Rational Functions

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.

Analyzing the Rational Function

    The following factors will be used to analyze the behavior of rational functions;
  1. The zero(es), which we will obtain by finding the values of x that make the numberator zero (N(x) = 0). The zeroes represent the points where the rational function crosses or touches the x axis.
  2. The vertical asymptote(s), which we will determining the values of x that make the denominator equal to zero (D(x)=0). Vertical asymptotes can be thought of as vertical lines that the function approaches but does not touch. A rational function does not go through a vertical asymptote (doesn't take on that value of x).
  3. The horizon asymptote, which we will define by taking the highest power of x in the numberator, along with its coefficient and dividing it by the highest power of x in the denominator along with its coefficient. The rules for the calculations are given below.
  4. The y axis intercept, which is determined by setting x equal to zero and finding the corresponding value of y (if it exists).
  5. The height above or below the x axis of the function for various values of x. This will be used to determine if the function is above or below the x axis or some horizontal asymptote.

Zeroes - To determine the zeroes of the rational function take the numberator, set it equal to zero and solve for the values of x.

    Examples
  • N(x) = 2x - 3, set N(x) = 0, 2x - 3 = 0, x = 3/2 so the function crosses the x axis at x = 3/2
  • N(x) = x2 - x, set N(x) = 0, x2 - x = 0, factoring gives: x(x-2) = 0, so, x = 0 and x = 2 are zeroes of the function.
  • N(x) = 16, set N(x) = 0, 16 = 0, this is never true so there are no zeroes for this function (it does not cross or touch the x axis)
  • N(x) = 2x, set N(x) = 0, 2x = 0, x = 0, this function crosses the x axis at the origin.

Asymptotes - We have defined two polynomials that make up a rational function. The numerator polynomial is labeled 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. Examples:
  • Horizontal Asymptotes: We will consider three cases
    1. 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. Example:
    2. 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 in the numberator and denominator that have the highest powers of x. Example:
    3. Degree of N(x) higher than that of D(x). There is a special case when 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. Speaking more generally if the highest power of x in the numerator is greater thant he highest power of x in the denominator the rational function will not have a horizontal asymptote. Example:

Y axis intercept - To determine the y axis intercept set x equal to zero and solve for the value of y.

    Examples
  • r(x) = (2x - 3)/ (x2-1), set x = 0 and r(0) = (0 - 3)/(0 - 1) = 3, so the function intercepts the y axis at the point (0,3)
  • r(x) = 14/[(x - 7)(x-2)], set x =0 and r(0) = 14/[(-7)(-2)] = 1, so the function intercepts the y axis at the point (0,1)
  • r(x)= 3x - 5/[x (x-1)], set x = 0 and r(0) = -5/[0 (-1)] = -5/0 = undefined, this function does not intercept the y axis.