The slope of a curve can become very steep but never will be vertical.
The asymptotes of a curve, or graph of a function, are lines that the curve approaches but never touches. Asymptotes come in three varieties: vertical, horizontal and oblique (diagonal). Introductory units on asymptotes typically cover only vertical and horizontal asymptotes, because oblique asymptotes are more complicated to calculate. Vertical asymptotes are vertical lines located at x values for which the function is undefined. Horizontal asymptotes are lines representing the values that the function approaches at very small or very large values of x.
Instructions
Vertical Asymptotes
1. Simplify the function until it is expressed as one fraction. For example, if the function is y = 1/(x^2 - x - 2) * (x - 2), the simplified form would be y = (x - 2)/(x^2 - x - 2).
2. Factor the function. For example, y = (x - 2)/(x^2 - x - 2) factors to y = (x - 2)/[(x + 1)(x - 2)]. The "x - 2" term in the numerator and denominator cancel, yielding y = 1/(x + 1).
3. Calculate the values of x for which the denominator of the reduced function is equal to zero. For example, if the function is y = 1/(x + 1), the denominator of the function would equal zero when x = -1. The vertical asymptote is therefore located at x = -1. Note that the function is also undefined at x = 2 because the denominator of the original function equals zero when x = 2. However, because the "x - 2" factor canceled out when the fraction was reduced, there is no asymptote at x = 2, only an infinitesimally narrow gap in the curve.
Horizontal Asymptotes
4. Identify the leading term in the numerator and denominator. The leading term is the term with the variable raised to the highest power. For example, in the function y = (x - 2)/(x^2 - x - 2), the leading term is x in the numerator and x^2 in the denominator.
5. Write a new fraction with only the leading terms from the numerator and denominator. For example, you would write x/(x^2) for the function y = (x - 2)/(x^2 - x - 2).
6. Reduce the new fraction. For example, you would reduce x/(x^2) to 1/x.
7. Input very large positive and negative numbers into the reduced fraction to determine the y values that the function approaches at extreme values of x. For example, the graph of y = 1/x approaches but never touches zero at very large positive and negative values of x. Thus, y = 0 is the horizontal asymptote.
Tags: numerator denominator, example function, leading term, term numerator, term numerator denominator, that function, very large
