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AP Calculus AB : Functions, Graphs, and Limits

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Example Questions

Example Question #2 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(10x + 11)}{(8x - 2)}$

$\frac{(10x - 11)}{(8x + 2)}$

$\frac{(8x + 2)}{(10x - 11)}$

$\frac{(8x - 2)}{(10x + 11)}$

Correct answer:

$\frac{(8x - 2)}{(10x + 11)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-1.1\\&\text{We find a zero denominator for the function: }\\&\frac{(8x - 2)}{(10x + 11)}\end{align*}$

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Example Question #1 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$-\frac{(4x - 1)}{(x + 11)}$

$-\frac{(x - 11)}{(4x + 1)}$

$-\frac{(x + 11)}{(4x - 1)}$

$-\frac{(4x + 1)}{(x - 11)}$

Correct answer:

$-\frac{(x - 11)}{(4x + 1)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-0.3\\&\text{We find a zero denominator for the function: }\\&-\frac{(x - 11)}{(4x + 1)}\end{align*}$

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Example Question #1 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

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

Possible Answers:

$\frac{(3x + 3)}{(8x - 18)}$

$-\frac{(3x + 3)}{(8x - 18)}$

$\frac{(8x - 18)}{(3x + 3)}$

$-\frac{(8x - 18)}{(3x + 3)}$

Correct answer:

$-\frac{(8x - 18)}{(3x + 3)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-2.67\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(8x - 18)}{(3x + 3)}\end{align*}$

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Example Question #11 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

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

Possible Answers:

$\frac{(6x - 5)}{(10x + 1)}$

$\frac{(10x + 1)}{(6x - 5)}$

$\frac{(10x - 1)}{(6x + 5)}$

$\frac{(6x + 5)}{(10x - 1)}$

Correct answer:

$\frac{(10x - 1)}{(6x + 5)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-0.8\\&\text{We find a zero denominator for the function: }\\&\frac{(10x - 1)}{(6x + 5)}\end{align*}$

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Example Question #11 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$\frac{(8x - 18)}{(16x - 12)}$

$\frac{(16x - 12)}{(8x - 18)}$

$-\frac{(16x - 12)}{(8x - 18)}$

$-\frac{(8x - 18)}{(16x - 12)}$

Correct answer:

$\frac{(16x - 12)}{(8x - 18)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=2\\&\text{This matches the ratio of coefficients for the function: }\\&\frac{(16x - 12)}{(8x - 18)}\end{align*}$

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Example Question #13 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

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

Possible Answers:

$\frac{(11x - 18)}{(15x - 17)}$

$\frac{(15x - 17)}{(11x - 18)}$

$-\frac{(11x - 18)}{(15x - 17)}$

$-\frac{(15x - 17)}{(11x - 18)}$

Correct answer:

$-\frac{(11x - 18)}{(15x - 17)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-0.73\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(11x - 18)}{(15x - 17)}\end{align*}$

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Example Question #11 : Functions, Graphs, And Limits

Asymptoteplot

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

Possible Answers:

$\frac{(17x + 4)}{(11x - 18)}$

$\frac{(11x + 18)}{(17x - 4)}$

$\frac{(11x - 18)}{(17x + 4)}$

$\frac{(17x - 4)}{(11x + 18)}$

Correct answer:

$\frac{(11x - 18)}{(17x + 4)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-0.2\\&\text{We find a zero denominator for the function: }\\&\frac{(11x - 18)}{(17x + 4)}\end{align*}$

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Example Question #11 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

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

Possible Answers:

$\frac{(18x + 17)}{(12x - 11)}$

$-\frac{(12x - 11)}{(18x + 17)}$

$\frac{(12x - 11)}{(18x + 17)}$

$-\frac{(18x + 17)}{(12x - 11)}$

Correct answer:

$-\frac{(18x + 17)}{(12x - 11)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-1.5\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(18x + 17)}{(12x - 11)}\end{align*}$

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Example Question #11 : Asymptotic And Unbounded Behavior

Asymptoteplot

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

Possible Answers:

$-\frac{(10x + 13)}{(x + 1)}$

$-\frac{(x - 1)}{(10x - 13)}$

$-\frac{(10x - 13)}{(x - 1)}$

$-\frac{(x + 1)}{(10x + 13)}$

Correct answer:

$-\frac{(x + 1)}{(10x + 13)}$

Explanation:

$\begin{align*}&\text{Observation of the data points shows that there’s a sharp increase}\\&\text{and decrease on either side of a particular x-value. This type}\\&\text{of behavior is observed when there’s a vertical asymptote. An}\\&\text{asymptote is a value that a function may approach, but will}\\&\text{never actually attain. In the case of vertical asymptotes, this}\\&\text{behavior occurs if the function approaches infinity for a given}\\&\text{x-value, often when a zero value appears in a denominator. Noting}\\&\text{this rule, the above function has a zero in the denominator}\\&\text{at a definite point centered approximately around:}\\&x=-1.3\\&\text{We find a zero denominator for the function: }\\&-\frac{(x + 1)}{(10x + 13)}\end{align*}$

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Example Question #17 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Asymptoteplot

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

Possible Answers:

$\frac{(5x - 15)}{(19x - 15)}$

$\frac{(19x - 15)}{(5x - 15)}$

$-\frac{(5x - 15)}{(19x - 15)}$

$-\frac{(19x - 15)}{(5x - 15)}$

Correct answer:

$-\frac{(19x - 15)}{(5x - 15)}$

Explanation:

$\begin{align*}&\text{Notice that as x increases, the points of data graphed appears}\\&\text{to level out, flattening towards a certain value. This value}\\&\text{is what is known as a horizontal asymptote. An asymptote is}\\&\text{a value that a function approaches, but never actually reaches.}\\&\text{Think of a horizontal asymptote as the limit of a function as}\\&\text{x approaches infinity. In such a case, as x approaches infinity,}\\&\text{any constants added or subtracted in the numerator and denominator}\\&\text{become irrelevant. What matters is the power of x in the denominator}\\&\text{and the numerator; if those are the same, then the coefficients}\\&\text{define the asymptote. We see that the function flattens towards:}\\&f(\infty)=-3.8\\&\text{This matches the ratio of coefficients for the function: }\\&-\frac{(19x - 15)}{(5x - 15)}\end{align*}$

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AP Calculus AB : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #2 : Asymptotic And Unbounded Behavior

Example Question #1 : Asymptotic And Unbounded Behavior

Example Question #1 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #11 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #11 : Asymptotic And Unbounded Behavior

Example Question #13 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #11 : Functions, Graphs, And Limits

Example Question #11 : Asymptotic Behavior In Terms Of Limits Involving Infinity

Example Question #11 : Asymptotic And Unbounded Behavior

Example Question #17 : Asymptotic Behavior In Terms Of Limits Involving Infinity

All AP Calculus AB Resources

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