Is it possible to not have a vertical asymptote

The graph actually crosses its asymptote at one point. This can never happen with a vertical asymptote. Example 3.

Now an example where the numerator is one degree higher than the denominator. The numerator is a second degree polynomial while the denominator is of the first degree.

We use long division and divide the numerator by the denominator. We can now rewrite f x :. The graph is shown below. If we want to speculate on further possibilities we can see that if the degree of the numerator is 2 degrees greater than that of the denominator then the graph goes out of the coordinate system following a parabolic curve and so on.

Example 4. How To: Given a rational function, find the domain. Set the denominator equal to zero. The domain is all real numbers except those found in Step 2. Show Solution Begin by setting the denominator equal to zero and solving. How To: Given a rational function, identify any vertical asymptotes of its graph. Factor the numerator and denominator. Note any restrictions in the domain of the function.

Reduce the expression by canceling common factors in the numerator and the denominator. Note any values that cause the denominator to be zero in this simplified version. These are where the vertical asymptotes occur. Note any restrictions in the domain where asymptotes do not occur. A rational function has at most one horizontal or oblique asymptote , and possibly many vertical asymptotes. Vertical asymptotes occur only when the denominator is zero. In other words, vertical asymptotes occur at singularities, or points at which the rational function is not defined.

To find the horizontal asymptote , we note that the degree of the numerator is two and the degree of the denominator is one. The three rules that horizontal asymptotes follow are based on the degree of the numerator, n, and the degree of the denominator, m. Rational function. In mathematics, a rational function is any function which can be defined by a rational fraction, i. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K.

Process for Graphing a Rational Function Find the intercepts, if there are any. Find the vertical asymptotes by setting the denominator equal to zero and solving. Find the horizontal asymptote, if it exists, using the fact above.

The vertical asymptotes will divide the number line into regions. Sketch the graph. A horizontal asymptote is a y-value on a graph which a function approaches but does not actually reach. Rule 2: If the degree of the numerator is greater than the degree of the denominator, then there is no horizontal asymptote.

A slant oblique asymptote occurs when the polynomial in the numerator is a higher degree than the polynomial in the denominator. To find the slant asymptote you must divide the numerator by the denominator using either long division or synthetic division. Examples: Find the slant oblique asymptote. It should be noted that, if the degree of the numerator is larger than the degree of the denominator by more than one, the end behavior of the graph will mimic the behavior of the reduced end behavior fraction.

For instance, if we had the function. The horizontal asymptote of a rational function can be determined by looking at the degrees of the numerator and denominator. Find the horizontal asymptote and interpret it in context of the problem. Both the numerator and denominator are linear degree 1. Because the degrees are equal, there will be a horizontal asymptote at the ratio of the leading coefficients. In the numerator, the leading term is t , with coefficient 1.

In the denominator, the leading term is 10 t , with coefficient The horizontal asymptote will be at the ratio of these values:. First, note that this function has no common factors, so there are no potential removable discontinuities.

The function will have vertical asymptotes when the denominator is zero, causing the function to be undefined. The numerator has degree 2, while the denominator has degree 3. A rational function will have a y -intercept when the input is zero, if the function is defined at zero. A rational function will not have a y -intercept if the function is not defined at zero.

Likewise, a rational function will have x -intercepts at the inputs that cause the output to be zero. Since a fraction is only equal to zero when the numerator is zero, x -intercepts can only occur when the numerator of the rational function is equal to zero. We can find the y -intercept by evaluating the function at zero.