// Numbas version: exam_results_page_options {"name": "Graphing logarithms with horizontal and vertical transformations and base>1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Graphing $y=a\\log_{b}(\\pm x+d)+c$

"}, "tags": [], "functions": {"graph1": {"parameters": [["quad", "number"]], "language": "javascript", "definition": "var div = Numbas.extensions.jsxgraph.makeBoard('300px','300px',{boundingBox:[-12,12,12,-12],grid:true,axis:false});\nvar board = div.board;\n\n// create the x-axis.\nvar xaxis = board.create('line',[[0,0],[1,0]], { strokeColor: 'black', fixed: true});\nvar xticks = board.create('ticks',[xaxis,2],{\n drawLabels: true,\n label: {offset: [-4, -10]},\n minorTicks: 0\n});\n\n// create the y-axis\nvar yaxis = board.create('line',[[0,0],[0,1]], { strokeColor: 'black', fixed: true });\nvar yticks = board.create('ticks',[yaxis,2],{\ndrawLabels: true,\nlabel: {offset: [-20, 0]},\nminorTicks: 0\n});\n\na = Numbas.jme.unwrapValue(scope.variables.a);\nb = Numbas.jme.unwrapValue(scope.variables.b);\nc = Numbas.jme.unwrapValue(scope.variables.c);\nd = Numbas.jme.unwrapValue(scope.variables.d);\ns = Numbas.jme.unwrapValue(scope.variables.s);\n\n\n\nif(quad==1){board.create('functiongraph',[function(x){ return a*Math.log(s*x+d)/Math.log(b)+c}],{strokeWidth:2});}\nif(quad==2){board.create('functiongraph',[function(x){ return a*Math.log(-s*x-d)/Math.log(b)+c}],{strokeWidth:2});}\nif(quad==3){board.create('functiongraph',[function(x){ return -a*Math.log(s*x+d)/Math.log(b)-c}],{strokeWidth:2});}\nif(quad==4){board.create('functiongraph',[function(x){ return -a*Math.log(-s*x-d)/Math.log(b)-c}],{strokeWidth:2});}\n\nreturn div;", "type": "html"}}, "name": "Graphing logarithms with horizontal and vertical transformations and base>1", "ungrouped_variables": ["b", "a", "c", "d", "s"], "parts": [{"variableReplacementStrategy": "originalfirst", "gaps": [{"variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "marks": 1, "answersimplification": "fractionNumbers, simplifyFractions, basic, unitFactor", "showFeedbackIcon": true, "answer": "{s}*{b}^(-{c/a})-{s}*{d}", "vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "checkingaccuracy": 0.001, "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "marks": 1, "showFeedbackIcon": true, "answer": "0", "vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "type": "jme", "checkingtype": "absdiff"}], "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "

The $x$-intercept of $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.

\n

Note: You should input the exact answer, not just a decimal approximation.

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Another important point on the curve is ${\\large(}\\simplify[fractionNumbers]{{(b-d)/s}},$ [[0]]$\\large)$.

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$y$ increases without bound.

", "

$y$ decreases without bound.

", "

$y$ approaches $\\var{c}$.

", "

$y$ approaches $0$.

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As $x$ increases decreases without bound,

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The vertical asymptote of $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ is $x=$ [[0]].

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{graph1(1)}

", "

{graph1(2)}

", "

{graph1(3)}

", "

{graph1(4)}

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Which graph best represents $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$?

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a

", "templateType": "anything", "definition": "random(-4..4 except [0,1,b])", "name": "a", "group": "Ungrouped variables"}, "s": {"description": "", "templateType": "anything", "definition": "random(-1,1)", "name": "s", "group": "Ungrouped variables"}, "d": {"description": "", "templateType": "anything", "definition": "random(-4..4 except 0)", "name": "d", "group": "Ungrouped variables"}}, "extensions": ["jsxgraph"], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "

This question assumes you understand the definition and the laws of logarithms.

\n

a) To find the $x$-intercept, substitute $y=0$ into the equation and solve for $x$:

\n

\\begin{align*}&&0&=\\simplify{{a}log({s}x+{d},{b})+{c}}\\\\\\implies&& \\var{-c}&=\\simplify{{a}log({s}x+{d},{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{-{c/a}}&=\\simplify{log({s}x+{d},{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})}&=\\simplify{{s}x+{d}}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic,unitDenominator,unitFactor]{({b}^(-{c/a})-{d})/{s}}&=x\\end{align*}

\n

Therefore, the $x$-intercept is the point $\\left(\\simplify[fractionNumbers,simplifyFractions,basic,unitDenominator,unitFactor]{({b}^(-{c/a})-{d})/{s}},0\\right)$.

\n

b) Substitute $x=\\simplify[fractionNumbers]{{(b-d)/s}}$ into the equation: $y=\\simplify[!collectNumbers,basic,fractionNumbers]{{a}log(({(b-d)})+{d},{b})+{c}}=\\simplify[!collectNumbers,basic]{{a}log({b},{b})+{c}}=\\var{a+c}$. Therefore, another easily found point is $\\left(\\simplify[fractionNumbers]{{(b-d)/s}},\\var{a+c}\\right)$.

\n

\n

c) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).

\n

\n

c) As $x$ gets smaller and smaller (decreases without bound, or approaches negative infinity) $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).

\n

\n

\n

d) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ the closer $\\simplify{{s}x+{d}}$ gets to zero (approaching from the right), the smaller larger $y$ gets (without bound). In other words, as $x$ approaches $\\simplify[fractionNumbers]{{-d/s}}$ from the right left, $y$ approaches negative infinity. This means that the asymptote for $y=\\simplify{{a}log({s}x+{d},{b})+{c}}$ is the line $x=\\simplify[fractionNumbers]{{-d/s}}$.

\n

e) Given all the information above, it should be clear that the graph should look like

\n

{graph1(1)}

", "statement": "

The following questions will gauge your understanding of logarithms and how to graph them.

\n

The logarithm you will be working with for this question is \$y=\\simplify{{a}log({s}x+{d},{b})+{c}}.\$

", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}