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{"name": "Jamie's copy of Graphing exponentials with vertical transformations and base>1 ", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "statement": "<p>The following questions will gauge your understanding of exponentials and how to graph them.&nbsp;</p>\n<p>The exponential you will be working with for this question is \\[y=\\simplify{{a}{b}^x+{c}}.\\]</p>\n<p></p>\n<p></p>", "advice": "<p>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,&nbsp;the $y$-intercept is the point $(0,\\var{a+c})$.</p>\n<p>b) Substitute $x=1$ into the equation:&nbsp;$y=\\simplify[!collectNumbers, basic]{{a}{b}^1+{c}}=\\var{a*b+c}$. Therefore, another easily found point is $(1,\\var{a*b+c})$.</p>\n<p>c) As $x$ gets larger and larger&nbsp;(increases without bound, or approaches infinity) $y=\\simplify{{a}{b}^x+{c}}$ gets <span data-jme-visible=\"a&lt;0\">smaller and smaller (decreases without bound, or approaches negative infinity).</span> <span data-jme-visible=\"a&gt;0\">larger and larger (increases without bound, or approaches infinity).</span></p>\n<p>d) 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&nbsp;the curve $\\simplify{{a}{b}^x+{c}}$ the smaller $x$ gets, the closer $y$ gets to $\\var{c}$.&nbsp;In other words as $x$ approaches negative infinity, $y$ approaches $\\var{c}$. So&nbsp;the asymptote for $y=\\simplify{{a}{b}^x+{c}}$ is the line $y=\\var{c}$.</p>\n<p>e) Given all the information above, it should be clear that the graph should look like&nbsp;</p>\n<p>{graph1(1)}</p>", "variables": {"b": {"name": "b", "definition": "random(2..10)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "a": {"name": "a", "definition": "random(-3..3 except [0,1,b])", "group": "Ungrouped variables", "templateType": "anything", "description": "<p>a</p>"}, "c": {"name": "c", "definition": "random(-4..3 except [0,a])", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "rulesets": {}, "tags": [], "preamble": {"js": "", "css": ""}, "extensions": ["jsxgraph"], "name": "Jamie's copy of Graphing exponentials with vertical transformations and base>1 ", "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 Math.pow(b,x)}],{strokeWidth:2});}\nif(quad==2){board.create('functiongraph',[function(x){ return Math.pow(1/b,x)}],{strokeWidth:2});}\nif(quad==3){board.create('functiongraph',[function(x){ return -Math.pow(1/b,x)}],{strokeWidth:2});}\nif(quad==4){board.create('functiongraph',[function(x){ return -Math.pow(b,x)}],{strokeWidth:2});}\n\nreturn div;", "parameters": [["quad", "number"]], "language": "javascript"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "ungrouped_variables": ["b", "a", "c"], "metadata": {"description": "<p>Graphing $y=ab^x+c$</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "parts": [{"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true, "gaps": [{"maxValue": "0", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacements": [], "allowFractions": false, "minValue": "0", "type": "numberentry", "showFeedbackIcon": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "scripts": {}, "correctAnswerFraction": false}, {"maxValue": "{a+c}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacements": [], "allowFractions": false, "minValue": "{a+c}", "type": "numberentry", "showFeedbackIcon": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "scripts": {}, "correctAnswerFraction": false}], "variableReplacements": [], "scripts": {}, "prompt": "<p>The $y$-intercept of $y=\\simplify{{a}{b}^x+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.</p>"}, {"variableReplacementStrategy": "originalfirst", "type": "gapfill", "showFeedbackIcon": true, "marks": 0, "showCorrectAnswer": true, "gaps": [{"maxValue": "{a*b+c}", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "variableReplacements": [], "allowFractions": true, "minValue": "{a*b+c}", "type": "numberentry", "showFeedbackIcon": true, "marks": 1, "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "scripts": {}, "correctAnswerFraction": true}], "variableReplacements": [], "scripts": {}, "prompt": "<p>Another easily found point on the curve is&nbsp;${\\large(}1,$&nbsp;[[0]]$\\large)$.</p>"}, {"displayType": "radiogroup", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "shuffleChoices": true, "matrix": ["if(a>0,1,0)", "if(a<0,1,0)", 0], "variableReplacements": [], "prompt": "<p>As $x$ increases without bound,&nbsp;</p>", "choices": ["<p>$y$ increases without bound.</p>", "<p>$y$ decreases without bound.</p>", "<p>$y$ approaches $\\var{c}$.</p>"], "minMarks": 0, "type": "1_n_2", "showFeedbackIcon": true, "marks": 0, "distractors": ["", "", ""], "displayColumns": "1", "scripts": {}, "maxMarks": 0}, {"displayType": "radiogroup", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "shuffleChoices": true, "matrix": ["1", 0, 0, 0], "variableReplacements": [], "prompt": "<p>Which graph best represents $y=\\simplify{{a}{b}^x+{c}}$?</p>", "choices": ["<p>{graph1(1)}</p>", "<p>{graph1(2)}</p>", "<p>{graph1(3)}</p>", "<p>{graph1(4)}</p>"], "minMarks": 0, "type": "1_n_2", "showFeedbackIcon": true, "marks": 0, "distractors": ["", "", "", ""], "displayColumns": "2", "scripts": {}, "maxMarks": 0}], "type": "question", "contributors": [{"name": "Jamie Wood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/79/"}]}]}], "contributors": [{"name": "Jamie Wood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/79/"}]}