Algebra-2

To graph the function , we can analyze it from the following aspects:

 

 1. Asymptotes

Vertical Asymptote:

We determine the vertical asymptote by setting the denominator equal to zero. For the function , when we solve , we get . So, the vertical asymptote of this function is .

 

Horizontal Asymptote:

Here, both the numerator and the denominator are polynomials of degree . According to the method of finding horizontal asymptotes for rational functions (by comparing the coefficients of the highest degree terms when the degrees are the same), the coefficient of in the numerator is , and that in the denominator is also . Therefore, the horizontal asymptote is .

 

 2. Intercepts

y-intercept:

To find the -intercept, we substitute into the function. So, . That means the -intercept is the point .

 

x-intercept:

To find the -intercept, we set . That is, . Since a fraction is zero when its numerator is zero while the denominator isn't zero, we solve and get . Hence, the -intercept is .

 

 3. Monotonicity

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