Graphing $y=ab^{\\pm x+d}+c$
"}, "variables": {"s": {"definition": "random(-1,1)", "group": "Ungrouped variables", "name": "s", "description": "", "templateType": "anything"}, "d": {"definition": "random(-4..4 except 0)", "group": "Ungrouped variables", "name": "d", "description": "", "templateType": "anything"}, "b": {"definition": "random(2..6)", "group": "Ungrouped variables", "name": "b", "description": "", "templateType": "anything"}, "a": {"definition": "random(-4..4 except [0,1,b])", "group": "Ungrouped variables", "name": "a", "description": "a
", "templateType": "anything"}, "c": {"definition": "random(-6..6 except [0,a,b])", "group": "Ungrouped variables", "name": "c", "description": "", "templateType": "anything"}}, "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}^({s}x+{d})+{c}}.\\]
", "variable_groups": [], "rulesets": {}, "name": "Graphing exponentials with horizontal and vertical transformations and base>1 ", "advice": "a) To find the $y$-intercept, substitute $x=0$ into the equation: $y=\\simplify[!collectNumbers,basic,fractionNumbers]{{a}{b}^{d}+{c}}=\\var[fractionNumbers]{a*b^d+c}$. Therefore, the $y$-intercept is the point $(0,\\var[fractionNumbers]{a*b^d+c})$.
\nb) Substitute $x=\\simplify[fractionNumbers]{{-d/s}}$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}{b}^0+{c}}=\\var{a+c}$. Therefore, another easily found point is $\\left(\\simplify[fractionNumbers]{{-d/s}},\\var{a+c}\\right)$.
\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity). closer and closer to $\\var{c}$.
\nd) An asymptote is a line or curve that approaches a given curve arbitrarily closely.
\n\nAs $x$ gets very large, $\\simplify{{b}^({s}x+{d})}$ gets close to zero, $\\simplify{{a}{b}^({s}x+{d})}$ gets close to zero, and $\\simplify{{a}{b}^({s}x+{d})+{c}}$ gets close to $\\var{c}$. That is, for the curve $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ the larger $x$ gets, the closer $y$ gets to $\\var{c}$. In other words as $x$ approaches infinity, $y$ approaches $\\var{c}$. So the asymptote for $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ is the line $y=\\var{c}$.
\n\nAs $x$ gets very small, $\\simplify{{b}^({s}x+{d})}$ gets close to zero, $\\simplify{{a}{b}^({s}x+{d})}$ gets close to zero, and $\\simplify{{a}{b}^({s}x+{d})+{c}}$ gets close to $\\var{c}$. That is, for the curve $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ 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 $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ 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": [{"prompt": "The $y$-intercept of $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.
", "showFeedbackIcon": true, "gaps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": true, "maxValue": "0", "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "minValue": "0", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": false, "scripts": {}}, {"answer": "{a*b^d+c}", "variableReplacementStrategy": "originalfirst", "showpreview": true, "showFeedbackIcon": true, "type": "jme", "showCorrectAnswer": true, "variableReplacements": [], "vsetrange": [0, 1], "vsetrangepoints": 5, "checkingtype": "absdiff", "checkingaccuracy": 0.001, "marks": 1, "checkvariablenames": false, "answersimplification": "fractionNumbers", "scripts": {}, "expectedvariablenames": []}], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "variableReplacements": []}, {"prompt": "Another important point on the curve is ${\\large(}\\simplify[fractionNumbers]{{-d/s}},$ [[0]]$\\large)$.
", "showFeedbackIcon": true, "gaps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": true, "type": "numberentry", "showCorrectAnswer": true, "maxValue": "{a+c}", "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "minValue": "{a+c}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": true, "scripts": {}}], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "variableReplacements": []}, {"showFeedbackIcon": true, "choices": ["$y$ increases without bound.
", "$y$ decreases without bound.
", "$y$ approaches $\\var{c}$.
"], "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "type": "1_n_2", "showCorrectAnswer": true, "variableReplacements": [], "matrix": ["if(s>0 and a>0,1,0)", "if(s>0 and a<0,1,0)", "if(s<0,1,0)"], "distractors": ["", "", ""], "displayColumns": "1", "maxMarks": 0, "minMarks": 0, "marks": 0, "prompt": "As $x$ increases without bound,
", "scripts": {}}, {"prompt": "The horizontal asymptote of $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ is $y=$ [[0]].
", "showFeedbackIcon": true, "gaps": [{"showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": true, "maxValue": "{c}", "variableReplacements": [], "correctAnswerStyle": "plain", "mustBeReduced": false, "minValue": "{c}", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "allowFractions": false, "scripts": {}}], "variableReplacementStrategy": "originalfirst", "marks": 0, "showCorrectAnswer": true, "type": "gapfill", "scripts": {}, "variableReplacements": []}, {"showFeedbackIcon": true, "choices": ["{graph1(1)}
", "{graph1(2)}
", "{graph1(3)}
", "{graph1(4)}
"], "shuffleChoices": true, "variableReplacementStrategy": "originalfirst", "displayType": "radiogroup", "type": "1_n_2", "showCorrectAnswer": true, "variableReplacements": [], "matrix": ["1", 0, 0, 0], "distractors": ["", "", "", ""], "displayColumns": "2", "maxMarks": 0, "minMarks": 0, "marks": 0, "prompt": "Which graph best represents $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$?
", "scripts": {}}], "functions": {"graph1": {"parameters": [["quad", "number"]], "type": "html", "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);\nd = Numbas.jme.unwrapValue(scope.variables.d);\ns = Numbas.jme.unwrapValue(scope.variables.s);\n\n\n\nif(quad==1){board.create('functiongraph',[function(x){ return a*(Math.pow(b,s*x+d))+c}],{strokeWidth:2});}\nif(quad==2){board.create('functiongraph',[function(x){ return a*(Math.pow(1/b,s*x+d))+c}],{strokeWidth:2});}\nif(quad==3){board.create('functiongraph',[function(x){ return -a*(Math.pow(1/b,s*x+d))+c}],{strokeWidth:2});}\nif(quad==4){board.create('functiongraph',[function(x){ return -a*(Math.pow(b,s*x+d))+c}],{strokeWidth:2});}\n\nreturn div;"}}, "tags": [], "ungrouped_variables": ["b", "a", "c", "d", "s"], "variablesTest": {"maxRuns": 100, "condition": ""}, "extensions": ["jsxgraph"], "preamble": {"js": "", "css": ""}, "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/"}]}