a
", "templateType": "anything", "definition": "random(-3..3 except [0,1,b])", "name": "a", "group": "Ungrouped variables"}, "b": {"description": "", "templateType": "anything", "definition": "random(2..10)", "name": "b", "group": "Ungrouped variables"}, "c": {"description": "", "templateType": "anything", "definition": "random(-4..3 except [0,a])", "name": "c", "group": "Ungrouped variables"}}, "ungrouped_variables": ["b", "a", "c"], "preamble": {"js": "", "css": ""}, "variablesTest": {"maxRuns": 100, "condition": ""}, "tags": [], "rulesets": {}, "functions": {"graph1": {"type": "html", "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\nif(quad==1){board.create('functiongraph',[function(x){ return a*Math.pow(b,x)+c}],{strokeWidth:2});}\nif(quad==2){board.create('functiongraph',[function(x){ return a*Math.pow(1/b,x)+c}],{strokeWidth:2});}\nif(quad==3){board.create('functiongraph',[function(x){ return -a*Math.pow(1/b,x)+c}],{strokeWidth:2});}\nif(quad==4){board.create('functiongraph',[function(x){ return -a*Math.pow(b,x)+c}],{strokeWidth:2});}\n\nreturn div;", "parameters": [["quad", "number"]], "language": "javascript"}}, "statement": "The following questions will gauge your understanding of exponentials and how to graph them.
\nThe exponential you will be working with for this question is \\[y=\\simplify{{a}{b}^x+{c}}.\\]
\n\n", "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "Graphing $y=ab^x+c$
"}, "name": "Graphing exponentials with vertical transformations and base>1 ", "extensions": ["jsxgraph"], "advice": "a) To find the $y$-intercept, substitute $x=0$ into the equation: $y=\\simplify[!collectNumbers,basic]{{a}{b}^0+{c}}=\\simplify[!collectNumbers, basic]{{a}+{c}}=\\var{a+c}$. Therefore, the $y$-intercept is the point $(0,\\var{a+c})$.
\nb) Substitute $x=1$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}{b}^1+{c}}=\\var{a*b+c}$. Therefore, another easily found point is $(1,\\var{a*b+c})$.
\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}{b}^x+{c}}$ 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. Because as $x$ gets very small, $\\simplify{{b}^x}$ gets close to zero, $\\simplify{{a}{b}^x}$ gets close to zero, and $\\simplify{{a}{b}^x+{c}}$ gets close to $\\var{c}$. That is, for the curve $\\simplify{{a}{b}^x+{c}}$ the smaller $x$ gets, the closer $y$ gets to $\\var{c}$. In other words as $x$ approaches negative infinity, $y$ approaches $\\var{c}$. So the asymptote for $y=\\simplify{{a}{b}^x+{c}}$ is the line $y=\\var{c}$.
\ne) Given all the information above, it should be clear that the graph should look like
\n{graph1(1)}
", "parts": [{"scripts": {}, "showFeedbackIcon": true, "prompt": "The $y$-intercept of $y=\\simplify{{a}{b}^x+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.
", "marks": 0, "type": "gapfill", "gaps": [{"scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "minValue": "0", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "marks": 1, "type": "numberentry", "correctAnswerStyle": "plain", "maxValue": "0", "allowFractions": false, "mustBeReduced": false, "correctAnswerFraction": false}, {"scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "minValue": "{a+c}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "marks": 1, "type": "numberentry", "correctAnswerStyle": "plain", "maxValue": "{a+c}", "allowFractions": false, "mustBeReduced": false, "correctAnswerFraction": false}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": []}, {"scripts": {}, "showFeedbackIcon": true, "prompt": "Another easily found point on the curve is ${\\large(}1,$ [[0]]$\\large)$.
", "marks": 0, "type": "gapfill", "gaps": [{"scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "minValue": "{a*b+c}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "marks": 1, "type": "numberentry", "correctAnswerStyle": "plain", "maxValue": "{a*b+c}", "allowFractions": true, "mustBeReduced": false, "correctAnswerFraction": true}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": []}, {"scripts": {}, "minMarks": 0, "shuffleChoices": true, "choices": ["$y$ increases without bound.
", "$y$ decreases without bound.
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"], "distractors": ["", "", ""], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "displayType": "radiogroup", "matrix": ["if(a>0,1,0)", "if(a<0,1,0)", 0], "prompt": "As $x$ increases without bound,
", "marks": 0, "type": "1_n_2", "displayColumns": "1", "maxMarks": 0}, {"scripts": {}, "showFeedbackIcon": true, "prompt": "The horizontal asymptote of $y=\\simplify{{a}{b}^x+{c}}$ is $y=$ [[0]].
", "marks": 0, "type": "gapfill", "gaps": [{"scripts": {}, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "minValue": "{c}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": [], "showFeedbackIcon": true, "marks": 1, "type": "numberentry", "correctAnswerStyle": "plain", "maxValue": "{c}", "allowFractions": false, "mustBeReduced": false, "correctAnswerFraction": false}], "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "variableReplacements": []}, {"scripts": {}, "minMarks": 0, "shuffleChoices": true, "choices": ["{graph1(1)}
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", "marks": 0, "type": "1_n_2", "displayColumns": "2", "maxMarks": 0}], "variable_groups": [], "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}