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{"name": "Asymptotes and intercepts", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"b": {"description": "", "group": "Ungrouped variables", "name": "b", "definition": "random(-12..12 except [0, a])", "templateType": "anything"}, "a": {"description": "", "group": "Ungrouped variables", "name": "a", "definition": "random(-12..12 except 0)", "templateType": "anything"}, "k": {"description": "", "group": "Ungrouped variables", "name": "k", "definition": "random(-12..12 except 0)", "templateType": "anything"}}, "name": "Asymptotes and intercepts", "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={k}/(x-{a})+{b}}\\]</p>\n<p>This means the equation of the vertical asymptote is $x=\\var{a}$.</p>\n<p>The <strong>horizontal asymptote</strong> corresponds to the value of $y$ that results from $x$ approaching infinity. As $x$ gets really really large, the fraction $\\simplify{{k}/(x-{a})}$ gets really really close to zero. The bigger $x$ gets, the closer&nbsp;$\\simplify{{k}/(x-{a})}$ gets to zero, and the closer $\\simplify[all]{y={k}/(x-{a})+{b}}$ gets to $y=\\var{b}$.</p>\n<p>This means the equation of the horizontal asymptote is&nbsp;$y=\\var{b}$.</p>\n<p>Notice that in our equation, $\\var{k}$ is multiplying the fraction $\\simplify[all]{1/(x-{a})}$. It is a general fact that because $\\var{k}$ is <span data-jme-visible=\"k&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=\"k&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=\"k&gt;0\">above</span> <span data-jme-visible=\"k&lt;0\">below</span> the horizontal asymptote $y=\\var{b}$.&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=\"k&lt;0\">above</span> <span data-jme-visible=\"k&gt;0\">below</span> the horizontal asymptote $y=\\var{b}$.&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;\">$\\simplify{{k}/(0-{a})+{b}}$</td>\n</tr>\n<tr>\n<td></td>\n<td>$=$</td>\n<td style=\"text-align: left;\">$\\simplify[fractionNumbers]{{-k/a+b}}$</td>\n</tr>\n</tbody>\n</table>\n<p>To find the <strong>$x$-intercept</strong>, let $y=0$ and solve for $x$:</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;\">$\\simplify{{k}/(x-{a})+{b}}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\var{-b}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{k}/(x-{a})}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{{-b}(x-{a})}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{k}$</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\">$\\simplify{x-{a}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\simplify{{k}/({-b})}$</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;\">$\\simplify[fractionNumbers]{{-k/b+a}}$</td>\n</tr>\n</tbody>\n</table>\n<p></p>", "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "variable_groups": [], "metadata": {"description": "<p>Identifying some of the basic properties (intercepts, asymptotes, quadrants) of a right hyperbola.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>You are given the equation of a right hyperbola &nbsp;\\[\\simplify[all]{y={k}/(x-{a})+{b}}.\\]</p>", "extensions": [], "ungrouped_variables": ["k", "a", "b"], "parts": [{"scripts": {}, "gaps": [{"scripts": {}, "maxValue": "{a}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "type": "numberentry", "minValue": "{a}", "marks": 1, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>The graph of this equation has a vertical asymptote at $x=$&nbsp;[[0]].</p>", "variableReplacements": [], "type": "gapfill", "marks": 0}, {"scripts": {}, "gaps": [{"scripts": {}, "maxValue": "{b}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "type": "numberentry", "minValue": "{b}", "marks": 1, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>The graph of this equation has a horizontal&nbsp;asymptote at $y=$&nbsp;[[0]].</p>", "variableReplacements": [], "type": "gapfill", "marks": 0}, {"scripts": {}, "gaps": [{"scripts": {}, "maxValue": "{b-k/a}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "type": "numberentry", "minValue": "{b-k/a}", "marks": 1, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>The $y$-intercept is at $y=$[[0]].</p>", "variableReplacements": [], "type": "gapfill", "marks": 0}, {"scripts": {}, "gaps": [{"scripts": {}, "maxValue": "{a-k/b}", "showCorrectAnswer": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "type": "numberentry", "minValue": "{a-k/b}", "marks": 1, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "variableReplacements": [], "correctAnswerFraction": true, "variableReplacementStrategy": "originalfirst"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>The $x$-intercept is at $x=$&nbsp;[[0]].</p>\n<p></p>", "variableReplacements": [], "type": "gapfill", "marks": 0}], "type": "question", "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}]}]}], "contributors": [{"name": "Denis Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1216/"}]}