// Numbas version: exam_results_page_options {"name": " Cubic graph - student finds equation", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"ungrouped_variables": ["a", "x2", "b", "y2", "c", "coeff2", "coeff1", "coeff0", "y"], "preamble": {"css": "", "js": ""}, "variable_groups": [], "metadata": {"description": "

Students are given a graph of a cubic, and are asked to find the equation for it.

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{eqnline(a,b,x2,y2)}

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The above graph shows a graph of a cubic equation, it is your task to find this equation.

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You are given the three points of the curve with the x axis, \$(\\var{x2},0)\$, \$(\\var{b},0)\$ and \$(\\var{a},0)\$, and the \$y\$-intercept at \$(0, \\var{y})\$ as indicated on the diagram.

", "name": " Cubic graph - student finds equation", "parts": [{"variableReplacementStrategy": "originalfirst", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacementStrategy": "originalfirst", "answer": "x^3+{coeff2}x^2+{coeff1}x+{coeff0}", "variableReplacements": [], "checkvariablenames": false, "showCorrectAnswer": true, "vsetrangepoints": 5, "vsetrange": [0, 1], "expectedvariablenames": [], "answersimplification": "all", "showpreview": true, "checkingaccuracy": 0.001, "checkingtype": "absdiff", "type": "jme", "marks": "2", "scripts": {}}], "prompt": "

Write the equation of the curve in the diagram.

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\$y=\\;\$[[0]]

", "scripts": {}}], "functions": {"eqnline": {"type": "html", "parameters": [["a", "number"], ["b", "number"], ["x2", "number"], ["y2", "number"]], "language": "javascript", "definition": "// This function creates the board and sets it up, then returns an\n// HTML div tag containing the board.\n \n// The line is described by the equation \n// y = a*x + b\n\n// This function takes as its parameters the coefficients a and b,\n// and the coordinates (x2,y2) of a point on the line.\n\n// First, make the JSXGraph board.\n// The function provided by the JSXGraph extension wraps the board up in \n// a div tag so that it's easier to embed in the page.\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',\n{boundingBox: [-13,22,13,-22],\n axis: false,\n showNavigation: false,\n grid: true\n});\n \n// div.board is the object created by JSXGraph, which you use to \n// manipulate elements\nvar board = div.board; \n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,2],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,2],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n// mark the two given points - one on the y-axis, and one at (x2,y2)\n\n\n\n\nboard.create('functiongraph',[function(x){ return (x-a)*(x-b)*(x-x2);},-13,13]);\n\nreturn div;"}}, "tags": [], "rulesets": {}, "variables": {"a": {"templateType": "anything", "description": "", "definition": "random(-4..4 except 0)", "name": "a", "group": "Ungrouped variables"}, "coeff0": {"templateType": "anything", "description": "", "definition": "-{a}{b}{x2}", "name": "coeff0", "group": "Ungrouped variables"}, "coeff1": {"templateType": "anything", "description": "", "definition": "({a}{b}+{b}{x2}+{a}{x2})", "name": "coeff1", "group": "Ungrouped variables"}, "c": {"templateType": "anything", "description": "", "definition": "random(-5..5 except [0,b,-b,a,-a])", "name": "c", "group": "Ungrouped variables"}, "coeff2": {"templateType": "anything", "description": "", "definition": "-({a}+{b}+{x2})", "name": "coeff2", "group": "Ungrouped variables"}, "x2": {"templateType": "anything", "description": "", "definition": "random(-4..4 except 0 except a except -a)", "name": "x2", "group": "Ungrouped variables"}, "b": {"templateType": "anything", "description": "", "definition": "random(-5..5 except [0,a,-a,x2])", "name": "b", "group": "Ungrouped variables"}, "y2": {"templateType": "anything", "description": "", "definition": "x2*a+b", "name": "y2", "group": "Ungrouped variables"}, "y": {"templateType": "anything", "description": "", "definition": "-a*b*x2", "name": "y", "group": "Ungrouped variables"}}, "extensions": ["jsxgraph"], "advice": "

We know that the graph crosses the \$x\$-axis at \$(\\var{x2},0)\$, \$(\\var{a},0)\$ and \$(\\var{b},0)\$. Since this is a cubic, we know our equations has three roots, and by the previous observation, they are at \$\\var{x2}\$, \$\\var{a}\$ and \$\\var{b}\$. Hence we can write our equation as \$\\simplify{y=(x-{a})(x-{b})(x-{x2})}\$ which simplifies to \$\\simplify{y=x^3+{coeff2}x^2+{coeff1}x+{coeff0}}\$.

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