Graphing $y=a\\log_{b}(x)+c$
"}, "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);\n\n\n\nif(quad==1){board.create('functiongraph',[function(x){ return a*Math.log(x)/Math.log(b)+c}],{strokeWidth:2});}\nif(quad==2){board.create('functiongraph',[function(x){ return a*Math.log(-x)/Math.log(b)+c}],{strokeWidth:2});}\nif(quad==3){board.create('functiongraph',[function(x){ return -a*Math.log(x)/Math.log(b)-c}],{strokeWidth:2});}\nif(quad==4){board.create('functiongraph',[function(x){ return -a*Math.log(-x)/Math.log(b)-c}],{strokeWidth:2});}\n\nreturn div;", "type": "html"}}, "name": "Graphing logarithms with vertical transformations and base>1", "ungrouped_variables": ["b", "a", "c"], "parts": [{"variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "The $x$-intercept of $y=\\simplify{{a}log(x,{b})+{c}}$ is the point $\\large($ [[0]], [[1]]$\\large)$.
\nNote: You should input the exact answer, not just a decimal approximation.
", "variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "answersimplification": "fractionNumbers, simplifyFractions,basic", "showCorrectAnswer": true, "checkvariablenames": false, "variableReplacements": [], "expectedvariablenames": [], "scripts": {}, "marks": 1, "showFeedbackIcon": true, "answer": "{b}^(-{c/a})", "vsetrangepoints": 5, "showpreview": true, "vsetrange": [0, 1], "checkingaccuracy": 0.001, "type": "jme", "checkingtype": "absdiff"}, {"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "0", "scripts": {}, "marks": 1, "showFeedbackIcon": true, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "0"}], "marks": 0}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "Another easily found point on the curve is ${\\large(}\\var{b},$ [[0]]$\\large)$.
", "variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": true, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "{a+c}", "scripts": {}, "marks": 1, "showFeedbackIcon": true, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": true, "maxValue": "{a+c}"}], "marks": 0}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", "", "", ""], "variableReplacements": [], "displayColumns": "1", "showCorrectAnswer": true, "showFeedbackIcon": true, "maxMarks": 0, "scripts": {}, "marks": 0, "matrix": ["if(a>0,1,0)", "if(a<0,1,0)", 0, 0], "displayType": "radiogroup", "shuffleChoices": true, "choices": ["$y$ increases without bound.
", "$y$ decreases without bound.
", "$y$ approaches $\\var{c}$.
", "$y$ approaches $0$.
"], "type": "1_n_2", "prompt": "As $x$ increases without bound,
", "minMarks": 0}, {"variableReplacementStrategy": "originalfirst", "scripts": {}, "showCorrectAnswer": true, "showFeedbackIcon": true, "type": "gapfill", "prompt": "The vertical asymptote of $y=\\simplify{{a}log(x,{b})+{c}}$ is $x=$ [[0]].
", "variableReplacements": [], "gaps": [{"variableReplacementStrategy": "originalfirst", "notationStyles": ["plain", "en", "si-en"], "variableReplacements": [], "showCorrectAnswer": true, "allowFractions": false, "correctAnswerStyle": "plain", "mustBeReducedPC": 0, "minValue": "0", "scripts": {}, "marks": 1, "showFeedbackIcon": true, "mustBeReduced": false, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "0"}], "marks": 0}, {"variableReplacementStrategy": "originalfirst", "distractors": ["", "", "", ""], "variableReplacements": [], "displayColumns": "2", "showCorrectAnswer": true, "showFeedbackIcon": true, "maxMarks": 0, "scripts": {}, "marks": 0, "matrix": ["1", 0, 0, 0], "displayType": "radiogroup", "shuffleChoices": true, "choices": ["{graph1(1)}
", "{graph1(2)}
", "{graph1(3)}
", "{graph1(4)}
"], "type": "1_n_2", "prompt": "Which graph best represents $y=\\simplify{{a}log(x,{b})+{c}}$?
", "minMarks": 0}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"c": {"description": "", "definition": "random(-4..4 except [0,a,-a])", "templateType": "anything", "name": "c", "group": "Ungrouped variables"}, "b": {"description": "", "definition": "random(2..10)", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}, "a": {"description": "a
", "definition": "random(-3..3 except [0,1,b])", "templateType": "anything", "name": "a", "group": "Ungrouped variables"}}, "extensions": ["jsxgraph"], "type": "question", "preamble": {"js": "", "css": ""}, "advice": "This question assumes you understand the definition and the laws of logarithms.
\na) To find the $x$-intercept, substitute $y=0$ into the equation and solve for $x$:
\n$\\begin{align*}&&0&=\\simplify{{a}log(x,{b})+{c}}\\\\\\implies&& \\var{-c}&=\\simplify{{a}log(x,{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{-{c/a}}&=\\simplify{log(x,{b})}\\\\\\implies&&\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})}&=x\\end{align*}$
\nTherefore, the $x$-intercept is the point $\\left(\\simplify[fractionNumbers,simplifyFractions,basic]{{b}^(-{c/a})},0\\right)$.
\nb) Substitute $x=\\var{b}$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}log({b},{b})+{c}}=\\simplify[!collectNumbers]{{a}*1+{c}}=\\var{a+c}$. Therefore, another easily found point is $(\\var{b},\\var{a+c})$.
\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $\\simplify{{a}log(x,{b})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).
\nd) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\simplify{{a}log(x,{b})+{c}}$ the closer $x$ gets to zero (approaching from the right), the smaller larger $y$ gets (without bound). In other words, as $x$ approaches $0$ from the right, $y$ approaches negative infinity. This means that the asymptote for $y=\\log_{\\var{b}}(x)$ is the line $x=0$ (the $y$-axis).
\ne) 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.
\nThe logarithm you will be working with for this question is \\[y=\\simplify{{a}log(x,{b})+{c}}.\\]
\n\n", "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/"}]}