Section 9-4

Rational Functions

The first few problems in this section review simplifying rational expressions. (Recall that a rational expression is an expression a/b where a and b are polynomials.)

A rational function is a function f(x) = a, where a is a rational expression involving the variable x.

A SPECIAL TYPE OF RATIONAL FUNCTION: HYPERBOLAS

The "basic" hyperbola is the function

This function has a graph which consists of two disjoint parts. Note that 0 is not in the domain. Note also that the function approaches 0 asymptotically as x grows infinitely large (or infinitely negative), and that it approaches infinity as x approaches 0 from the positive side, negative infinity as x approaches 0 from the negative side.

The center of the hyperbola is the point of rotational symmetry. In the above example, the center is (0, 0).

The graph of the rational function

is a hyperbola whose center is (h, k). The constant a controls the "steepness" with which the graph approaches the asymptotes.