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AP Calculus BC : Asymptotic and Unbounded Behavior

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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:

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

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

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

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

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 #1 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

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

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

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

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

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 #1 : 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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Example Question #11 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

$-\frac{(20x + 10)}{(12x - 10)}$

$-\frac{(12x - 10)}{(20x + 10)}$

$\frac{(20x + 10)}{(12x - 10)}$

$\frac{(12x - 10)}{(20x + 10)}$

Correct answer:

$-\frac{(20x + 10)}{(12x - 10)}$

Explanation:

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

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Example Question #12 : Identifying Asymptotes Graphically

Asymptoteplot

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

Possible Answers:

$-\frac{(16x - 9)}{(3x + 18)}$

$-\frac{(16x + 9)}{(6x - 18)}$

$-\frac{(3x + 18)}{(16x - 9)}$

$-\frac{(6x - 18)}{(16x + 9)}$

Correct answer:

$-\frac{(6x - 18)}{(16x + 9)}$

Explanation:

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

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AP Calculus BC : Asymptotic and Unbounded Behavior

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

Example Question #1 : Identifying Asymptotes Graphically

Example Question #1 : Identifying Asymptotes Graphically

Example Question #1 : Identifying Asymptotes Graphically

Example Question #11 : Identifying Asymptotes Graphically

Example Question #12 : Identifying Asymptotes Graphically

All AP Calculus BC Resources

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