Algebra-2

To graph the function , we'll analyze it in the following steps:

 

 1. Factorization

First, factor both the numerator and the denominator:

The numerator

The denominator

 

So the function can be rewritten as:

 

For , we can cancel out the common factor , and we obtain:

 

 2. Asymptotes and Holes

Hole:

Since we canceled out the factor from the numerator and the denominator, there is a hole in the graph at . To find the -coordinate of the hole, we substitute into the simplified function . We get . So, there is a hole at the point .

 

Vertical Asymptote:

Set the denominator of the simplified function , we find . So, the vertical asymptote is .

 

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 determine 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 . Thus, the horizontal asymptote is .

 

 3. Intercepts

x-intercept:

Set , for the simplified function , we have , which implies

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