Algebra-2

Here's how we can analyze the given function to determine which graph is correct:

1. Simplify the function

First, we factor the numerator and the denominator:

The numerator

The denominator

 

So the function can be written as:

 

We can cancel out the common factor (with the condition ), and the simplified function is:

 

 2. Find the vertical asymptote

The vertical asymptote occurs when the denominator of the simplified function is equal to zero.

Set , we get . So the vertical asymptote is

 

 3. Find the horizontal asymptote

To find the horizontal asymptote, we consider the degrees of the polynomial in the numerator and the denominator.

 

Since the degree of the numerator and the denominator are the same (both are of degree ), we divide the leading coefficients.

 

The leading coefficient of the numerator is

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