// Numbas version: exam_results_page_options
{"name": "Graphing: semi-circles centred at the origin", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variables": {"r": {"definition": "shuffle(1..12)[0..5]", "templateType": "anything", "group": "Ungrouped variables", "name": "r", "description": ""}}, "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "<p>Recognising the equation of a semi-circle centred at the origin. Top, bottom, left and right.</p>"}, "statement": "<p>Being able to identify the equation of a semi-circle can come in handy, not just as an exercise in graphing, relations or functions, but also for more advanced things like integration using trigonometric substitution.&nbsp;</p>", "name": "Graphing: semi-circles centred at the origin", "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": [], "rulesets": {}, "tags": ["circles", "Graphing", "graphing", "semi-circles"], "advice": "<p>The equation of a circle centred at the origin of radius $r$ is normally written as \\[x^2+y^2=r^2.\\]&nbsp;</p>\n<p>Solving for $y$ gives the equation \\[y=\\pm\\sqrt{r^2-x^2}.\\]</p>\n<p>This is the equation for the same circle, it is just written differently. Notice we needed to&nbsp;take the plus or minus <g class=\"gr_ gr_15 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling\" id=\"15\" data-gr-id=\"15\">square-root</g> to undo the square. As you would expect, the plus sign allows&nbsp;$y$ to have positive values and the minus&nbsp;sign allows&nbsp;$y$ to have positive values. This means the equation \\[y=\\sqrt{r^2-x^2}\\] is the top half of the circle centred at the origin with radius $r$, and \\[y=-\\sqrt{r^2-x^2}\\] is the bottom half of the circle.</p>\n<p>Solving the original equation of the circle for $x$ gives the equation&nbsp;\\[x=\\pm\\sqrt{r^2-y^2}.\\]</p>\n<p>Again, this is the equation for the same circle, it is just written differently. Notice the plus or minus sign giving both positive and negative $x$ values. If we only want the <g class=\"gr_ gr_20 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling multiReplace\" id=\"20\" data-gr-id=\"20\">right hand</g> side of the circle we restrict ourselves to the positive $x$ values and so the equation of that semi-circle would be \\[x=\\sqrt{r^2-y^2}.\\]</p>\n<p>However, if we only want the <g class=\"gr_ gr_21 gr-alert gr_spell gr_inline_cards gr_run_anim ContextualSpelling multiReplace\" id=\"21\" data-gr-id=\"21\">left hand</g> side of the circle we restrict ourselves to the negative $x$ values, and our equation would be \\[x=-\\sqrt{r^2-y^2}.\\]</p>\n<p></p>", "functions": {}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "parts": [{"shuffleAnswers": true, "maxMarks": 0, "scripts": {}, "marks": 0, "warningType": "none", "minMarks": 0, "choices": ["<p>The&nbsp;left half of a&nbsp;circle</p>", "<p>The right&nbsp;half of a&nbsp;circle</p>", "<p>The top&nbsp;half of a&nbsp;circle</p>", "<p>The bottom&nbsp;half of a&nbsp;circle</p>", "<p>An entire circle</p>"], "type": "m_n_x", "variableReplacements": [], "matrix": [["1", 0, 0, 0, 0], [0, "1", 0, 0, 0], [0, 0, "1", 0, 0], [0, 0, 0, "1", 0], [0, 0, 0, 0, "1"]], "maxAnswers": 0, "layout": {"expression": "", "type": "all"}, "prompt": "<p>Match the equations with the descriptions by clicking on the corresponding buttons.</p>", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showFeedbackIcon": true, "displayType": "radiogroup", "answers": ["<p>$x=-\\sqrt{\\var{r[0]}^2-y^2}$</p>", "<p>$x=\\sqrt{\\var{r[1]}^2-y^2}$</p>", "<p>$y=\\sqrt{\\var{r[2]}^2-x^2}$</p>", "<p>$y=-\\sqrt{\\var{r[3]}^2-x^2}$</p>", "<p>$y=\\pm\\sqrt{\\var{r[4]}^2-x^2}$</p>"], "minAnswers": 0, "shuffleChoices": true}], "ungrouped_variables": ["r"], "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}