Graphs of rational equations can generate three types of asymptotes: horizontal asymptotes, vertical asymptotes and slant asymptotes. A horizontal asymptotes will always have a slope of 0, while that of a vertical asymptote will be infinity. But the slope of a slant asymptote depends on the polynomials in the equation. Slant asymptotes only appear in graphs of equations in which the degree of the numerator is exactly one higher than the degree of the denominator.
Write the polynomials in the numerator and the denominator in standard form by combining like terms and ordering terms by descending degree. For example, the equation y = (3 - 6x^2 + 2x) / (x - 3 + 2x) is written y = (-6x^2 + 2x + 3) / (3x - 3) in standard form.
Compare the degrees of the polynomials in the numerator and the denominator. The equation will only have a slant asymptote if the degree of the polynomial in the numerator is one greater than the degree of the polynomial in the denominator. In the example equation, the degree in the numerator is 2 and the degree in the denominator is 1, so there will be a slant asymptote.
Calculate the slope of the slant asymptote by dividing the leading coefficient of the polynomial in the numerator by the leading coefficient of the polynomial in the denominator. In the example, the leading coefficients are -6 and 3, so the slope of the slant asymptote is -6/3, or -2.
Verify the slope of the asymptote by plugging in very large positive and negative values of x. The value of the equation should approach the value of the equation y = mx, where m is the slope of the asymptote, as x goes to positive and negative infinity. For example, plug the value x = 20 into the example equation to get y = -2,357/57 or -41.35. This value is very close to -40, the value of y = -2x when x = 20.