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}}.\\]
", "extensions": ["jsxgraph"], "parts": [{"showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 0, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReduced": false, "minValue": "0", "correctAnswerFraction": false, "maxValue": "0", "allowFractions": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 1, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0}, {"answer": "{a*b^d+c}", "vsetrangepoints": 5, "checkingtype": "absdiff", "variableReplacementStrategy": "originalfirst", "checkvariablenames": false, "showCorrectAnswer": true, "vsetrange": [0, 1], "answersimplification": "fractionNumbers", "variableReplacements": [], "scripts": {}, "expectedvariablenames": [], "marks": 1, "checkingaccuracy": 0.001, "type": "jme", "showpreview": true, "showFeedbackIcon": true}], "prompt": "The $y$-intercept of $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.
", "type": "gapfill", "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 0, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain", "mustBeReduced": false, "minValue": "{a+c}", "correctAnswerFraction": true, "maxValue": "{a+c}", "allowFractions": true, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "marks": 1, "type": "numberentry", "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0}], "prompt": "Another important point on the curve is ${\\large(}\\simplify[fractionNumbers]{{-d/s}},$ [[0]]$\\large)$.
", "type": "gapfill", "variableReplacementStrategy": "originalfirst"}, {"minMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "prompt": "As $x$ increases without bound,
", "maxMarks": 0, "displayColumns": "1", "showCorrectAnswer": true, "distractors": ["", "", ""], "matrix": ["if(s>0 and a>0,1,0)", "if(s>0 and a<0,1,0)", "if(s<0,1,0)"], "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "choices": ["$y$ increases without bound.
", "$y$ decreases without bound.
", "$y$ approaches $\\var{c}$.
"], "marks": 0, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}, {"minMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "prompt": "Which graph best represents $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$?
", "maxMarks": 0, "displayColumns": "2", "showCorrectAnswer": true, "distractors": ["", "", "", ""], "matrix": ["1", 0, 0, 0], "showFeedbackIcon": true, "variableReplacements": [], "scripts": {}, "choices": ["{graph1(1)}
", "{graph1(2)}
", "{graph1(3)}
", "{graph1(4)}
"], "marks": 0, "type": "1_n_2", "variableReplacementStrategy": "originalfirst"}], "rulesets": {}, "name": "Jamie's copy of Graphing exponentials with horizontal and vertical transformations and base>1 ", "variables": {"d": {"definition": "random(-4..4 except 0)", "name": "d", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"definition": "random(-4..4 except [0,1,b])", "name": "a", "description": "a
", "group": "Ungrouped variables", "templateType": "anything"}, "b": {"definition": "random(2..6)", "name": "b", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "s": {"definition": "random(-1,1)", "name": "s", "description": "", "group": "Ungrouped variables", "templateType": "anything"}, "c": {"definition": "random(-6..6 except [0,a,b])", "name": "c", "description": "", "group": "Ungrouped variables", "templateType": "anything"}}, "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)}
", "tags": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "Graphing $y=ab^{\\pm x+d}+c$
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