Question

Find the holes, asymptotes, and zeros of the following function:

Answer 1

Hello!

Holes will occur where the same factor appears in the numerator and denominator. Restrictions should be listed after simplifying the function.

The zeros of the graph is when the graph crosses the x-axis. It can be found by replacing y with zero, and solving for x.

Vertical and horizontal asymptotes: brainly.com/question/10678557. :)

Firstly, let's find the holes of the graph.

$\frac{x^{2}-7x+6}{2x^{2}+8x+8}$ can be factored into: $\frac{(x-6)(x-1)}{2(x+2)(x+2)}$. According to the factors, there are no holes that occur in this function.

Let's find the zeros! I'll make this easy and I'll post a graph. :)

The zeros are x = 6, and x = 1.

Vertical Asymptote:

2x² + 8x + 8 = 0

2(x + 2)(x + 2)

x + 2 = 0 (subtract two from both sides)

x = -2

Horizontal Asymptote:

If the degree of the numerator = degree of denominator, then y = leading coefficient of numerator / leading coefficient of denominator is the horizontal asymptote. This occurs in this problem.

y = 1/2

Therefore, the function $\frac{x^{2}-7x+6}{2x^{2}+8x+8}$, has no holes, has a vertical asymptote of

x = -2, a horizontal asymptote of y = 1/2 and zeros at (1, 0) and (6,0), or x = 1, and x = 6.