// Numbas version: exam_results_page_options {"name": "nuExam04 - Exponentials", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"statement": "

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

\n

The exponential you will be working with for this question is \\[y=\\var{b}^x.\\]

\n

", "ungrouped_variables": ["b"], "preamble": {"css": "", "js": ""}, "advice": "

a) To find the \$y\$-intercept, substitute \$x=0\$ into the equation: \$y=\\var{b}^0=1\$. Therefore, the \$y\$-intercept is the point \$(0,1)\$.

\n

b) Substitute \$x=1\$ into the equation: \$y=\\var{b}^1=\\var{b}\$. Therefore, another easily found point is \$(1,\\var{b})\$.

\n

c) Let's investigate what happens to the value of \$y\$ when we add 1 to the value of \$x\$:

\n

\\[\\var{b}^{x+1}=\\var{b}^x\\var{b}^1=\\var{b}^x\\var{b}\\]  That is, the old \$y\$ value is multiplied by \$\\var{b}\$, so we can say that \$y\$ is increased by a factor of \$\\var{b}\$.

\n

d) Since \$y=\\var{b}^x\$ is an exponential and as \$x\$ increases \$y\$ increases without bound, we call this exponential growth.

\n

e) An asymptote is a line or curve that approaches a given curve arbitrarily closely. For the curve \$y=\\var{b}^x\$ the smaller \$x\$ gets, the closer \$y\$ gets to \$0\$. In other words as \$x\$ approaches negative infinity, \$y\$ approaches \$0\$. This means that the asymptote for \$y=\\var{b}^x\$ is the line \$y=0\$ (the \$x\$-axis).

\n

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

\n

{graph1(1)}

", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "functions": {"graph1": {"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;", "language": "javascript", "type": "html", "parameters": [["quad", "number"]]}}, "variables": {"b": {"definition": "random(2..10)", "description": "", "templateType": "anything", "name": "b", "group": "Ungrouped variables"}}, "tags": [], "extensions": ["jsxgraph"], "name": "nuExam04 - Exponentials", "metadata": {"description": "

The easiest type of exponential to graph where the base is greater than 1 and no transformations take place.

", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "parts": [{"showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "useCustomName": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "minValue": "0", "allowFractions": false, "scripts": {}, "marks": 1, "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "unitTests": [], "maxValue": "0", "type": "numberentry", "customName": "", "customMarkingAlgorithm": ""}, {"notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": false, "useCustomName": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "minValue": "1", "allowFractions": false, "scripts": {}, "marks": 1, "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "unitTests": [], "maxValue": "1", "type": "numberentry", "customName": "", "customMarkingAlgorithm": ""}], "unitTests": [], "variableReplacements": [], "type": "gapfill", "customName": "", "customMarkingAlgorithm": "", "prompt": "

The \$y\$-intercept of \$y=\\var{b}^x\$ is the point \$\\large(\$[[0]], [[1]]\$\\large)\$.

"}, {"showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerFraction": true, "useCustomName": false, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "showFractionHint": true, "minValue": "{b}", "allowFractions": true, "scripts": {}, "marks": 1, "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "unitTests": [], "maxValue": "{b}", "type": "numberentry", "customName": "", "customMarkingAlgorithm": ""}], "unitTests": [], "variableReplacements": [], "type": "gapfill", "customName": "", "customMarkingAlgorithm": "", "prompt": "

Another easily found point on the curve is \${\\large(}1,\$ [[0]]\$\\large)\$.

"}, {"showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "sortAnswers": false, "scripts": {}, "marks": 0, "showCorrectAnswer": true, "gaps": [{"matrix": [0, 0, 0, 0, "1", 0], "showFeedbackIcon": true, "unitTests": [], "showCellAnswerState": true, "variableReplacementStrategy": "originalfirst", "useCustomName": false, "maxMarks": 0, "scripts": {}, "marks": 0, "choices": ["

increases by 1.

", "

decreases by 1.

", "

increases by {b-1}.

", "

decreases by {b-1}.

", "

increases by a factor of {b}.

", "

decreases by a factor of {b}.

"], "showCorrectAnswer": true, "minMarks": 0, "distractors": ["", "", "", "", "", ""], "shuffleChoices": true, "displayColumns": 0, "displayType": "dropdownlist", "variableReplacements": [], "type": "1_n_2", "customName": "", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true}], "unitTests": [], "variableReplacements": [], "type": "gapfill", "customName": "", "customMarkingAlgorithm": "", "prompt": "

Given \$y=\\var{b}^x\$, everytime \$x\$ increases by 1, \$y\$  [[0]].

"}, {"unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "displayColumns": 0, "scripts": {}, "choices": ["

exponential growth

", "

exponential decay

"], "matrix": ["1", 0], "shuffleChoices": false, "displayType": "radiogroup", "variableReplacements": [], "useCustomName": false, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "showCellAnswerState": true, "type": "1_n_2", "maxMarks": 0, "marks": 0, "showCorrectAnswer": true, "minMarks": 0, "distractors": ["", ""], "customName": "", "prompt": "

Would \$y=\\var{b}^x\$ best be described as exponential decay or exponential growth?

"}, {"unitTests": [], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "displayColumns": "2", "scripts": {}, "choices": ["

{graph1(1)}

", "

{graph1(2)}

", "

{graph1(3)}

", "

{graph1(4)}

"], "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "displayType": "radiogroup", "variableReplacements": [], "useCustomName": false, "customMarkingAlgorithm": "", "showFeedbackIcon": true, "showCellAnswerState": true, "type": "1_n_2", "maxMarks": 0, "marks": 0, "showCorrectAnswer": true, "minMarks": 0, "distractors": ["", "", "", ""], "customName": "", "prompt": "

Which graph best represents \$y=\\var{b}^x\$?

"}], "type": "question", "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}]}], "contributors": [{"name": "Michael Proudman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/269/"}, {"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}, {"name": "Maria Aneiros", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3388/"}]}