the horizontal&nbsp;direction, this means we&nbsp;shift horizontally&nbsp;$\\var{abs(hsh)}$ unit<span data-jme-visible=\"abs(hsh)&lt;&gt;1\">s</span> to the&nbsp;<span data-jme-visible=\"hsh&gt;0\">left</span> <span data-jme-visible=\"hsh&lt;0\">right</span>.</span></p>\n<p><span data-jme-visible=\"hsc&lt;&gt;0 &amp;&amp; hsc&lt;&gt;1\">Notice that&nbsp;$x$ is modified twice before the function $f$ gets to it.&nbsp;If we want the same $y$ value as the original graph we need to feed $f$ with the same input. Let $x_o$ be the old $x$ value and $x_n$ be the new $x$ value. To get the same $y$ values we want $f(x_o)=\\simplify[fractionNumbers, unitFactor, basic]{f({hsc}x_n+{hsh})}$, that is, we require $x_o=\\simplify[fractionNumbers, unitFactor, basic]{{hsc}x_n+{hsh}}$. Rearranging we see $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$.&nbsp;</span></p>\n<p><span data-jme-visible=\"hsc&lt;&gt;0 &amp;&amp; hsc&lt;&gt;1\"><strong>Method one (shift then scale)</strong></span></p>\n<p><span data-jme-visible=\"hsc&lt;&gt;0 &amp;&amp; hsc&lt;&gt;1\">The equation $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$ tells us to take the old $x$ <span data-jme-visible=\"hsh&gt;0\">subtract $\\var{abs(hsh)}$ from it,</span>&nbsp;<span data-jme-visible=\"hsh&lt;0\">add $\\var{abs(hsh)}$ to it,</span>&nbsp;then&nbsp;multiply this result by&nbsp;$\\simplify[fractionNumbers, unitFactor, basic]{{recip}}$.</span></p>\n<p><span data-jme-visible=\"hsc&lt;&gt;0 &amp;&amp; hsc&lt;&gt;1\"><strong>Method two (scale then shift)</strong></span></p>\n<p><span data-jme-visible=\"hsc&lt;&gt;0 &amp;&amp; hsc&lt;&gt;1\">We could rewrite&nbsp;$x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$ as&nbsp;$x_n=\\simplify[fractionNumbers, unitFactor, basic, simplifyFractions]{{recip}*x_o-{hsh/hsc}}$. This equation tells us to take the old $x$ value, multiply it by $\\simplify[fractionNumbers, unitFactor, basic]{{recip}}$, then <span data-jme-visible=\"hsh/hsc&gt;0\">subtract $\\simplify[fractionNumbers, unitFactor, basic]{{abs(hsh/hsc)}}$&nbsp;from this result.</span>&nbsp;<span data-jme-visible=\"hsh/hsc&lt;0\">add $\\simplify[fractionNumbers, unitFactor, basic]{{abs(hsh/hsc)}}$ to this result.</span></span></p>\n<p></p>\n<p><span data-jme-visible=\"vsc&lt;&gt;1\">The $\\simplify[fractionNumbers,all]{{vsc}f(x)}$ part of the equation means that the&nbsp;$y$ value of each point on the graph will be <span data-jme-visible=\"abs(vsc)=2\">$\\var{vsc}$ times what they were before.</span>&nbsp; <span data-jme-visible=\"vsc=0.5\">half of what they were before.</span><span data-jme-visible=\"vsc=-0.5\">&nbsp;a negative half of what they were before.</span>&nbsp;<span data-jme-visible=\"vsc=-1\">the negative of what it was before.</span> Since the $y$ value is displayed in the vertical direction, this means we&nbsp;stretch or&nbsp;scale vertically by a factor of $\\var{vsc}$.&nbsp;</span></p>\n<p><span data-jme-visible=\"vsh&lt;&gt;0\">The $\\simplify[fractionNumbers,all]{f(x)+{vsh}}$ part of the equation means that the $y$ value of each point on the graph will be $\\var{abs(vsh)}$ unit<span data-jme-visible=\"abs(vsh)&lt;&gt;1\">s</span>&nbsp;<span data-jme-visible=\"vsh&gt;0\">greater</span> <span data-jme-visible=\"vsh&lt;0\">less</span> than they were before. Since the $y$ value is displayed in the vertical direction, this means we&nbsp;shift vertically by $\\var{vsh}$ unit<span data-jme-visible=\"abs(vsh)&lt;&gt;1\">s</span>.</span></p>"}], "marks": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "stepsPenalty": "10", "variableReplacements": [], "prompt": "<div id=\"dragpoint\"></div>\n<p>El punto&nbsp;$A$ $(-2,\\var{yo0})$ se debe mover a&nbsp;$\\big($[[0]],[[1]]$\\big)$.<br/>El punto $B$ $(-1,\\var{yo1})$ se debe mover a $\\big($[[2]],[[3]]$\\big)$.<br/>El punto $C$ $(0,\\var{yo2})$ se debe mover a $\\big($[[4]],[[5]]$\\big)$.<br/>El punto $D$ $(1,\\var{yo3})$ se debe mover a $\\big($[[6]],[[7]]$\\big)$.<br/>El punto $E$ $(2,\\var{yo4})$ se debe mover a $\\big($[[8]],[[9]]$\\big)$.</p>"}], "statement": "<p>La gr&aacute;fica de color Negro corresponde a la funci&oacute;n&nbsp;$y=f(x)$. Mueva o ingrese las coordenadas de los puntos de color rojo, de manera que estos representen la gr&aacute;fica de color Rojo simbolizada por $y=\\simplify[fractionNumbers,all]{{vsc}f({hsc}x+{hsh})+{vsh}}$.</p>", "ungrouped_variables": ["selector", "vsh", "hsh", "vsc", "hsc", "yo", "yn", "xo", "xn", "yo0", "yo1", "yo2", "yo3", "yo4", "maxx", "maxy", "recip"], "advice": "", "variable_groups": [], "extensions": ["jsxgraph"], "variablesTest": {"maxRuns": 100, "condition": ""}, "type": "question", "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}]}], "contributors": [{"name": "Marlon Arcila", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/321/"}]}