We know that the graph crosses the $x$-axis at both $(\\var{a},0)$ and $(\\var{b},0)$. Since this is a quadratic, we know our equations has two roots, and by the previous observation, they are at $\\var{a}$ and $\\var{b}$. Hence we can write our equation as $\\simplify{y=(x-{a})(x-{b})}$ which simplifies to $\\simplify{y=x^2-({a}+{b})x+({a}*{b})}$.
\n\nTo find the coefficients of the turning point of the quadratic, we know the x-coordinate of the turning point will correspond to the solution to $dy/dx=0$. So we get $\\simplify{2x-({a}+{b})}=0$ hence $\\simplify{x=({a}+{b})/2}$. We substitute this value of x back into the equation of the quadratic to find the corresponding y-coordinate.
", "rulesets": {}, "showQuestionGroupNames": false, "variables": {"c": {"description": "", "definition": "a*b", "templateType": "anything", "name": "c", "group": "Ungrouped variables"}, "b": {"description": "", "definition": "random(-5..5 except [0,a,-a])", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "a": {"description": "", "definition": "random(-4..4 except 0)", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}, "x2": {"description": "", "definition": "random(-3..3 except -1..1)", "templateType": "anything", "name": "x2", "group": "Ungrouped variables"}, "y2": {"description": "", "definition": "x2*a+b", "templateType": "anything", "name": "y2", "group": "Ungrouped variables"}}, "functions": {"eqnline": {"type": "html", "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);},-13,13]);\n\nreturn div;", "parameters": [["a", "number"], ["b", "number"], ["x2", "number"], ["y2", "number"]]}}, "statement": "{eqnline(a,b,x2,y2)}
\nThe above graph shows a graph of a quadratic equation, it is your task to find this equation.
\nYou are given the two points of the curve with the x axis, $(\\var{b},0)$ and $(\\var{a},0)$, and the $y$-intercept at $(0,\\var{c})$ as indicated on the diagram.
", "tags": [], "variable_groups": [], "parts": [{"marks": 0, "type": "gapfill", "prompt": "Write the equation of the line in the diagram.
\n$y=\\;$[[0]]
", "gaps": [{"marks": 1, "type": "jme", "variableReplacements": [], "answersimplification": "all", "answer": "x^2-({a}+{b})x+{a}{b}", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showpreview": true, "vsetrange": [0, 1], "checkvariablenames": false, "checkingtype": "absdiff", "expectedvariablenames": [], "scripts": {}, "showCorrectAnswer": true}], "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}, {"marks": 0, "type": "gapfill", "prompt": "Find the coordinates of the turning point of this quadratic
\n$x=$[[0]]
\n$y=$[[1]]
", "gaps": [{"marks": 1, "type": "jme", "checkvariablenames": false, "answer": "({a}+{b})/2", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showpreview": true, "vsetrange": [0, 1], "variableReplacements": [], "checkingtype": "absdiff", "expectedvariablenames": [], "scripts": {}, "showCorrectAnswer": true}, {"marks": 1, "type": "jme", "checkvariablenames": false, "answer": "(-{b}^2-{a}^2+2{a}{b})/4", "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "checkingaccuracy": 0.001, "showpreview": true, "vsetrange": [0, 1], "variableReplacements": [], "checkingtype": "absdiff", "expectedvariablenames": [], "scripts": {}, "showCorrectAnswer": true}], "variableReplacements": [], "scripts": {}, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "extensions": ["jsxgraph"], "preamble": {"js": "", "css": ""}, "question_groups": [{"name": "", "pickQuestions": 0, "pickingStrategy": "all-ordered", "questions": []}], "metadata": {"description": "Students enter equation and turning point
", "licence": "Creative Commons Attribution 4.0 International"}, "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}]}], "contributors": [{"name": "Adrian Jannetta", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/164/"}]}