Question

determine the asymptotes of the function graphed below.

a. there is a horizontal asymptote at ( y = - 2 ) and a vertical asymptote at ( x = 0 ).
b. there is a horizontal asymptote at ( y = - 2 ).
c. there are horizontal asymptotes at ( y = - 3 ) and ( y = - 1 ).
d. there is a vertical asymptote at ( y = 0 ) and a horizontal asymptote at ( x = - 2 ).

Explanation:

Step1: Recall the definitions of asymptotes

A vertical asymptote is a vertical line \(x = a\) where the function approaches infinity or negative infinity as \(x\) approaches \(a\). A horizontal asymptote is a horizontal line \(y = b\) which the function approaches as \(x\to\pm\infty\).

Step2: Analyze the graph for vertical asymptote

Looking at the graph, as \(x\) approaches \(0\) from the left and right, the function values go to positive and negative infinity. So, \(x = 0\) is a vertical asymptote.

Step3: Analyze the graph for horizontal asymptote

As \(x\to\pm\infty\), the function approaches \(y=-2\). So, \(y = - 2\) is a horizontal asymptote.

Answer:

A. There is a horizontal asymptote at \(y = - 2\) and a vertical asymptote at \(x = 0\).