Math · College algebra · Concept
Rational functions: asymptotes and holes
A rational function is a quotient of polynomials, f(x) = p(x)/q(x). Factor first: a factor that cancels leaves a hole, and a zero of the denominator that remains gives a vertical asymptote. Comparing the degrees of p and q gives the horizontal asymptote, or a slant asymptote, that describes the end behavior.
The domain: no dividing by zero
f(x) = p(x)/q(x) is undefined wherever q(x) = 0. Find those x-values first; they are excluded from the domain whatever happens later.
Holes and vertical asymptotes
Factor p and q. If a factor x − c cancels, the graph has a hole at x = c: f is undefined there, but it approaches the value of the simplified expression. If x − c is still in the denominator after cancelling, the graph has a vertical asymptote x = c: the values grow without bound as x approaches c.
End behavior from the degrees
Far to the left and right, the leading terms of p and q dominate.
| Degrees | End behavior |
|---|---|
| deg p < deg q | Horizontal asymptote y = 0 |
| deg p = deg q | Horizontal asymptote y = (leading coefficient of p)/(leading coefficient of q) |
| deg p = deg q + 1 | Slant asymptote: the quotient from polynomial division |
| deg p > deg q + 1 | No horizontal or slant asymptote |
Intercepts
The x-intercepts are the zeros of the simplified numerator; the zero of a cancelled factor is a hole, not an intercept. The y-intercept is f(0), if 0 is in the domain.
A graph may cross a horizontal asymptote
A horizontal asymptote describes the far left and far right only, and the graph can cross it in between. A graph never crosses a vertical asymptote, because the function is undefined there.
Common mistakes
- Calling every zero of the denominator a vertical asymptote: a cancelled factor gives a hole.
- Cancelling a factor and forgetting its exclusion: the simplified formula is defined at the hole, but the function is not.
- Using the ratio of leading coefficients when the degrees differ; that ratio is the asymptote only when the degrees are equal.
- Believing a graph can never cross its horizontal asymptote.
Key terms
- Rational expression
- A fraction whose numerator and denominator are polynomials. Any value that makes the original denominator zero stays excluded, even if that factor cancels when you simplify.
- Vertical asymptote
- A vertical line x = a that the graph approaches as the function’s values grow without bound near a, often where a denominator is zero. The graph is not joined across it.
- Horizontal asymptote
- A horizontal line y = L that the graph approaches as x → ∞ or x → −∞. Unlike a vertical asymptote, the graph can cross it.
- Removable discontinuity
- A hole in a graph: the function has a limit at the point but is undefined there or has a different value. Redefining the value to match the limit removes the hole.
- Slant asymptote
- A slanted line y = mx + b that a graph approaches as x → ±∞. A rational function has one when the numerator’s degree is one more than the denominator’s, and long division finds it.
- End behavior
- How a function behaves as x tends toward positive or negative infinity. For a polynomial, the degree and leading coefficient determine the eventual directions of both ends.
Work through an example
Find the domain, hole, asymptotes and intercepts of f(x) = (x² − 1)/(x² − x − 2).
Graph a rational function with a hole →Sources and scope
Authored study material. Tool results depend on the stated inputs and model assumptions.
Try in the workspace
Open the example inputs, change a value and keep a useful result on your board.
See the hole and asymptotes in Graph Explore the hole in the limit explorer Check the simplification in Math Open worked example on a board Asymptotes and holes in Math ReferenceYour existing work stays on this device. Examples open as editable copies.