The vertical asymptote corresponds to the value of $x$ that results in attempting to divide by $0$. For the equation
\n\\[\\simplify[all]{y=({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}\\]
\nThis means the equation of the vertical asymptote is $x=\\var{a}$.
\nThe oblique asymptote corresponds to the value of $y$ that results from $x$ approaching infinity. Let's look at two methods that determine this limit:
\nMethod 1 - Breaking up the fraction
\nWe 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 by dealing with the $x^2$ term and then the $x$ term:
\n$\\begin{align}\\simplify[all]{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}&=\\simplify[all]{({b}x(x-{a})+{a*b}x-{a*b+c}x+{a*c+d})/(x-{a})}\\\\&=\\simplify[all]{({b}x(x-{a})-{c}x+{a*c+d})/(x-{a})}\\\\&=\\simplify{({b}x(x-{a})-{c}(x-{a})-{a*c}+{a*c+d})/(x-{a})}\\\\&=\\simplify{({b}x(x-{a})-{c}(x-{a})+{d})/(x-{a})}\\\\&=\\simplify{({b}x(x-{a}))/(x-{a})-({c}(x-{a}))/(x-{a})+{d}/(x-{a})}\\\\&=\\simplify{{b}x-{c}+{d}/(x-{a})}\\end{align}$
\nNow as $x$ gets very large $\\simplify{{d}/(x-{a})}$ gets very close to $0$ and so $y$ approaches $\\simplify{{b}x-{c}}$.
\n\n
Method 2 - Polynomial long division
\nThere 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 here. The result of the division (in this case $\\simplify{{b}x-{c}}$) will be the oblique asymptote.
\n\nThis means the equation of the horizontal asymptote is $y=\\simplify{{b}x-{c}}$.
\nNotice 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 positive the graph will be in the top right and bottom left parts of the plane. negative the graph will be in the top left and bottom right parts of the plane. We can see this as follows:
\nTo find the $y$-intercept, let $x=0$ and solve for $y$:
\n| $y$ | \n$=$ | \n$\\displaystyle\\simplify[!collectNumbers]{({b}0^2-{a*b+c}0+{a*c+d})/(0-{a})}$ | \n
| \n | $=$ | \n$\\displaystyle\\simplify[fractionNumbers]{{-c-d/a}}$ | \n
To find the $x$-intercept, let $y=0$ and solve for $x$ (here we use the quadratic equation):
\n| $0$ | \n$=$ | \n$\\displaystyle\\simplify{({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}$ | \n
| $0$ | \n$=$ | \n$\\displaystyle\\simplify{({b}x^2-{a*b+c}x+{a*c+d})}$ | \n
| $x$ | \n$=$ | \n$\\displaystyle\\frac{-(\\var{-a*b-c})\\pm\\sqrt{(\\var{-a*b-c})^2-4(\\var{b})(\\var{a*c+d})}}{2(\\var{b})}$ | \n
Since this includes the square root of a negative number there are no $x$-intercepts.
\nTherefore, the $x$-intercept is $x=\\simplify[fractionNumbers,simplifyFractions,unitDenominator]{({a*b+c})/{2*b}}$.
\nTherefore, the $x$-intercepts are $x=\\simplify{({a*b+c}-sqrt({des}))/{2*b}}$ and $x=\\simplify{({a*b+c}+sqrt({des}))/{2*b}}$.
", "rulesets": {}, "metadata": {"description": "Identifying some of the basic properties (intercepts, asymptotes, quadrants) of a rational function (quadratic over linear)
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "c", "d", "des"], "statement": "Data l'equazione \\[\\simplify[all]{y=({b}x^2-{a*b+c}x+{a*c+d})/(x-{a})}.\\]
", "variables": {"c": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-12..12 except 0)", "name": "c"}, "a": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-12..12 except 0)", "name": "a"}, "d": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-12..12 except 0)", "name": "d"}, "des": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "a^2*b^2+c^2-2*a*b*c-4*b*d", "name": "des"}, "b": {"description": "", "group": "Ungrouped variables", "templateType": "anything", "definition": "random(-12..12 except 0)", "name": "b"}}, "preamble": {"css": "", "js": ""}, "name": "riccardo's copy of Rational functions: Quadratic over linear", "tags": [], "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "showFeedbackIcon": true, "type": "numberentry", "minValue": "{a}", "scripts": {}, "correctAnswerFraction": false, "showCorrectAnswer": true, "mustBeReduced": false, "variableReplacements": [], "maxValue": "{a}", "allowFractions": false}], "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "prompt": "Il grafico di questa equazione ha un asintoto verticale di equazione $x=$ [[0]].
"}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "checkingtype": "absdiff", "vsetrangepoints": 5, "scripts": {}, "expectedvariablenames": [], "answer": "{b}x-{c}", "showCorrectAnswer": true, "checkvariablenames": false, "checkingaccuracy": 0.001, "showpreview": true, "type": "jme"}], "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "prompt": "Il grafico di questa equazione ha un asintoto obliquo di equazione $y=$ [[0]].
"}, {"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "distractors": ["", "", "", ""], "displayType": "checkbox", "warningType": "none", "marks": 0, "minMarks": 0, "matrix": ["if(d<0, 1, 0)", "if(d>0, 1, 0)", "if(d>0, 1, 0)", "if(d<0, 1, 0)"], "minAnswers": 0, "shuffleChoices": false, "maxAnswers": 0, "scripts": {}, "maxMarks": 0, "showCorrectAnswer": true, "choices": ["top left
", "top right
", "bottom left
", "bottom right
"], "variableReplacements": [], "displayColumns": "2", "type": "m_n_2", "prompt": "The two asymptotes break the plane up into four parts: top right, top left, bottom left and bottom right.
\n\nThe graph is in which of these parts of the plane?
"}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "gaps": [{"variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "marks": 1, "showFeedbackIcon": true, "type": "numberentry", "minValue": "{-c-d/a}", "scripts": {}, "correctAnswerFraction": true, "showCorrectAnswer": true, "mustBeReduced": false, "variableReplacements": [], "maxValue": "{-c-d/a}", "allowFractions": true}], "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "prompt": "The $y$-intercept is at $y=$[[0]].
"}, {"variableReplacementStrategy": "originalfirst", "scripts": {"mark": {"script": "var gaps = this.gaps;\n// if the student said \"yes\" to gap 2, then mark their factorisation\nif(gaps[0].zero) {\n this.setCredit(gaps[0].credit*credit*gaps[0].marks);\n// otherwise, work out the credit based on the amounts awarded for the first three gaps\n} else {\n if(gaps[0].one){\n this.setCredit((gaps[0].credit*gaps[0].marks+gaps[1].credit*credit*gaps[1].marks)/(gaps[0].marks+gaps[1].marks))\n } else {\n this.setCredit((gaps[0].credit*gaps[0].marks+gaps[2].credit*gaps[2].marks+gaps[3].credit*gaps[3].marks)/(gaps[0].marks+gaps[2].marks+gaps[3].marks));\n }}", "order": "instead"}, "constructor": {"script": "// override the default 'submit' method for the part\n// we only want to submit gap 1 if the answer to gap 0 is \"one\", and submit gaps 2 and 3 if the answer to gap 0 is \"two\"\nthis.submit = Numbas.util.extend(function() {\n this.gaps[0].submit();\n this.gaps[1].submit();\n this.gaps[2].submit();\n this.gaps[3].submit();\n if(this.gaps[0].one) {\n this.gaps[1].submit();\n } else {\n this.gaps[1].answered = true;\n }\n if(this.gaps[0].two) {\n this.gaps[2].submit();\n this.gaps[3].submit();\n } else {\n this.gaps[2].answered = true;\n this.gaps[3].answered = true;\n }\n},Part.prototype.submit);", "order": "after"}}, "gaps": [{"variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true, "distractors": ["", "", ""], "displayType": "radiogroup", "marks": 0, "minMarks": 0, "matrix": ["if(des<0,1,0)", "if(des=0,1,0)", "if(des>0,1,0)"], "shuffleChoices": false, "scripts": {"mark": {"script": "// apply the normal marking algorithm for this part\nthis.__proto__.mark.apply(this);\n// store whether the student said \"yes\" in an attribute that's easier to access\nthis.zero = this.ticks[0][0];\nthis.one = this.ticks[0][1];\nthis.two = this.ticks[0][2];", "order": "instead"}}, "maxMarks": 0, "showCorrectAnswer": true, "choices": ["0
", "1
", "2
"], "variableReplacements": [], "displayColumns": 0, "type": "1_n_2"}, {"variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "marks": 1, "showFeedbackIcon": true, "checkingtype": "absdiff", "vsetrangepoints": 5, "type": "jme", "scripts": {}, "expectedvariablenames": [], "answer": "{a*b+c}/{2*b}", "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacements": [], "showpreview": true, "answersimplification": "simplifyFractions"}, {"variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "checkingtype": "absdiff", "vsetrangepoints": 5, "scripts": {}, "expectedvariablenames": [], "answer": "({a*b+c}-sqrt({des}))/{2*b}", "showCorrectAnswer": true, "checkvariablenames": false, "checkingaccuracy": 0.001, "showpreview": true, "type": "jme"}, {"variableReplacementStrategy": "originalfirst", "vsetrange": [0, 1], "variableReplacements": [], "marks": 1, "showFeedbackIcon": true, "checkingtype": "absdiff", "vsetrangepoints": 5, "scripts": {}, "expectedvariablenames": [], "answer": "({a*b+c}+sqrt({des}))/{2*b}", "showCorrectAnswer": true, "checkvariablenames": false, "checkingaccuracy": 0.001, "showpreview": true, "type": "jme"}], "showCorrectAnswer": true, "marks": 0, "showFeedbackIcon": true, "variableReplacements": [], "type": "gapfill", "prompt": "How many $x$-intercepts are there? [[0]]
\n\nThe $x$-intercept is $x=$ [[1]].
\nThe $x$-intercepts are $x=$ [[2]] and $x=$ [[3]] (in ascending order)
\nNote: You could input $\\frac{1-\\sqrt{133}}{2}$ by typing (1-sqrt(133))/2
\n