The graph of a function $y=f(x)$ is shown below. Move the red points so the red curve represents $y=\\simplify[fractionNumbers,all]{{vsc}f({hsc}x+{hsh})+{vsh}}$.
", "variablesTest": {"condition": "", "maxRuns": 100}, "rulesets": {"std": ["all", "fractionNumbers"]}, "ungrouped_variables": ["selector", "vsh", "hsh", "vsc", "hsc", "yo", "yn", "xo", "xn", "yo0", "yo1", "yo2", "yo3", "yo4", "maxx", "maxy", "recip"], "advice": "", "name": "Lovkush's copy of Graphs: Two random transformations", "question_groups": [{"questions": [], "pickQuestions": 0, "name": "", "pickingStrategy": "all-ordered"}], "variable_groups": [], "functions": {}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "notes": "", "description": "Horizontal and vertical shifts and scales of a random cubic spline
"}, "variables": {"maxx": {"definition": "max(map(abs(a),a,xn)+5)+1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "maxx"}, "yo": {"definition": "repeat(random(-5..5),5)", "templateType": "anything", "description": "the (random) original y values which relate to the x values
", "group": "Ungrouped variables", "name": "yo"}, "selector": {"definition": "shuffle([0,0,1,1])", "templateType": "anything", "description": "order is ['vsh','hsh','vsc','hsc'] 1 is on 0 is off
", "group": "Ungrouped variables", "name": "selector"}, "yo2": {"definition": "yo[2]", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "yo2"}, "vsh": {"definition": "if(selector[0]=1,random(-3..3#0.5 except 0),0)\n", "templateType": "anything", "description": "vertical shift
", "group": "Ungrouped variables", "name": "vsh"}, "maxy": {"definition": "max(map(abs(a),a,yn)+5)+1", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "maxy"}, "yo3": {"definition": "yo[3]", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "yo3"}, "yn": {"definition": "map(vsc*y+vsh,y,yo)", "templateType": "anything", "description": "new y values after the transformation
", "group": "Ungrouped variables", "name": "yn"}, "yo1": {"definition": "yo[1]", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "yo1"}, "yo4": {"definition": "yo[4]", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "yo4"}, "vsc": {"definition": "if(selector[2]=1,random(-2,-1,-0.5,0.5,2),1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "vsc"}, "xn": {"definition": "map((x-hsh)/hsc,x,xo)", "templateType": "anything", "description": "new transformed x values
", "group": "Ungrouped variables", "name": "xn"}, "hsc": {"definition": "if(selector[3]=1,random(-2,-1,-0.5,0.5,2),1)", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "hsc"}, "xo": {"definition": "list(-2..2)", "templateType": "anything", "description": "original x values
", "group": "Ungrouped variables", "name": "xo"}, "hsh": {"definition": "if(selector[1]=1,random(-3..3 except 0),0)", "templateType": "anything", "description": "horizontal shift
", "group": "Ungrouped variables", "name": "hsh"}, "recip": {"definition": "1/hsc", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "recip"}, "yo0": {"definition": "yo[0]", "templateType": "anything", "description": "", "group": "Ungrouped variables", "name": "yo0"}}, "preamble": {"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;iThe point $A$ was $(-2,\\var{yo0})$ but it is now $\\big($[[0]],[[1]]$\\big)$.
The point $B$ was $(-1,\\var{yo1})$ but it is now $\\big($[[2]],[[3]]$\\big)$.
The point $C$ was $(0,\\var{yo2})$ but it is now $\\big($[[4]],[[5]]$\\big)$.
The point $D$ was $(1,\\var{yo3})$ but it is now $\\big($[[6]],[[7]]$\\big)$.
The point $E$ was $(2,\\var{yo4})$ but it is now $\\big($[[8]],[[9]]$\\big)$.
Suppose $x$ is substituted into the function $y=\\simplify[fractionNumbers,all]{{vsc}f({hsc}x+{hsh})+{vsh}}$. The order of operations tells us what happens to $x$, in our case the order is, $\\var{hsc}$ multiplies it , $\\var{hsh}$ is added to it, $f$ is taken of it, $\\var{vsc}$ multiplies it , $\\var{vsh}$ is added to it. In this case there are two transformations to the original graph.
\nThe $\\simplify[fractionNumbers,all]{f({hsc}x)}$ part of the equation means that to get the same $y$ value as the original graph the new $x$ value will have to be $2$ times $-2$ times the negative of half of the negative half of what it was before (so that when you multiply the new $x$ value by $\\var{hsc}$ you get the old one).
\nSince the $x$ value is displayed in the horizontal direction, this means we stretch or scale horizontally by a factor of $\\simplify[fractionNumbers,all]{{recip}}$, or in other words, compress horizontally by a factor of $\\simplify[fractionNumbers,all]{{hsc}}$.
\nThe $\\simplify[fractionNumbers,all]{f(x+{hsh})}$ means that to get the same $y$ value as the original graph the new $x$ value will have to be $\\var{abs(hsh)}$ units less greater than the original $x$ value (so that when you add $\\var{hsh}$ to subtract $\\var{abs(hsh)}$ from the new $x$ value you get the old one).
\nSince the $x$ value is displayed in the horizontal direction, this means we shift horizontally $\\var{abs(hsh)}$ units to the left right.
\nNotice that $x$ is modified twice before the function $f$ gets to it. 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 $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$.
\nMethod one (shift then scale)
\nThe equation $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$ tells us to take the old $x$ subtract $\\var{abs(hsh)}$ from it, add $\\var{abs(hsh)}$ to it, then multiply this result by $\\simplify[fractionNumbers, unitFactor, basic]{{recip}}$.
\nMethod two (scale then shift)
\nWe could rewrite $x_n=\\simplify[fractionNumbers, unitFactor, basic]{{recip}*(x_o-{hsh})}$ as $x_n=\\simplify[fractionNumbers, unitFactor, basic, simplifyFractions]{{recip}*x_o-{hsh/hsc}}$. This equation tells us to take the old $x$ value, multiply it by $\\simplify[fractionNumbers, unitFactor, basic]{{recip}}$, then subtract $\\simplify[fractionNumbers, unitFactor, basic]{{abs(hsh/hsc)}}$ from this result. add $\\simplify[fractionNumbers, unitFactor, basic]{{abs(hsh/hsc)}}$ to this result.
\n\nThe $\\simplify[fractionNumbers,all]{{vsc}f(x)}$ part of the equation means that the $y$ value of each point on the graph will be $\\var{vsc}$ times what they were before. half of what they were before. a negative half of what they were before. the negative of what it was before. Since the $y$ value is displayed in the vertical direction, this means we stretch or scale vertically by a factor of $\\var{vsc}$.
\nThe $\\simplify[fractionNumbers,all]{f(x)+{vsh}}$ part of the equation means that the $y$ value of each point on the graph will be $\\var{abs(vsh)}$ units greater less than they were before. Since the $y$ value is displayed in the vertical direction, this means we shift vertically by $\\var{vsh}$ units.
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