// Numbas version: exam_results_page_options
{"name": "Graphing: cubic", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Graphing: cubic", "variablesTest": {"condition": "", "maxRuns": 100}, "showQuestionGroupNames": false, "statement": "<p>You are given the equation&nbsp;$y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$.&nbsp;</p>", "advice": "", "type": "question", "question_groups": [{"pickingStrategy": "all-ordered", "name": "", "questions": [], "pickQuestions": 0}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "parts": [{"scripts": {}, "displayType": "radiogroup", "marks": 0, "stepsPenalty": "1", "prompt": "<p>This equation, or its graph, can be described as a</p>", "maxMarks": 0, "steps": [{"scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>An equation of the form $y=ax^3+bx^2+cx+d$ is&nbsp;known as&nbsp;a cubic, or a cubic polynomial. If we expand&nbsp;$y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$ we will see it is a cubic.</p>", "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "type": "1_n_2", "distractors": ["", "", "", "", "", ""], "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "choices": ["<p>straight line</p>", "<p>parabola/quadratic</p>", "<p>cubic</p>", "<p>hyperbola</p>", "<p>circle</p>", "<p>quartic</p>"], "shuffleChoices": true, "matrix": [0, "0", "1", 0, 0, 0], "minMarks": 0}, {"scripts": {}, "displayType": "radiogroup", "marks": 0, "stepsPenalty": "1", "prompt": "<p>As we move to the far left of the graph, the graph</p>", "maxMarks": 0, "steps": [{"scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.</p>\n<p>The leading term (the term that includes the highest power) determines the long term behaviour of a&nbsp;polynomial.</p>\n<p>By expanding $y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$ we see that the leading term is $\\simplify[all,fractionNumbers]{{a}x^3}$.&nbsp;</p>\n<p>As we go far to the left of the graph $x$ is negative, and so $\\simplify[all,fractionNumbers]{{a}x^3}$&nbsp;<span data-jme-visible=\"a&gt;0\">is negative. That is, the graph goes downwards.</span>&nbsp;<span data-jme-visible=\"a&lt;0\">is positive. That is, the graph goes upwards.</span>&nbsp;</p>\n<p></p>", "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "type": "1_n_2", "showCorrectAnswer": true, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "choices": ["<p>goes upwards.</p>", "<p>goes downwards.</p>"], "shuffleChoices": false, "matrix": "lleading", "minMarks": 0}, {"scripts": {}, "displayType": "radiogroup", "marks": 0, "stepsPenalty": "1", "prompt": "<p>As we move to the far right&nbsp;of the graph, the graph</p>", "maxMarks": 0, "steps": [{"scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>What happens to the graph as you go far to the left or right is called the long term behaviour of a graph.</p>\n<p>The leading term (the term that includes the highest power) determines the long term behaviour of a&nbsp;polynomial.</p>\n<p>By expanding $y=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})}$ we see that the leading term is $\\simplify[all,fractionNumbers]{{a}x^3}$.&nbsp;</p>\n<p>As we go far to the right&nbsp;of the graph $x$ is&nbsp;positive, and so&nbsp;$\\simplify[all,fractionNumbers]{{a}x^3}$ <span data-jme-visible=\"a&lt;0\">is negative. That is, the graph goes downwards.</span>&nbsp;<span data-jme-visible=\"a&gt;0\">is positive. That is, the graph goes upwards.</span>&nbsp;</p>\n<p></p>", "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "type": "1_n_2", "showCorrectAnswer": true, "displayColumns": 0, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "choices": ["<p>goes upwards.</p>", "<p>goes downwards.</p>"], "shuffleChoices": false, "matrix": "rleading", "minMarks": 0}, {"scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "minValue": "{yint}", "allowFractions": true, "marks": 1, "type": "numberentry", "correctAnswerFraction": true, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "maxValue": "{yint}"}], "prompt": "<p>The $y$-intercept of the graph is $y=$[[0]].</p>", "steps": [{"scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>The $y$-intercept is the value of $y$ when $x=0$, that is, the value of $y$ where the graph hits&nbsp;the $y$-axis. To find it, substitute $x=0$ into our equation:</p>\n<p>\\[y=\\simplify[unitFactor,basic,fractionNumbers]{(0-{d})({a}0^2+{b}0+{c})}=\\var{yint}.\\]</p>\n<p></p>", "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"scripts": {}, "expectedvariablenames": [], "marks": 1, "checkingtype": "absdiff", "answer": "{xints}", "showCorrectAnswer": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "checkingaccuracy": 0.001, "vsetrangepoints": 5}], "prompt": "<p>The set of $x$-intercepts of the graph&nbsp;would be&nbsp;[[0]].</p>\n<p><strong>Note:&nbsp;</strong>If there are no intercepts, enter<strong> <strong>set()</strong></strong></p>\n<p><span>If there is only one intercept, say $x=5$, enter <strong><strong>set(5)</strong></strong></span></p>\n<p><span>If there are two intercepts, say $x=-2$ and $x=1.5$, enter <strong>set(-2,1.5)</strong></span></p>\n<p><span>If there are three intercepts, say $x=-2$,&nbsp;$x=1.5$ and $x=5$, enter<strong> <strong>set(-2,1.5,5)</strong></strong></span></p>\n<p></p>", "steps": [{"scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>The $x$-intercept is the value of $x$ when $y=0$, that is, the value of $x$ where the graph hits&nbsp;the $x$-axis. To find it, substitute $y=0$ into our equation:</p>\n<p>\\[0=\\simplify[all,fractionNumbers]{(x-{d})({a}x^2+{b}x+{c})} \\]</p>\n<p>Recall, if a product is zero then one of the factors must be zero, therefore</p>\n<p>\\[\\simplify[all,fractionNumbers]{x-{d}=0}\\quad \\text{or}\\quad\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}=0}.\\]</p>\n<p>Solving the&nbsp;first equation says that one of the $x$-intercepts is $x=\\var{d}$.</p>\n<p>For the second equation we will use the quadratic formula. Recall for $ax^2+bx+c=0$, the solutions&nbsp;(if they exist) are given by \\[x=\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}.\\]</p>\n<ul>\n<li>If $b^2-4ac&gt;0$ then there will be two more $x$-intercepts.</li>\n<li>If&nbsp;$b^2-4ac=0$ then there will be one more $x$-intercept.</li>\n<li>If $b^2-4ac&lt;0$ then there will be no more $x$-intercepts since the square root of a negative number is not a real number.</li>\n</ul>\n<p>For the equation&nbsp;$y=\\simplify[all,fractionNumbers]{{a}x^2+{b}x+{c}}$, we have $b^2-4ac=\\simplify[basic,unitFactor,fractionNumbers]{{b}^2-4{a}{c}}=\\var{disc}$ and so there <span data-jme-visible=\"switch=-1\">are no more $x$-intercepts.</span> <span data-jme-visible=\"switch=0\">is one more $x$-intercept:</span>&nbsp;<span data-jme-visible=\"switch=1\">are two more $x$-intercepts:</span>&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\"><span data-jme-visible=\"switch&lt;&gt;-1\">$x$</span></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"switch&lt;&gt;-1\">$=$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"switch&lt;&gt;-1\">$\\dfrac{-b}{2a}\\pm\\dfrac{\\sqrt{b^2-4ac}}{2a}$</span></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"switch&lt;&gt;-1\">&nbsp;</span></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"switch&lt;&gt;-1\">$=$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"switch&lt;&gt;-1\">$\\simplify[basic,unitFactor,fractionNumbers]{{axis_x}}\\pm\\simplify[basic,unitFactor,fractionNumbers]{sqrt{{disc}}/({2*a})}$</span></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\"></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"switch&lt;&gt;-1\">&nbsp;</span></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"switch&lt;&gt;-1\">$=$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"switch&lt;&gt;-1\"><span data-jme-visible=\"switch=0\">$\\var{axis_x}$</span> &nbsp;<span data-jme-visible=\"switch=1\">$\\var{xint0}, \\, \\var{xint1}$</span></span></td>\n</tr>\n</tbody>\n</table>\n<p></p>", "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1"}, {"scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"scripts": {}, "variableReplacements": [], "showCorrectAnswer": true, "minValue": "2", "allowFractions": false, "marks": 1, "type": "numberentry", "correctAnswerFraction": false, "showPrecisionHint": false, "variableReplacementStrategy": "originalfirst", "maxValue": "2"}], "prompt": "<p>Given the degree of a polynomial is $3$, the maximum number of possible 'bends' or 'turns' in the graph is&nbsp;[[0]].</p>", "steps": [{"scripts": {}, "variableReplacements": [], "marks": 0, "prompt": "<p>A degree $n$ polynomial has at most $n-1$ bends in its graph.</p>", "type": "information", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst"}], "type": "gapfill", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "stepsPenalty": "1"}], "functions": {}, "tags": ["cubic", "graphing", "polynomial", "Polynomial", "polynomials", "sketching"], "ungrouped_variables": ["switch", "axis_x", "a", "b", "thres", "s", "c", "lleading", "rleading", "axis_y", "xints", "disc", "xint0", "xint1", "d", "yint"], "rulesets": {}, "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "extensions": [], "variables": {"a": {"description": "", "name": "a", "templateType": "anything", "definition": "random(-6..6 except 0)", "group": "Ungrouped variables"}, "rleading": {"description": "", "name": "rleading", "templateType": "anything", "definition": "[if(a>0,1,0),if(a<0,1,0)]", "group": "Ungrouped variables"}, "thres": {"description": "", "name": "thres", "templateType": "anything", "definition": "b^2/(4a)", "group": "Ungrouped variables"}, "xint1": {"description": "", "name": "xint1", "templateType": "anything", "definition": "axis_x+s/(2*a)", "group": "Ungrouped variables"}, "disc": {"description": "", "name": "disc", "templateType": "anything", "definition": "b^2-4*a*c", "group": "Ungrouped variables"}, "axis_y": {"description": "", "name": "axis_y", "templateType": "anything", "definition": "a*axis_x^2+b*axis_x+c", "group": "Ungrouped variables"}, "xint0": {"description": "", "name": "xint0", "templateType": "anything", "definition": "axis_x-s/(2*a)", "group": "Ungrouped variables"}, "s": {"description": "", "name": "s", "templateType": "anything", "definition": "random(map(n^2,n,1..12))*2*a", "group": "Ungrouped variables"}, "d": {"description": "", "name": "d", "templateType": "anything", "definition": "random(-10..10 except [xint0, xint0])", "group": "Ungrouped variables"}, "lleading": {"description": "", "name": "lleading", "templateType": "anything", "definition": "[if(a<0,1,0),if(a>0,1,0)]", "group": "Ungrouped variables"}, "xints": {"description": "", "name": "xints", "templateType": "anything", "definition": "if(switch=-1,set(d) ,if(switch=0,set(d,axis_x),set(d,axis_x-s/(2*a),axis_x+s/(2*a))))", "group": "Ungrouped variables"}, "c": {"description": "", "name": "c", "templateType": "anything", "definition": "if(switch=0, thres,if(switch<0,(b^2+s^2)/(4a),(b^2-s^2)/(4a)))", "group": "Ungrouped variables"}, "yint": {"description": "", "name": "yint", "templateType": "anything", "definition": "-d*c", "group": "Ungrouped variables"}, "switch": {"description": "", "name": "switch", "templateType": "anything", "definition": "random(-1,0,1)", "group": "Ungrouped variables"}, "axis_x": {"description": "", "name": "axis_x", "templateType": "anything", "definition": "random(-4..4#0.5)", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "templateType": "anything", "definition": "-2*a*axis_x", "group": "Ungrouped variables"}}, "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}]}]}], "contributors": [{"name": "steve kilgallon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/268/"}]}