Asymptote Lesson A Rational expression is the quotient of two polynomials. An asymptote is a line that a graph approaches but does not reach as the values for x or y become very large or very small. For Example if you graphed the line y = (x + 3) / (x2 - x - 6) you will see two vertical asymptotes The bottom factors to (x + 2)(x - 3) so either -2 or 3 will make the denominator 0, which it can not be. The graphs will approach those lines (x = -2 and x = 3), but it will not reach them. So in general, to find the vertical asymptotes you find what values for x make the denominator 0. Finding the Horizontal Asymptote We can find the horizontal asymptotes by : If the numerator has a smaller degree than the denominator, then the horizontal asymptote will be y = 0. The graph will approach that line but wont't cross it. (for example : y = 3x2 / x3 If the degree if the numerator and the denominator is the same, then the horizontal asymptote will be the quotient coefficient of the highest powered variables. For example y = (4x2 + 3) / (x2 - 7) then coefficients will be 4 / 1 so the horizontal asymptote will be y = 4. If the numerator has a higher degree then the denominator, then there is no horizontal asymptote. y = 1 /(x-3) + 5 The vertical asymptote is 3 (x cannot be 3) ,br> The horizontal asymptote is 5. Since x cannot be 3, 1/(x-3) will always be greater or smaller than 5, but it won't be equal to 5. g(x) = 1 / (x+8) - 1 The vertical asymptote is -8 (x cannot be -8) The horizontal asymptote is -1 (y will never be equal to -1 since 1/(x+8) will never be equal to 0.