Introduction to Rational Functions and Asymptotes
Section 1: Introduction to Rational Functions
A graphing calculator would be helpful. You can also use DESMOS. However, students should be able to logically determine the behavior of a rational function around the singular points and its end-behavior.
A rational function R is the ratio P/Q of two polynomials P and Q. Thus the domain of R is the set of all values of x for which Q(x) is not zero. The points a for which Q(x) = 0 are called singular points. How does R behave around a singular point a? If the values of R become unboundedly large then R has a vertical asymptote x - a. On the other hand if R approaches a finite number then the graph of R has a hole at that singular point.
The end-behavior of R is the study of values of R when x becomes unboundedly large in either direction. If the degree of P in the numerator is less than or equal to the degree of Q in the denominator, then the values approach a finite number L. The line y = L is a horizontal asymptote to R.
If the degree of P is more than the degree of Q, then we can write R = q + r/Q, where q is the quotient on (long) dividing Q into P and r is remainder. In this case, values of R become unboundedly large along the quotient function q. If q is linear then it is called slant asymptote. If q is nonlinear then it is called curve asymptote.
Review: Polynomial factoring and zeros, sign chart, long division method.
- Find the domain of a rational function.
- Study the behavior of the function around singular points.
- Identify the vertical asymptotes and holes.
- Study the end-behavior and identify horizontal asymptote(s) or curved/slant asymptotes any.
- Draw the graph using the above information.