// Numbas version: exam_results_page_options
{"name": "Graphing exponentials of the form y=b^x with 0<b<1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Graphing exponentials of the form y=b^x with 0<b<1", "advice": "<p><strong>Note:</strong> if we so desire we can&nbsp;rewrite this exponential with a negative index since&nbsp;$\\left(\\simplify[fractionNumbers]{{b}}\\right)^x=\\left(\\simplify[fractionNumbers]{{1/b}}\\right)^{-x}$.</p>\n<p>a) To find the $y$-intercept, substitute $x=0$ into the equation: $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^0=1$.&nbsp;Therefore,&nbsp;the $y$-intercept is the point $(0,1)$.</p>\n<p>b)&nbsp;Substitute $x=1$ into the equation:&nbsp;$y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^1=\\simplify[fractionNumbers]{{b}}$. Therefore, another easily found point is $\\left(0,\\simplify[fractionNumbers]{{b}}\\right)$.</p>\n<p>c) Let's investigate what happens to the value of $y$ when we add 1 to the value of $x$:</p>\n<p>\\[\\left(\\simplify[fractionNumbers]{{b}}\\right)^{x+1}=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x\\left(\\simplify[fractionNumbers]{{b}}\\right)^1=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x\\left(\\simplify[fractionNumbers]{{b}}\\right)\\] &nbsp;That is, the old $y$ value is multiplied by $\\simplify[fractionNumbers]{{b}}$ (which actually decreases the $y$ value since $\\simplify[fractionNumbers]{{b}}&lt;1$).</p>\n<p>d) Since $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ is an exponential and as $x$ increases $y$ decreases to 0, we call this <em>exponential decay</em>.</p>\n<p>e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ the larger&nbsp;$x$ gets, the closer $y$ gets to $0$. In other words as $x$ approaches infinity, $y$ approaches $0$. This means that the asymptote for $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ is the line $y=0$ (the $x$-axis).</p>\n<p>f) Given all the information above, it should be clear that the graph should look like&nbsp;</p>\n<p>{graph1(1)}</p>", "variable_groups": [], "ungrouped_variables": ["b"], "functions": {"graph1": {"language": "javascript", "parameters": [["quad", "number"]], "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\nb = Numbas.jme.unwrapValue(scope.variables.b);\n\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;", "type": "html"}}, "extensions": ["jsxgraph"], "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "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=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x.\\]</p>", "preamble": {"js": "", "css": ""}, "tags": [], "parts": [{"showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "<p>The $y$-intercept of $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ is the point $\\large($[[0]], [[1]]$\\large)$.</p>", "type": "gapfill", "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "0", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": false, "minValue": "0", "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}, {"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "1", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": true, "minValue": "1", "correctAnswerFraction": true, "variableReplacements": [], "mustBeReduced": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "<p>Another easily found point on the curve is&nbsp;${\\large(}1,$&nbsp;[[0]]$\\large)$.</p>", "type": "gapfill", "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "{b}", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": true, "minValue": "{b}", "correctAnswerFraction": true, "variableReplacements": [], "mustBeReduced": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}, {"showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "<p>Given $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$, everytime&nbsp;$x$ increases by 1, $y$ &nbsp;[[0]].</p>", "type": "gapfill", "gaps": [{"variableReplacements": [], "choices": ["<p>increases by 1.</p>", "<p>decreases by 1.</p>", "<p>increases by $\\simplify[fractionNumbers]{{b}}$.&nbsp;</p>", "<p>decreases by $\\simplify[fractionNumbers]{{b}}$.</p>", "<p>is multiplied by $\\simplify[fractionNumbers]{{b}}$.</p>", "<p>is divided&nbsp;by $\\simplify[fractionNumbers]{{b}}$.</p>"], "maxMarks": 0, "scripts": {}, "displayType": "radiogroup", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "shuffleChoices": true, "showFeedbackIcon": true, "marks": 0, "displayColumns": "1", "matrix": [0, 0, 0, 0, "1", 0], "distractors": ["", "", "", "", "", ""], "minMarks": 0}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}, {"variableReplacements": [], "choices": ["<p>exponential growth</p>", "<p>exponential&nbsp;decay</p>"], "maxMarks": 0, "scripts": {}, "prompt": "<p>Would&nbsp;$y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ best be described as exponential decay or exponential growth?</p>", "displayType": "radiogroup", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "shuffleChoices": false, "showFeedbackIcon": true, "marks": 0, "displayColumns": 0, "matrix": ["0", "1"], "distractors": ["", ""], "minMarks": 0}, {"showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "<p>The horizontal asymptote of&nbsp;$y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$ is $y=$&nbsp;[[0]].</p>", "type": "gapfill", "gaps": [{"correctAnswerStyle": "plain", "scripts": {}, "maxValue": "0", "mustBeReducedPC": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": false, "minValue": "0", "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true}, {"variableReplacements": [], "choices": ["<p>{graph1(1)}</p>", "<p>{graph1(2)}</p>", "<p>{graph1(3)}</p>", "<p>{graph1(4)}</p>"], "maxMarks": 0, "scripts": {}, "prompt": "<p>Which graph best represents $y=\\left(\\simplify[fractionNumbers]{{b}}\\right)^x$?</p>", "displayType": "radiogroup", "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "type": "1_n_2", "shuffleChoices": true, "showFeedbackIcon": true, "marks": 0, "displayColumns": "2", "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""], "minMarks": 0}], "rulesets": {}, "metadata": {"description": "<p>Graphing exponentials with a&nbsp;base between 0 and 1&nbsp;and no transformations take place.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "variables": {"b": {"templateType": "anything", "description": "", "definition": "random(1/10,1/5,3/10,2/5,1/2,3/5,7/10,4/5,9/10,1/3,1/4,1/6,1/7,1/8)", "group": "Ungrouped variables", "name": "b"}}, "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/"}]}