A rational function is the ratio of two polynomials N(x)/D(x) which have no common factors. The denominator polynomial cannot be equal to zero. In this lesson we will look at the long range and short range behavior of rational functions.
Asymptotes (Inspired by Analytic Geometry, Ross R. Middlemiss, McGraw-Hill Book Company, Inc, 1955)
We will define two polynomials that make up a fraction. The numerator polynomial will be N(x) and the denominator polynomial will be D(x).
Vertical Asymptotes: To find the vertical asymptotes set D(x) = 0 and solve for the values of x. The figure provides as example of this:
Horizontal Asymptotes: There are three possible cases
- Case I. Degree of N(x) lower than degree of D(x). In this case the value of the fraction will approach zero as x becomes larger and larger in absolute value, and the graph will be asymptotic to the x-axis.
- Case II. Degree of N(x) same as that of D(x). In this case, as x becomes larger and larger in absolute value, the value of the fraction approaches a number which is equal to the ratio of the coefficients of the terms that have the highest powers of x.
- Case III. Degree of N(x) higher than that of D(x). We will only deal with the case where the Degree of N(x) is one higher than the degree of D(x). In that case you will have a slant asymptote. To find the asymptote divide N(x) by D(x) to get the equation of the asymptote.