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{"name": "Rational functions: Quadratic over linear", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Rational functions: Quadratic over linear", "tags": [], "metadata": {"description": "<p>Identifying some of the basic properties (intercepts, asymptotes, quadrants) of a&nbsp;rational function (quadratic over linear)</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>You are given the equation &nbsp;\\[\\simplify[all]{y=({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}.\\]</p>", "advice": "<p>The <strong>vertical asymptote</strong> corresponds to the value of $x$ that results in attempting to divide by $0$. For the equation</p>\n<p>\\[\\simplify[all]{y=({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}\\]</p>\n<p>This means the equation of the vertical asymptote is $x=\\var{a}$.</p>\n<p>The <strong>oblique&nbsp;asymptote</strong> corresponds to the value of $y$ that results from $x$ approaching infinity. Let's look at two methods that determine this limit:</p>\n<p style=\"padding-left: 30px;\"><strong>Method 1 - Breaking up the fraction</strong></p>\n<p style=\"padding-left: 30px;\">We can rewrite the numerator of $\\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}$ so that it includes multiples of the denominator. We do this term by term. We first write $\\simplify{{b}x^2}$ in terms of $\\simplify{x-{a}}$:</p>\n<p style=\"padding-left: 30px; text-align: center;\">$\\simplify{{b}x^2={b}x(x-{a})+{a*b}x}$</p>\n<p style=\"padding-left: 30px;\">and so the next term is now $\\simplify[!collectNumbers]{{a*b}x-{a*b+c}x=-{c}x}$. We now write this in terms of $\\simplify{x-{a}}$:</p>\n<p style=\"padding-left: 30px; text-align: center;\">$\\simplify{-{c}x=-{c}(x-{a})-{a*c}}$</p>\n<p style=\"padding-left: 30px;\">and so our next term is now $\\simplify[!collectNumbers]{-{a*c}+{a*c+d}={d}}$. Now we can rewrite the numerator and break up our fraction:</p>\n<p style=\"padding-left: 30px;\">$\\begin{align}\\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}&amp;=\\simplify{({b}x(x-{a})-{c}(x-{a})+{d})/(x-{a})}\\\\&amp;=\\simplify{({b}x(x-{a}))/(x-{a})-({c}(x-{a}))/(x-{a})+{d}/(x-{a})}\\\\&amp;=\\simplify{({b}x-{c})+{d}/(x-{a})}\\end{align}$</p>\n<p style=\"padding-left: 30px;\">Now as $x$ gets very large $\\simplify{{d}/(x-{a})}$ gets very close to $0$ and so $y$ approaches $\\simplify{{b}x-{c}}$.</p>\n<p><strong></strong></p>\n<p style=\"padding-left: 30px;\"><strong>Method 2 -&nbsp;Polynomial long division</strong></p>\n<p style=\"padding-left: 30px;\">There is a procedure called polynomial long division, if you are comfortable with long division of numbers then it isn't too different. You can see an example of the procedure <a href=\"https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/dividing-polynomials-1\">here</a>.&nbsp;The result of the division (in this case $\\simplify{{b}x-{c}}$) will be the oblique asymptote. The remainder will be $\\var{d}$, which 'remains' to be divided by $\\simplify{x-{a}}$. Just like in the method above, you will be able to say \\[y=\\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}=\\simplify{({b}x-{c})+{d}/(x-{a})}\\] <br/>so as $x$ gets very large $\\simplify{{d}/(x-{a})}$ gets very close to $0$ and so $y$ approaches $\\simplify{{b}x-{c}}$.&nbsp;</p>\n<p></p>\n<p>This means the equation of the horizontal asymptote is&nbsp;$y=\\simplify{{b}x-{c}}$.</p>\n<p>Notice that from Method 1 and Method 2 above we found that our equation can be written as $\\simplify{{b}x-{c}+{d}/(x-{a})}$, and so we can see that $\\var{d}$ is multiplying the fraction $\\simplify{1/(x-{a})}$. It is a general fact that because $\\var{d}$ is <span data-jme-visible=\"d&gt;=0\">positive the graph will be in the top right and bottom left parts of the plane.</span>&nbsp;&nbsp;<span data-jme-visible=\"d&lt;=0\">negative&nbsp;the graph will be in the top left and bottom right&nbsp;parts of the plane.</span>&nbsp;We can see this as follows:</p>\n<ul>\n<li>By substituting into the equation an $x$ value that is to the right of the vertical asymptote&nbsp;$x=\\var{a}$,&nbsp;we will find the resulting $y$ value is <span data-jme-visible=\"d&gt;=0\">above</span> <span data-jme-visible=\"d&lt;=0\">below</span> the oblique&nbsp;asymptote $y=\\simplify{{b}x-{c}+{d}}$.&nbsp;</li>\n<li>By substituting into the equation an $x$ value that is to the left of the vertical asymptote&nbsp;$x=\\var{a}$,&nbsp;we will find the resulting $y$ value is <span data-jme-visible=\"d&lt;=0\">above</span> <span data-jme-visible=\"d&gt;=0\">below</span> the oblique&nbsp;asymptote $y=\\simplify{{b}x-{c}+{d}}$.&nbsp;</li>\n</ul>\n<p>To find the <strong>$y$-intercept</strong>, let $x=0$ and solve for $y$:</p>\n<table>\n<tbody>\n<tr>\n<td>$y$</td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle\\simplify[!collectNumbers]{({b}0^2-{a*b+c}0+{a*c+d})/(0-{a})}$</td>\n</tr>\n<tr>\n<td></td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle\\simplify[fractionNumbers]{{-c-d/a}}$</td>\n</tr>\n</tbody>\n</table>\n<p>To find the <strong>$x$-intercept</strong>, let $y=0$ and solve for $x$ (here we use the quadratic equation):</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$0$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle\\simplify{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$0$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle\\simplify{({b}x^2-{a*b+c}x+{a*c+d})}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$x$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\displaystyle\\frac{-(\\var{-a*b-c})\\pm\\sqrt{(\\var{-a*b-c})^2-4(\\var{b})(\\var{a*c+d})}}{2(\\var{b})}$</td>\n</tr>\n</tbody>\n</table>\n<p><span data-jme-visible=\"des&lt;0\">Since this includes the square root of a negative number&nbsp;there are no $x$-intercepts.</span></p>\n<p><span data-jme-visible=\"des=0\">Therefore, the $x$-intercept is $x=\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{({a*b+c})/{2*b}}$.</span></p>\n<p><span data-jme-visible=\"des&gt;0\">Therefore, the $x$-intercepts are $x=\\simplify{({a*b+c}-sqrt({des}))/{2*b}}$ and $x=\\simplify{({a*b+c}+sqrt({des}))/{2*b}}$.</span></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(-12..12 except 0)", "description": "", "templateType": 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"b", "c", "d", "des", "xint"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The graph of this equation has a vertical asymptote at $x=$&nbsp;[[0]].</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{a}", "maxValue": "{a}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The graph of this equation has an oblique asymptote at $y=$&nbsp;[[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b}x-{c}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"type": "m_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The two asymptotes&nbsp;break the plane up into four parts: top right, top left, bottom left and bottom right.</p>\n<p></p>\n<p>The graph is in which of these parts of the plane?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "checkbox", "displayColumns": "2", "minAnswers": 0, "maxAnswers": 0, "warningType": "none", "showCellAnswerState": true, "markingMethod": "sum ticked cells", "choices": ["<p>top&nbsp;left &nbsp; &nbsp; &nbsp; &nbsp;</p>", "<p>top right</p>", "<p>bottom left</p>", "<p>bottom right</p>"], "matrix": ["if(d<0, 1, 0)", "if(d>0, 1, 0)", "if(d>0, 1, 0)", "if(d<0, 1, 0)"], "distractors": ["", "", "", ""]}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The $y$-intercept is at $y=$[[0]].</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{-c-d/a}", "maxValue": "{-c-d/a}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the $x$ intercepts.</p>\n<p>If there are none, enter <strong>set()</strong>. If there is one, say $x=1$, enter <strong>set(1)</strong>. If there are two, say x=1 and x=2, enter <strong>set(1, 2)</strong>, etc.</p>\n<p>You could input an expression such as $\\frac{1-\\sqrt{133}}{2}$ by typing <strong>(1-sqrt(133))/2</strong></p>\n<p>The set of $x$ intercepts is <strong>[[0]]</strong>.</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{xint}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "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/"}]}