Identifying some of the basic properties (intercepts, asymptotes, quadrants) of a right hyperbola except this time the equation will be written like y=(ax+b)/(cx+d)
"}, "rulesets": {}, "statement": "You are given the equation \\[\\simplify[all]{y=({b}x+{k-a*b})/(x-{a})}.\\]
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", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerStyle": "plain", "showFeedbackIcon": true, "marks": 1, "maxValue": "{a}", "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "numberentry", "mustBeReduced": false, "minValue": "{a}", "variableReplacements": []}], "type": "gapfill", "variableReplacements": []}, {"showFeedbackIcon": true, "marks": 0, "prompt": "The graph of this equation has a horizontal asymptote at $y=$ [[0]].
", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerStyle": "plain", "showFeedbackIcon": true, "marks": 1, "maxValue": "{b}", "allowFractions": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "numberentry", "mustBeReduced": false, "minValue": "{b}", "variableReplacements": []}], "type": "gapfill", "variableReplacements": []}, {"matrix": ["if(k<0, 1, 0)", "if(k>0, 1, 0)", "if(k>0, 1, 0)", "if(k<0, 1, 0)"], "showFeedbackIcon": true, "marks": 0, "variableReplacements": [], "warningType": "none", "displayColumns": "2", "maxAnswers": 0, "maxMarks": 0, "distractors": ["", "", "", ""], "displayType": "checkbox", "minAnswers": 0, "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?
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", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerStyle": "plain", "showFeedbackIcon": true, "marks": 1, "maxValue": "{b-k/a}", "allowFractions": true, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "numberentry", "mustBeReduced": false, "minValue": "{b-k/a}", "variableReplacements": []}], "type": "gapfill", "variableReplacements": []}, {"showFeedbackIcon": true, "marks": 0, "prompt": "The $x$-intercept is at $x=$ [[0]].
\n", "scripts": {}, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"correctAnswerStyle": "plain", "showFeedbackIcon": true, "marks": 1, "maxValue": "{a-k/b}", "allowFractions": true, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "type": "numberentry", "mustBeReduced": false, "minValue": "{a-k/b}", "variableReplacements": []}], "type": "gapfill", "variableReplacements": []}], "ungrouped_variables": ["k", "a", "b"], "advice": "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+{k-a*b})/(x-{a})}\\]
\nThis means the equation of the vertical asymptote is $x=\\var{a}$.
\nThe horizontal 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 $\\simplify[all]{({b}x+{k-a*b})/(x-{a})}$ so that it includes a multiple of the denominator as such:
\n$\\begin{align}\\simplify[all]{({b}x+{k-a*b})/(x-{a})}&=\\simplify[all]{({b}(x-{a})+{k})/(x-{a})}\\\\&=\\simplify{({b}(x-{a}))/(x-{a})+{k}/(x-{a})}\\\\&=\\simplify{{b}+{k}/(x-{a})}\\end{align}$
\nNow as $x$ gets very large $\\simplify{{k}/(x-{a})}$ gets very close to $0$ and so $y$ approaches $\\var{b}$.
\n\n
Method 2 - Dividing by the dominant term
\nSince $\\simplify{({b}x+{k-a*b})/(x-{a})}$ is a fraction be can rewrite it by dividing the numerator and denominator by the same value, in this case we will divide by $x$:
\n$\\begin{align}\\simplify[all]{({b}x+{k-a*b})/(x-{a})}&=\\simplify[all]{({b}x/x+{k-a*b}/x)/(x/x-{a}/x)}\\\\&=\\simplify[all]{({b}+{k-a*b}/x)/(1-{a}/x)}\\end{align}$
\nNow as $x$ gets very large both $\\simplify{{k-a*b}/x}$ and $\\simplify{{a}/x}$ gets very close to zero and so $y$ approaches $\\var{b}$.
\n\nThis means the equation of the horizontal asymptote is $y=\\var{b}$.
\nNotice that from Method 1 above we found that our equation can be written as $\\simplify[all]{y={b}+{k}/(x-{a})}$, and so we can see that $\\var{k}$ is multiplying the fraction $\\simplify[all]{1/(x-{a})}$. It is a general fact that because $\\var{k}$ 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$\\simplify[!collectNumbers]{({b}0+{k-a*b})/(0-{a})}$ | \n
| \n | $=$ | \n$\\simplify[fractionNumbers]{{-k/a+b}}$ | \n
To find the $x$-intercept, let $y=0$ and solve for $x$:
\n| $0$ | \n$=$ | \n$\\simplify{({b}x+{k-a*b})/(x-{a})}$ | \n
| $0$ | \n$=$ | \n$\\simplify{({b}x+{k-a*b})}$ | \n
| $\\simplify{{-b}x}$ | \n$=$ | \n$\\var{k-a*b}$ | \n
| $x$ | \n$=$ | \n$\\simplify[fractionNumbers]{{-k/b+a}}$ | \n