3.5 - Rational Functions and Asymptotes

A rational function is a function that can be written as the ratio of two polynomials where the denominator isn't zero.

f(x) = p(x) / q(x)


The domain of a rational function is all real values except where the denominator, q(x) = 0.


The roots, zeros, solutions, x-intercepts (whatever you want to call them) of the rational function will be the places where p(x) = 0. That is, completely ignore the denominator. Whatever makes the numerator zero will be the roots of the rational function, just like they were the roots of the polynomial function earlier.

If you can write it in factored form, then you can tell whether it will cross or touch the x-axis at each x-intercept by whether the multiplicity on the factor is odd or even.

Vertical Asymptotes

An asymptote is a line that the curve approaches but does not cross. The equations of the vertical asymptotes can be found by finding the roots of q(x). Completely ignore the numerator when looking for vertical asymptotes, only the denominator matters.

If you can write it in factored form, then you can tell whether the graph will be asymptotic in the same direction or in different directions by whether the multiplicity is even or odd.

Asymptotic in the same direction means that the curve will go up or down on both the left and right sides of the vertical asymptote. Asymptotic in different directions means that the one side of the curve will go down and the other side of the curve will go up at the vertical asymptote.

Horizontal Asymptotes

A horizontal line is an asymptote only to the far left and the far right of the graph. "Far" left or "far" right is defined as anything past the vertical asymptotes or x-intercepts. Horizontal asymptotes are not asymptotic in the middle. It is okay to cross a horizontal asymptote in the middle.

The location of the horizontal asymptote is determined by looking at the degrees of the numerator (n) and denominator (m).


Sometimes, a factor will appear in the numerator and in the denominator. Let's assume the factor (x-k) is in the numerator and denominator. Because the factor is in the denominator, x=k will not be in the domain of the function. This means that one of two things can happen. There will either be a vertical asymptote at x=k, or there will be a hole at x=k.

Let's look at what will happen in each of these cases.

Oblique Asymptotes

When the degree of the numerator is exactly one more than the degree of the denominator, the graph of the rational function will have an oblique asymptote. Another name for an oblique asymptote is a slant asymptote.

To find the equation of the oblique asymptote, perform long division (synthetic if it will work) by dividing the denominator into the numerator. As x gets very large (this is the far left or far right that I was talking about), the remainder portion becomes very small, almost zero. So, to find the equation of the oblique asymptote, perform the long division and discard the remainder.