Identifying Asymptotes Graphically - AP Calculus BC

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point:}$\&x=\frac{2}{17}$\&$\text{We find a zero denominator for the function: }$\&$\frac{(13x - 9)}{(17x - 2)}$\end{align}$

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-$\frac{3}{8}$\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(6x - 18)}{(16x + 9)}$\end{align}$

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Question

$\begin{align}&$\text{Which of the four following functions is depicted in the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point centered around:}$\&x=1.3\&$\text{We find a zero denominator for the function: }$\&$\frac{(4x + 18)}{(14x - 19)}$\end{align}$

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Question

$\begin{align}&$\text{Which of the four following functions is depicted in the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-0.8\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(5x + 12)}{(6x - 10)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=0.3\&$\text{This matches the ratio of coefficients for the function: }$\&$\frac{(3x - 2)}{(10x - 11)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-1.12\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(19x + 9)}{(17x - 5)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point centered approximately around:}$\&x=-0.3\&$\text{We find a zero denominator for the function: }$\&-$\frac{(x - 13)}{(15x + 4)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point centered approximately around:}$\&x=0.1\&$\text{We find a zero denominator for the function: }$\&$\frac{(9x - 11)}{(15x - 1)}$\end{align}$

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=0.71\&$\text{This matches the ratio of coefficients for the function: }$\&$\frac{(12x - 20)}{(17x)}$\end{align}$

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=1.78\&$\text{This matches the ratio of coefficients for the function: }$\&$\frac{(16x - 13)}{(9x - 4)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=0.2\&$\text{This matches the ratio of coefficients for the function: }$\&$\frac{(3x + 14)}{(15x - 16)}$\end{align}$

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-1.67\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(20x + 10)}{(12x - 10)}$\end{align}$

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point:}$\&x=\frac{2}{17}$\&$\text{We find a zero denominator for the function: }$\&$\frac{(13x - 9)}{(17x - 2)}$\end{align}$

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Question

$\begin{align}&$\text{Decide which of the following functions, most likely represents the graph above.}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-$\frac{3}{8}$\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(6x - 18)}{(16x + 9)}$\end{align}$

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Question

$\begin{align}&$\text{Which of the four following functions is depicted in the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point centered around:}$\&x=1.3\&$\text{We find a zero denominator for the function: }$\&$\frac{(4x + 18)}{(14x - 19)}$\end{align}$

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Question

$\begin{align}&$\text{Which of the four following functions is depicted in the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-0.8\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(5x + 12)}{(6x - 10)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=0.3\&$\text{This matches the ratio of coefficients for the function: }$\&$\frac{(3x - 2)}{(10x - 11)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Notice that as x increases, the function graphed appears to}$\&$\text{level out, flattening towards a certain value. This value is}$\&$\text{what is known as a horizontal asymptote. An asymptote is a value}$\&$\text{that a function approaches, but never actually reaches. Think}$\&$\text{of a horizontal asymptote as the limit of a function as x approaches}$\&$\text{infinity. In such a case, as x approaches infinity, any constants}$\&$\text{added or subtracted in the numerator and denominator become}$\&$\text{irrelevant. What matters is the power of x in the denominator}$\&$\text{and the numerator; if those are the same, then the coefficients}$\&$\text{define the asymptote. We see that the function flattens towards:}$\&f(\infty)=-1.12\&$\text{This matches the ratio of coefficients for the function: }$\&-$\frac{(19x + 9)}{(17x - 5)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point centered approximately around:}$\&x=-0.3\&$\text{We find a zero denominator for the function: }$\&-$\frac{(x - 13)}{(15x + 4)}$\end{align}$

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Question

$\begin{align}&$\text{Of the following functions, which most likely represents the graph above?}$\end{align}$

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Answer

$\begin{align}&$\text{Observation of the graph shows that there’s a sharp increase}$\&$\text{and decrease on either side of a particular x-value. This type}$\&$\text{of behavior is observed when there’s a vertical asymptote. An}$\&$\text{asymptote is a value that a function may approach, but will}$\&$\text{never actually attain. In the case of vertical asymptotes, this}$\&$\text{behavior occurs if the function approaches infinity for a given}$\&$\text{x-value, often when a zero value appears in a denominator. Noting}$\&$\text{this rule, the above function has a zero in the denominator}$\&$\text{at a definite point centered approximately around:}$\&x=0.1\&$\text{We find a zero denominator for the function: }$\&$\frac{(9x - 11)}{(15x - 1)}$\end{align}$

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