Write the equation of the line in the diagram. The line described by your equation will also be drawn on the diagram.
\n$y=\\;$[[0]]
", "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "variableReplacements": [], "showFeedbackIcon": true, "marks": 0, "customMarkingAlgorithm": "", "type": "gapfill", "scripts": {}, "unitTests": [], "gaps": [{"answer": "{a}x+{b}", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "showPreview": true, "variableReplacements": [], "vsetRange": [0, 1], "checkVariableNames": false, "showFeedbackIcon": true, "marks": 1, "failureRate": 1, "expectedVariableNames": [], "customMarkingAlgorithm": "", "type": "jme", "scripts": {}, "unitTests": [], "answerSimplification": "all", "vsetRangePoints": 5, "checkingType": "absdiff", "extendBaseMarkingAlgorithm": true}], "extendBaseMarkingAlgorithm": true}], "metadata": {"description": "There are copious comments in the definition of the function eqnline about the voodoo needed to have a JSXGraph diagram interact with the input box for a part.
", "licence": "Creative Commons Attribution 4.0 International"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"description": "", "name": "a", "group": "Ungrouped variables", "definition": "random(-4..4 except 0)", "templateType": "anything"}, "x2": {"description": "", "name": "x2", "group": "Ungrouped variables", "definition": "random(-3..3 except -1..1)", "templateType": "anything"}, "y2": {"description": "", "name": "y2", "group": "Ungrouped variables", "definition": "x2*a+b", "templateType": "anything"}, "b": {"description": "", "name": "b", "group": "Ungrouped variables", "definition": "random(-6..6 except [0,a])", "templateType": "anything"}}, "functions": {"eqnline": {"parameters": [["a", "number"], ["b", "number"], ["x2", "number"], ["y2", "number"]], "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: [-13,16,13,-16],\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]], { var yticks = board.create('ticks',[yaxis,2],{\nstrokeColor: 'black', fixed: true });\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\n// create the static line based on the coefficients a and b\nvar line1 = board.create('line',[[0,b],[1,a+b]],{fixed:true, strokeWidth: 1});\n\n// mark the two given points - one on the y-axis, and one at (x2,y2)\nvar p1 = board.create('point',[0,b],{fixed:true, size:3, name: 'P_1', face: 'cross'});\nvar p2 = board.create('point',[x2,y2],{fixed:true, size:3, name: 'P_2', face: 'cross'});\n\n// Now we can do the clever stuff with the student's answer!\n// We'll add a curve to the board which is a plot of a function we provide.\n// That function will parse the student's input and evaluate it.\n\n// The variable `studentExpression` will store the parsed version of\n// the student's expression.\nvar studentExpression;\n\n// This function evaluates the student's expression at a given point `t`.\nfunction makestudentline(x){\n // Create a JME scope with the variable x set to the given value.\n var nscope = new Numbas.jme.Scope([\nNumbas.jme.builtinScope,\n{variables: {x: new Numbas.jme.types.TNum(x)}}\n ]);\n \n // If the student's input has been parsed, evaluate it\n if(studentExpression) {\ntry {\n var val = Numbas.jme.evaluate(studentExpression,nscope).value;\n return val;\n}\ncatch(e) {\n // If there was an error evaluating the student's expression\n // (wrong variables, or some other weirdness)\n // throw an error\n throw(e)\n}\n }\n // Otherwise, if the student's expression hasn't been parsed\n // (they haven't written anything, or they wrote bad syntax)\n // return 0\n else {\nreturn 0;\n }\n}\nvar studentline = board.create('functiongraph', \n [makestudentline,-13,13],\n {strokeColor:'black', strokeWidth: 3, visible: false}\n );\n\n// This is where some voodoo happens.\n// Because the HTML for the question is inserted into the page after the function eqnline\n// is called, we need to wait until the 'question-html-attached' event is fired\n// to do the interaction with the student input box.\n// So:\n\n// When the question is inserted into the page\nquestion.signals.on('HTMLAttached',function(e) {\n \n // Create a Knockout.js observable\n ko.computed(function(){\n// Get the student's input string from part 0, gap 0.\nvar studentString = question.parts[0].gaps[0].display.studentAnswer();\n\n// Try to parse it as a JME expression\ntry {\n studentExpression = Numbas.jme.compile(studentString,scope);\n \n // If the student didn't write anything, compile returns null\n if(studentExpression === null)\nthrow(new Error('no expression'));\n \n // If everything worked, show the line and update it\n // (this calls makestudentline on a few points)\n studentline.showElement();\n studentline.updateCurve();\n}\ncatch(e) {\n // If something went wrong, hide the curve\n studentExpression = null;\n studentline.hideElement();\n}\n\nboard.update();\n });\n}); \n\nreturn div;", "type": "html", "language": "javascript"}}, "name": "Michael's copy of Use student input in a JSXGraph diagram", "advice": "\nFirst Method.
\nYou are given that the line goes through $(0,\\var{b})$ and $(-1,\\var{b-a})$ and the equation of the line is of the form $y=ax+b$
\nHence:
\n1) At $x=0$ we have $y=\\var{b}$, and this gives $\\var{b}=a \\times 0 +b =b$ on putting $x=0$ into $y=ax+b$.
\nSo $b=\\var{b}$.
\n2) At $x=-1$ we have $y=\\var{b-a}$, and this gives $\\var{b-a}=a \\times (-1) +b =\\simplify[all,!collectNumbers]{-a+{b}}$ on putting $x=-1$ into $y=ax+b$.
\nOn rearranging we obtain $a=\\simplify[all,!collectNumbers]{{b}-{b-a}}=\\var{a}$.
\nSo $a=\\var{a}$.
\nSo the equation of the line is $\\simplify{y={a}*x+{b}}$.
\nSecond Method.
\nThe equation $y=ax+b$ tells us that the graph crosses the $y$-axis (when $x=0$) at $y=b$.
\nSo looking at the graph we immediately see that $b=\\var{b}$.
\n$a$ is the gradient of the line and is given by the change from $(-1,\\var{b-a})$ to $(0,\\var{b})$:
\n\\[a=\\frac{\\text{Change in y}}{\\text{Change in x}}=\\frac{\\simplify[all,!collectNumbers]{({b-a}-{b})}}{-1-0}=\\var{a}\\]
\n\n", "extensions": ["jsxgraph"], "statement": "{eqnline(a,b,x2,y2)}
\nThe above graph shows a line which has an equation of the form $y=ax+b$, where $a$ and $b$ are integers.
\nYou are given two points on the line, $(0,\\var{b})$ and $(\\var{x2},\\var{y2})$, as indicated on the diagram.
", "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Michael Hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1625/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Michael Hoffmann", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1625/"}]}