Graphing $y=ab^x+c$
"}, "extensions": ["jsxgraph"], "type": "question", "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}^x+{c}}.\\]
\n\n", "variable_groups": [], "preamble": {"js": "", "css": ""}, "variables": {"b": {"templateType": "anything", "name": "b", "description": "", "group": "Ungrouped variables", "definition": "random(2..10)"}, "c": {"templateType": "anything", "name": "c", "description": "", "group": "Ungrouped variables", "definition": "random(-4..3 except [0,a])"}, "a": {"templateType": "anything", "name": "a", "description": "a
", "group": "Ungrouped variables", "definition": "random(-3..3 except [0,1,b])"}}, "tags": [], "parts": [{"variableReplacements": [], "showFeedbackIcon": true, "prompt": "The $y$-intercept of $y=\\simplify{{a}{b}^x+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.
", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "0", "type": "numberentry", "maxValue": "0", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1}, {"variableReplacements": [], "minValue": "{a+c}", "type": "numberentry", "maxValue": "{a+c}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1}], "variableReplacementStrategy": "originalfirst", "marks": 0}, {"variableReplacements": [], "showFeedbackIcon": true, "prompt": "Another easily found point on the curve is ${\\large(}1,$ [[0]]$\\large)$.
", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{a*b+c}", "type": "numberentry", "maxValue": "{a*b+c}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "correctAnswerFraction": true, "mustBeReducedPC": 0, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": true, "showCorrectAnswer": true, "scripts": {}, "marks": 1}], "variableReplacementStrategy": "originalfirst", "marks": 0}, {"distractors": ["", "", ""], "matrix": ["if(a>0,1,0)", "if(a<0,1,0)", 0], "displayColumns": "1", "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "choices": ["$y$ increases without bound.
", "$y$ decreases without bound.
", "$y$ approaches $\\var{c}$.
"], "displayType": "radiogroup", "variableReplacements": [], "shuffleChoices": true, "prompt": "As $x$ increases without bound,
", "minMarks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "marks": 0}, {"variableReplacements": [], "showFeedbackIcon": true, "prompt": "The horizontal asymptote of $y=\\simplify{{a}{b}^x+{c}}$ is $y=$ [[0]].
", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"variableReplacements": [], "minValue": "{c}", "type": "numberentry", "maxValue": "{c}", "variableReplacementStrategy": "originalfirst", "mustBeReduced": false, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "mustBeReducedPC": 0, "showFeedbackIcon": true, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1}], "variableReplacementStrategy": "originalfirst", "marks": 0}, {"distractors": ["", "", "", ""], "matrix": ["1", 0, 0, 0], "displayColumns": "2", "type": "1_n_2", "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "choices": ["{graph1(1)}
", "{graph1(2)}
", "{graph1(3)}
", "{graph1(4)}
"], "displayType": "radiogroup", "variableReplacements": [], "shuffleChoices": true, "prompt": "Which graph best represents $y=\\simplify{{a}{b}^x+{c}}$?
", "minMarks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "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, the $y$-intercept is the point $(0,\\var{a+c})$.
\nb) Substitute $x=1$ into the equation: $y=\\simplify[!collectNumbers, basic]{{a}{b}^1+{c}}=\\var{a*b+c}$. Therefore, another easily found point is $(1,\\var{a*b+c})$.
\nc) As $x$ gets larger and larger (increases without bound, or approaches infinity) $y=\\simplify{{a}{b}^x+{c}}$ gets smaller and smaller (decreases without bound, or approaches negative infinity). larger and larger (increases without bound, or approaches infinity).
\nd) 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 the curve $\\simplify{{a}{b}^x+{c}}$ 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 $y=\\simplify{{a}{b}^x+{c}}$ 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)}
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