// Numbas version: exam_results_page_options {"name": "Graphing exponentials with vertical transformations and base>1", "extensions": ["jsxgraph"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"rulesets": {}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International", "description": "

Graphing \$y=ab^x+c\$

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

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

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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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As \$x\$ increases without bound,

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The horizontal asymptote of \$y=\\simplify{{a}{b}^x+{c}}\$ is \$y=\$ [[0]].

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

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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]{{a}{b}^0+{c}}=\\simplify[!collectNumbers, basic]{{a}+{c}}=\\var{a+c}\$. Therefore, the \$y\$-intercept is the point \$(0,\\var{a+c})\$.

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b) 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})\$.

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c) 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).

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d) 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}\$.

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

\n

{graph1(1)}

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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}^x+{c}}.\\]

\n

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