# Determining the equations of vertical and horizontal asymptotes

**vertical asymptotes**

The equations of vertical asymptotes appear in the form *x *= *h. *This equation of a line has only the *x *variable — no *y *variable — and the number *h. *A vertical asymptote occurs in the rational function if *f *(*x*) and *g *(*x*) have no common factors, and it appears at whatever values the denominator equals zero — *g *(*x*) = 0. (In other words, vertical asymptotes occur at values that don’t fall in the domain of the rational function.)

A *discontinuity *is a place where a rational function doesn’t exist — you find a break in the flow of the numbers being used in the function equation. A discontinuity is indicated by a numerical value that tells you where the function isn’t defined; this number isn’t in the domain of the function. You know that a function is *discontinuous *wherever a vertical asymptote appears in the graph because vertical asymptotes indicate breaks or gaps in the domain.

Also Read:

To find the vertical asymptotes of the function , for example, you first note that there’s no common factor in the numerator and denominator. Then you set the denominator equal to zero. Factoring *x ^{2} – 4x + 3 *= 0, you get (

*x*– 1)(

*x*– 3) = 0. The solutions are

*x*= 1 and

*x*= 3, which are the equations of the vertical asymptotes.

**Also Read : Find a quadratic equation that has given roots using reverse factoring and reverse completing the square**

**horizontal asymptotes**

The horizontal asymptote of a rational function has an equation that appears in the form *y *= *k. *This linear equation has only the variable *y *— no *x *— and the *k *is some number. A rational function has only one horizontal

asymptote — if it has one at all (some rational functions have no horizontal asymptotes, others have one, and none of them have more than one). A rational function has a horizontal asymptote when the degree (highest power) of *f *(*x*), the polynomial in the numerator, is less than or equal to the degree of *g *(*x*), the polynomial in the denominator.

Here’s a rule for determining the equation of a horizontal asymptote. The horizontal asymptote of is when n = m*, *meaning that the highest degrees of the polynomials are the same. The fraction here is made up of the lead coefficients of the two polynomials. When *n *< *m, *meaning that the degree in the numerator is less than the degree in the denominator, *y *= 0.

If you want to find the horizontal asymptote for , for example, you use the previously stated rules. Because 4 < 5, the horizontal asymptote is *y *= 0. Now look at what happens when the degree of the denominator is the same as the degree of the numerator. The horizontal asymptote of , is *y *= 3 (*a _{n} *over b

_{m}). The fraction formed by the lead coefficients is .