Identifying Asymptotes Graphically - AP Calculus BC

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Question

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

Answer

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

Compare your answer with the correct one above

Question

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

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

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

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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