Algebra-2

To graph the function , we'll analyze its key features as follows:

 

1. Simplification and Asymptotes

First, we can factor out a common factor in the numerator: .

 

Vertical Asymptote:

Set the denominator equal to zero, , we get . 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 . So, the horizontal asymptote is .

 

2. Intercepts

x-intercept:

Set , we have , which implies . Solving for , we get , . So, the -intercept is .

 

y-intercept:

Set in the function, we get . So, the -intercept is .

 

3. Monotonicity

We can use the quotient rule to find the derivative of the function to analyze its monotonicity. Let , and ,