Identifying Asymptotes Graphically

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AP Calculus BC › Identifying Asymptotes Graphically

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Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

Asymptoteplot

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

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

CORRECT

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

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

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

Explanation

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

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