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.
", "statement": "{eqnline(a,b,x2,y2,v,degrees)}
\nThe Blue graph shows a graph of a quadratic equation, $f(x)=sin(x)$ while v={v} and degrees = {degrees}; {degrees/180}
", "name": "David's copy of Sin graph wavelength and amplitude", "functions": {"eqnline": {"type": "html", "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('600px','600px',\n{boundingBox: [-6,1.75,6,-1.75],\n axis: false,\n showNavigation: true,\n //grid: true\n grid: true\n});\n\n\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});\n//var xticks = board.create('ticks',[xaxis,60],{\nvar xticks = board.create('ticks',[xaxis,1],{\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,0.5],{\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\n//board.create('functiongraph',[function(x){ return (x-a)*(x-b);},-8,8]);\n//board.create('functiongraph',[function(x){ return (x-a)*(x-b)+v;},-8,8],{ strokeColor: 'red'});\n\n//board.create('functiongraph',[function(x){ return Math.sin(x*(Math.PI/180));},-360,360]);\n//board.create('functiongraph',[function(x){ return Math.sin((x-60*v)*(Math.PI/180));},-360,360],{ strokeColor: 'red'});\n//board.create('functiongraph',[function(x){ return Math.sin((x+60*v)*(Math.PI/180));},-360,360],{ strokeColor: 'black'});\nboard.create('functiongraph',[function(x){ return Math.sin(x);},-7,7],{ strokeColor: 'blue'});\nboard.create('functiongraph',[function(x){ return Math.sin(x-(degrees/180)*Math.PI);},-7,7],{ strokeColor: 'red'});\nreturn div;", "language": "javascript", "parameters": [["a", "number"], ["b", "number"], ["x2", "number"], ["y2", "number"], ["v", "number"], ["degrees", "number"]]}}, "preamble": {"js": "", "css": ""}, "variable_groups": [], "rulesets": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Given th original formula the student enters the transformed formula
\nREADY TO USE
"}, "parts": [{"prompt": "The Blue graph has been transformed onto the red graph $g(x)$, type in the equation of the line red line.
\n$g(x)=\\;$[[0]]
", "gaps": [{"vsetRangePoints": 5, "answer": "sin(x-{degrees})", "answerSimplification": "all", "checkVariableNames": false, "failureRate": 1, "customMarkingAlgorithm": "", "checkingAccuracy": "0.01", "type": "jme", "vsetRange": [0, 1], "scripts": {}, "showFeedbackIcon": true, "marks": "2", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "checkingType": "absdiff", "variableReplacements": [], "showCorrectAnswer": true, "unitTests": [], "expectedVariableNames": [], "showPreview": true}], "customMarkingAlgorithm": "", "type": "gapfill", "scripts": {}, "showFeedbackIcon": true, "marks": 0, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showCorrectAnswer": true, "unitTests": [], "sortAnswers": false}], "ungrouped_variables": ["a", "x2", "b", "y2", "c", "v", "degrees"], "tags": [], "type": "question", "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}, {"name": "David Rickard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/451/"}]}]}], "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}, {"name": "David Rickard", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/451/"}]}