Algebra-2

Let's analyze the function  to determine which graph is correct:

 

1. Factorization

First, factor both the numerator and the denominator:

Numerator:

Denominator:

So the function can be written as . For , we can simplify this to .

 

 2. Asymptotes

Vertical Asymptote:

Set the denominator of the original (or unsimplified) form equal to zero, , which gives . So, there is a vertical asymptote at .

 

Horizontal Asymptote:

Since the degree of the polynomial in the numerator (degree ) is the same as the degree of the polynomial in the denominator (degree ), we find the horizontal asymptote by taking the ratio of the coefficients of the highest degree terms. The coefficient of  in the numerator is  and in the denominator is also . So, the horizontal asymptote is .

 

 3. Intercepts

x-intercept:

Set , for the simplified function , we have , which implies . So, the -intercept is .

 

y-intercept:

Set  in the simplified function , we get

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