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AP Calculus BC : Identifying Asymptotes Graphically

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Example Questions

Example Question #4 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(13x + 9)}{(17x + 2)}$

$\frac{(17x + 2)}{(13x + 9)}$

$\frac{(13x - 9)}{(17x - 2)}$

$\frac{(17x - 2)}{(13x - 9)}$

Correct answer:

$\frac{(13x - 9)}{(17x - 2)}$

Explanation:

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

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Example Question #5 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(4x + 18)}{(14x - 19)}$

$\frac{(4x - 18)}{(14x + 19)}$

$\frac{(14x - 19)}{(4x + 18)}$

$\frac{(14x + 19)}{(4x - 18)}$

Correct answer:

$\frac{(4x + 18)}{(14x - 19)}$

Explanation:

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

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Example Question #1 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

$\frac{(12x - 5)}{(10x + 6)}$

$-\frac{(5x + 12)}{(6x - 10)}$

$\frac{(18x + 6)}{(12x - 5)}$

$-\frac{(6x - 10)}{(5x + 12)}$

Correct answer:

$-\frac{(5x + 12)}{(6x - 10)}$

Explanation:

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

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Example Question #7 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(10x - 11)}{(3x - 2)}$

$-\frac{(10x - 11)}{(3x - 2)}$

$-\frac{(3x - 2)}{(10x - 11)}$

$\frac{(3x - 2)}{(10x - 11)}$

Correct answer:

$\frac{(3x - 2)}{(10x - 11)}$

Explanation:

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

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Example Question #8 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(17x - 5)}{(19x + 9)}$

$\frac{(19x + 9)}{(17x - 5)}$

$-\frac{(17x - 5)}{(19x + 9)}$

$-\frac{(19x + 9)}{(17x - 5)}$

Correct answer:

$-\frac{(19x + 9)}{(17x - 5)}$

Explanation:

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

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Example Question #9 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$-\frac{(15x - 4)}{(x + 13)}$

$-\frac{(x + 13)}{(15x - 4)}$

$-\frac{(15x + 4)}{(x - 13)}$

$-\frac{(x - 13)}{(15x + 4)}$

Correct answer:

$-\frac{(x - 13)}{(15x + 4)}$

Explanation:

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

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Example Question #10 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(9x - 11)}{(15x - 1)}$

$\frac{(9x + 11)}{(15x + 1)}$

$\frac{(15x + 1)}{(9x + 11)}$

$\frac{(15x - 1)}{(9x - 11)}$

Correct answer:

$\frac{(9x - 11)}{(15x - 1)}$

Explanation:

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

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Example Question #1 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

$\frac{(12x - 20)}{(17x)}$

$-\frac{(17x)}{(12x - 20)}$

$-\frac{(12x - 20)}{(17x)}$

$\frac{(17x)}{(12x - 20)}$

Correct answer:

$\frac{(12x - 20)}{(17x)}$

Explanation:

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

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Example Question #2 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

$-\frac{(9x - 4)}{(16x - 13)}$

$-\frac{(16x - 13)}{(9x - 4)}$

$\frac{(16x - 13)}{(9x - 4)}$

$\frac{(9x - 4)}{(16x - 13)}$

Correct answer:

$\frac{(16x - 13)}{(9x - 4)}$

Explanation:

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

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Example Question #3 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

$-\frac{(3x + 14)}{(15x - 16)}$

$\frac{(3x + 14)}{(15x - 16)}$

$\frac{(15x - 16)}{(3x + 14)}$

$-\frac{(15x - 16)}{(3x + 14)}$

Correct answer:

$\frac{(3x + 14)}{(15x - 16)}$

Explanation:

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

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AP Calculus BC : Identifying Asymptotes Graphically

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #4 : Asymptotic And Unbounded Behavior

Example Question #5 : Asymptotic And Unbounded Behavior

Example Question #1 : Identifying Asymptotes Graphically

Example Question #7 : Asymptotic And Unbounded Behavior

Example Question #8 : Asymptotic And Unbounded Behavior

Example Question #9 : Asymptotic And Unbounded Behavior

Example Question #10 : Asymptotic And Unbounded Behavior

Example Question #1 : Identifying Asymptotes Graphically

Example Question #2 : Identifying Asymptotes Graphically

Example Question #3 : Identifying Asymptotes Graphically

All AP Calculus BC Resources

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