Question

determine the asymptotes of the function graphed below. a. there is a vertical asymptote at ( y = 5 ) and a horizontal asymptote at ( x = 3 ). b. there is a horizontal asymptote at ( x = 5 ) and a vertical asymptote at ( y = 3 ). c. there is a vertical asymptote at ( x = 5 ) and a horizontal asymptote at ( y = 3 ). d. there is a horizontal asymptote at ( y = 5 ) and a vertical asymptote at ( x = 3 ).

Explanation:

Step1: Recall the definitions of vertical and horizontal 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 \(3\) (from the left and right), the function values go to positive or negative infinity. So, the vertical asymptote is \(x = 3\).

Step3: Analyze the graph for horizontal asymptote

As \(x\to\pm\infty\), the function approaches \(y = 5\). So, the horizontal asymptote is \(y = 5\).

Answer:

D. There is a horizontal asymptote at \(y = 5\) and a vertical asymptote at \(x = 3\).