Rational Functions

Rational Functions

Rational functions are those of the form:

y = f(x)/g(x)

Apart from finding the x-intercepts, y-intercepts and turning points of rational functions (which we have described in our Techniques section), an important property to consider are the asymptotes.

Vertical Asymptotes

Almost all graphs of rational functions have vertical asymptotes, i.e. asymptotes with equations of the form x = a, where a is a constant. To find the vertical asymptote, set g(x) = 0 and solve for x.

Example: In the graph of y = 1/x, x = 0 is the vertical asymptote.

Horizontal Asymptotes

Some graphs of rational functions have horizontal asymptotes. These are asymptotes with equations of the form y = a, where a is a constant. There are two categories of horizontal asymptotes.

When degree of numerator < degree of denominator

When the degree of f(x) is less than that of g(x), the horizontal asymptote is always y = 0. This can be verified by finding the limit as x tends to infinity. You'll find that y will tend to zero.

When degree of numerator = degree of denominator

When the degree of f(x) = degree of g(x), the horizontal asymptote is found by splitting the function y into its partial fractions:

y = c + fraction

The horizontal asymptote will be y = c, where c is a constant.

Oblique Asymptotes

Oblique asymptotes occur when the degree of the numerator is greater than that of the degree of the denominator. It can be linear, quadratic, cubic, quartic and so on. Here we focus on the linear oblique asymptote, but those of curves can be found similarly.

Similar to the horizontal asymptote, first split y into its partial fractions.

y = mx + c + fraction

y = mx + c is the oblique asymptote.