Graphing Rational Functions
A rational function is a function of the form y = p(x)/q(x) where both p(x) and q(x) are polynomials. Although polynomials are defined for all real values of x, rational functions are not defined for those values of x for which the denominator, q(x), is 0. The x-intercepts (if any) of y are the zeros of the numerator, p(x), since the function is zero only when it's numerator is 0. The most important feature that distinguishes the graphs of rational functions is the presence of asymptotes. Generally, asymptotes of a function are lines that the graph of the function gets closer and closer to (but does not actually touch), as one travels out along that line in either direction.
The vertical asymptotes for a rational function are determined by the zeros of the denominator (i.e. the values for which the denominator equals 0). You can find the vertical asymptotes by equating the denominator to 0 and solving, and then see if y approaches infinity or negative infinity on each side of the potential asymptote.
The horizontal asymptotes of a function can be found by dividing both the numerator and denominator of the rational function by the highest power of x that appears in the denominator. You will then likely produce at least one term of the form c/xn. As x approaches infinity (positive or negative), this term approaches zero, thus it can be eliminated from the expression, and you can solve for y to find the horizontal asymptotes.
Click here to see an example.
