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

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

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

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

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

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 $\\var{c}$.

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

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 $y=\\var{c}$.

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 $y=\\var{c}$.

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

{graph1(1)}

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

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The $y$-intercept is the point at which the graph intercepts the $y$-axis. The $y$-axis has equation $x=0$ so the $y$-intercept will always have an $x$ value of $0$, and the $y$ value will be the result of substituting $x=0$ into the equation $y=\\simplify[!noleadingminus]{{a}{b}^({s}x+{d})+{c}}$.

Substituting $x=0$ gives
\\begin{align}y&=\\simplify[!collectnumbers,!noleadingminus]{{a}{b}^({d})+{c}}\\\\&=\\simplify[fractionnumbers]{{a*b^d}+{c}}\\\\&=\\simplify[fractionnumbers]{{a*b^d+c}}\\end{align} and therefore the $y$-intercept is the point with coordinates $(0,\\var[fractionnumbers]{a*b^d+c})$.

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

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Substituting $x=1$ into $y=\\simplify[!noleadingminus]{{a}{b}^({s}x+{d})+{c}}$:

\\begin{align}y&=\\simplify[!collectnumbers, !noleadingminus]{{a}{b}^({s}+{d})+{c}}\\\\&=\\simplify[fractionnumbers]{{a*b^(s+d)}+{c}}\\\\&=\\simplify[fractionnumbers]{{a*b^(s+d)+c}}\\end{align}

Therefore, $(1,\\simplify[fractionnumbers]{{a*b^(s+d)+c}})$ is another point on the curve.

As $x$ increases without bound,

Since $\\var{b}$ is greater than $1$, as $x$ increases, the value of $\\var{b}^x$ grows exponentially, however, $\\var{b}^{-x}$ decays exponentially to $0$. Therefore, $\\simplify[!noleadingminus, all]{{a}{b}^({s}x+{d})+{c}}$ approaches $\\var{c}$. Furthermore, since $\\var{a}$ is positive, $y=\\simplify[!noleadingminus, all]{{a}{b}^({s}x+{d})+{c}}$ increases without bound. Furthermore, since $\\var{a}$ is negative, $y=\\simplify[!noleadingminus, all]{{a}{b}^({s}x+{d})+{c}}$ decreases without bound.Furthermore, since $\\var{a}$ is positive, $y=\\simplify[!noleadingminus, all]{{a}{b}^({s}x+{d})+{c}}$ approaches $\\var{c}$ from above. Furthermore, since $\\var{a}$ is negative, $y=\\simplify[!noleadingminus, all]{{a}{b}^({s}x+{d})+{c}}$ approaches $\\var{c}$ from below.

"}], "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["

$y$ increases without bound.

", "

$y$ decreases without bound.

", "

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

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

The horizontal asymptote is $y=\\var{c}$.

Based on the above, 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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