// Numbas version: finer_feedback_settings
{"name": "Equation of a line through two points", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>We find the equation of a straight line passing through two points by finding the gradient and the $y$-intercept of the line.</p>\n<h4>a)</h4>\n<p>We&nbsp;can find the gradient ($m$) using the points $A = (x_1,y_1)=(\\var{xa},\\var{ya})$ and $B = (x_2,y_2)=(\\var{xb},\\var{yb})$.</p>\n<p>As the definition of gradient is the ratio of vertical change ($y_2-y_1$) to horizontal change ($x_2-x_1$).<br/>The equation for gradient is,</p>\n<p>\\begin{align}<br/>m &amp;= \\frac{y_2-y_1}{x_2-x_1} \\\\[0.5em]<br/>&amp;= \\frac{\\simplify[!collectNumbers]{{yb}-{ya}}}{\\simplify[!collectNumbers]{{xb}-{xa}}} \\\\[0.5em]<br/>&amp;= \\frac{\\simplify[]{{yb}-{ya}}}{\\simplify{{xb}-{xa}}} \\\\[0.5em]<br/>&amp;=&nbsp;\\simplify[simplifyFractions,unitDenominator]{({yb-ya})/({xb-xa})}\\text{.}<br/>\\end{align}</p>\n<h4>b)</h4>\n<p>Rearranging the equation $y=mx+c$ and substituting either of the points gives</p>\n<p>\\[c = y_1-mx_1 \\quad \\mathrm{or} \\quad c = y_2-mx_2 \\,\\text{.} \\]</p>\n<p>We can then also use this equation with the other point's coordinates to check our answer.</p>\n<p>Let's use point $A$ first:</p>\n<p>\\[<br/>\\begin{align}<br/>c &amp;= y_1-mx_1 \\\\<br/>&amp;=&nbsp;\\var{ya}-\\var[fractionnumbers]{m}\\times\\var{xa} \\\\<br/>&amp; =&nbsp;\\simplify[fractionnumbers]{{ya-m*xa}}\\text{.}<br/>\\end{align}<br/>\\]</p>\n<p>We then check this against point $B$:</p>\n<p>\\[<br/>\\begin{align}<br/>y_2 &amp;= mx_2 + c \\\\[0.5em]<br/>&amp;= \\simplify[fractionNumbers]{{m}{xb}+{c}} \\\\[0.5em]<br/>&amp;= \\var[fractionnumbers]{m*xb+c}\\text{.}<br/>\\end{align}<br/>\\]</p>\n<h4>c)</h4>\n<p>We can now substitute these values for $m$ and $c$ into $y=mx+c$ &nbsp;to get:</p>\n<p>\\[y=\\simplify[!noLeadingMinus,fractionNumbers,unitFactor]{{m} x+ {c}}\\text{.}\\]</p>\n<p>The green line drawn on the graph represents the above line equation.</p>\n<p>{correctPoints()}</p>", "metadata": {"description": "<p>Use two points on a line graph to calculate the gradient and $y$-intercept and hence the equation of the straight line running through both points.</p>\n<p>The answer box for the third part plots the function which allows the student to check their answer against the graph before submitting.</p>\n<p>This particular example has a positive gradient.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "rulesets": {}, "preamble": {"css": "", "js": ""}, "variables": {"yb": {"description": "", "name": "yb", "group": "Ungrouped variables", "definition": "ya+random([2,4])", "templateType": "anything"}, "ya": {"description": "", "name": "ya", "group": "Ungrouped variables", "definition": "random(-4..2)", "templateType": "anything"}, "m": {"description": "", "name": "m", "group": "Ungrouped variables", "definition": "(ya-yb)/(xa-xb)", "templateType": "anything"}, "c": {"description": "", "name": "c", "group": "Ungrouped variables", "definition": "ya-m*xa", "templateType": "anything"}, "xb": {"description": "", "name": "xb", "group": "Ungrouped variables", "definition": "xa+random([2,4] except -xa)", "templateType": "anything"}, "xa": {"description": "", "name": "xa", "group": "Ungrouped variables", "definition": "random(-4..-1)", "templateType": "anything"}}, "parts": [{"useCustomName": false, "gaps": [{"useCustomName": false, "customName": "", "showCorrectAnswer": true, "unitTests": [], "marks": 1, "customMarkingAlgorithm": "", "maxValue": "m", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showFractionHint": true, "minValue": "m", "correctAnswerFraction": true, "showFeedbackIcon": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "allowFractions": true}], "showCorrectAnswer": true, "unitTests": [], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>Calculate the gradient, $m$, of the straight line between these two points.</p>\n<p>$m=$&nbsp;[[0]]</p>\n<p></p>"}, {"useCustomName": false, "gaps": [{"useCustomName": false, "customName": "", "showCorrectAnswer": true, "unitTests": [], "marks": 1, "customMarkingAlgorithm": "", "maxValue": "c", "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "showFractionHint": true, "minValue": "c", "correctAnswerFraction": false, "showFeedbackIcon": true, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "mustBeReduced": false, "extendBaseMarkingAlgorithm": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "allowFractions": false}], "showCorrectAnswer": true, "unitTests": [], "marks": 0, "sortAnswers": false, "showFeedbackIcon": true, "scripts": {}, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "customName": "", "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "customMarkingAlgorithm": "", "prompt": "<p>Use this gradient and the coordinates of the points to calculate the $y$-intercept, $c$.</p>\n<p>$c=$&nbsp;[[0]]</p>"}], "functions": {"correctPoints": {"type": "html", "parameters": [], "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[Math.min(-1,xa-2),Math.max(2,yb+2,c+1),Math.max(2,xb+2),Math.min(-1,ya-2,c-1)],grid: true});\nvar board = div.board;\nquestion.board = board;\n\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\n\n\n//ans(was tree) is defined at the end and nscope looks important\n//but they're both variables\n\nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"green\",setLabelText:'mx+c',visible: true, strokeWidth: 4, highlightStrokeColor: 'green'} )\n\n\n\nquestion.signals.on('HTMLAttached',function(e) {\nko.computed(function(){\n//define ans as this \ncorrect_line.updateCurve();\nboard.update();\n});\n  });\n\n\nreturn div;", "language": "javascript"}, "plotPoints": {"type": "html", "parameters": [], "definition": "//point coordinate variables\nvar xa = Numbas.jme.unwrapValue(scope.variables.xa);\nvar xb = Numbas.jme.unwrapValue(scope.variables.xb);\nvar ya = Numbas.jme.unwrapValue(scope.variables.ya);\nvar yb = Numbas.jme.unwrapValue(scope.variables.yb);\nvar m = Numbas.jme.unwrapValue(scope.variables.m);\nvar c = Numbas.jme.unwrapValue(scope.variables.c);\n\n//make board\nvar div = Numbas.extensions.jsxgraph.makeBoard('400px','400px',{boundingBox:[Math.min(-1,xa-2),Math.max(2,yb+2,c+1),Math.max(2,xb+2),Math.min(-1,ya-2,c-1)],grid: true});\nvar board = div.board;\n\n//points (with nice colors)\nvar a = board.create('point',[xa,ya],{name: 'A', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow', fixed: true, showInfobox: true});\nvar b = board.create('point',[xb,yb],{name: 'B', size: 7, fillColor: 'blue' , strokeColor: 'lightblue' , highlightFillColor: 'lightblue', highlightStrokeColor: 'yellow',fixed: true, showInfobox: true});\nvar correct_line = board.create('functiongraph',[function(x){ return m*x+c},-22,22], {strokeColor:\"blue\",setLabelText:'mx+c', strokeWidth: 4, highlightStrokeColor: 'green'} )\n\nreturn div;", "language": "javascript"}}, "variablesTest": {"maxRuns": 100, "condition": "\n"}, "statement": "<p>In this question we will identify&nbsp;the equation of the straight line passing&nbsp;through points &nbsp;$A=(\\var{xa},\\var{ya})$ and &nbsp;$B=(\\var{xb},\\var{yb})$ in the form $y = mx + c$.</p>\n<p>{plotPoints()}</p>", "extensions": ["jsxgraph"], "tags": [], "variable_groups": [], "ungrouped_variables": ["xa", "xb", "ya", "yb", "m", "c"], "name": "Equation of a line through two points", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Heather Driscoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1703/"}, {"name": "Philip Charlton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3474/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Bradley Bush", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1521/"}, {"name": "Aiden McCall", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1592/"}, {"name": "Heather Driscoll", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1703/"}, {"name": "Philip Charlton", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3474/"}]}