COLLEGE ALGEBRA
MATH 110
Southwestern   
Adventist University 
 
   Distance Education Lawrence E. Turner, Jr., Ph.D.  


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Graphing Rational Functions


Given a rational function which is the ratio of two polynomials:

   f(x)  =     p(x)  
q(x)

where p(x) is a n degree polynomial and q(x) is a m degree polynomial:

  p(x) = anxn + an−1xn−1 + ... + a1x + a0,        an ≠ 0.

and

  q(x) = bmxm + bm−1xm−1 + ... + b1x + b0,        bm ≠ 0.

To graph the function, perform:
  1. determine any vertical asymptotes
    • occur when the denominator, q(x), is zero and the numerator, p(x), is not zero
    • find the zeros of the denominator, q(x)
    • check to see if the numerator is also zero at these points
        if so, then the point is not a vertical asymptote but represents a common factor—cancel it out, but remember to exclude that point when plotting

  2. determine any horizontal asymptote
      occurs only when the degree of denominator is greater than or equal to the degree of numerator; that is, m ≥ n:

        m > n, horizontal asymptote at  y = 0

        m = n, horizontal asymptote at  y = an/bn

  3. determine any oblique asymptote
      occurs only when the degree of the denominator is exactly one less than the degree of the numerator; that is, m = n−1

        use long division to divide the numerator by the denominator to obtain:

           f(x)  =     p(x)     =   ax + d +    r(x)  
        q(x)q(x)

        the line y = cx+d is an oblique asymptote   (Note: r(x) is of less degree than q(x) so this part goes to zero as x gets large.)

  4. sketch the graph
    • draw the coordinate axes
    • add the vertical asymptotes
    • add the horizontal or oblique asymptote
    • evaluate the function at a few convenient points to determine what portions of the plane it lies in and get some idea of its behavior
    • notes:
      • one may obtain one horizontal or one oblique asymptote, never both
      • the graph can never cross a vertical asymptote
      • the graph may cross a horizontal or oblique asymptote near the origin—only when x gets large positive or negative does the graph approach a horizontal or oblique asymptote
 
© 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006, 2007, 2008 by Lawrence Turner