// Numbas version: exam_results_page_options {"name": "Jamie's copy of Graphing exponentials with horizontal and vertical transformations and base>1 ", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"js": "", "css": ""}, "ungrouped_variables": ["b", "a", "c", "d", "s"], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "

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

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The exponential you will be working with for this question is \$y=\\simplify{{a}{b}^({s}x+{d})+{c}}.\$

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The $y$-intercept of $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$ is the point $\\large($[[0]], [[1]]$\\large)$.

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Another important point on the curve is ${\\large(}\\simplify[fractionNumbers]{{-d/s}},$ [[0]]$\\large)$.

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As $x$ increases without bound,

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$y$ increases without bound.

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$y$ decreases without bound.

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$y$ approaches $\\var{c}$.

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Which graph best represents $y=\\simplify{{a}{b}^({s}x+{d})+{c}}$?

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{graph1(1)}

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{graph1(2)}

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{graph1(3)}

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{graph1(4)}

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a

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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 $(0,\\var[fractionNumbers]{a*b^d+c})$.

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b) 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 $\\left(\\simplify[fractionNumbers]{{-d/s}},\\var{a+c}\\right)$.

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c) 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 $\\var{c}$.

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d) An asymptote is a line or curve that approaches a given curve arbitrarily closely.

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As $x$ 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 $y=\\var{c}$.

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As $x$ 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 $y=\\var{c}$.

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

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{graph1(1)}

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Graphing $y=ab^{\\pm x+d}+c$

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