Question

determine the asymptotes of the function graphed below.

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

Explanation:

Step1: Recall the definition of vertical asymptote

A vertical asymptote is a vertical line \(x = a\) where the function approaches \(\pm\infty\) as \(x\) approaches \(a\).

Step2: Recall the definition of horizontal asymptote

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

Step3: Analyze the graph for vertical asymptote

Looking at the graph, as \(x\) approaches \(- 2\) from the left and right, the function approaches \(\pm\infty\). So \(x=-2\) is a vertical asymptote.

Step4: Analyze the graph for horizontal asymptote

As \(x\to\pm\infty\), the function approaches \(y = 0\). So \(y = 0\) is a horizontal asymptote.

Answer:

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