Determining the equations of vertical and horizontal 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.
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 x2 – 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.
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 (an over bm ). The fraction formed by the lead coefficients is .