Last updated: October 22, 2026 · Written by the Math Made Easy math teachers
Looking for algebra 2 rational functions? Then this Algebra 2 lesson on Rational Functions: Asymptotes, Holes, and Graphs explains each idea step by step, with worked examples.
Quick answer: A rational function is a quotient of polynomials. Its graph has a vertical asymptote where a denominator factor does not cancel, a hole where a factor cancels, and a horizontal asymptote set by the degrees. The form y = a/(x − h) + k has asymptotes x = h and y = k.
Standards: F-IF.C.7d, F-IF.B.4 · Level: Grade 11 Algebra 2
What is a rational function and where is it undefined?
A rational function is a quotient of two polynomials, f(x) = p(x)q(x). Its domain is every real number except the zeros of the denominator q. At each such number the graph has either a vertical asymptote or a hole.
A vertical asymptote is a vertical line the graph approaches but never reaches; it occurs at a zero of the denominator that does not cancel. A hole occurs at a zero that cancels with the numerator. A horizontal asymptote describes the behavior of f(x) as x becomes very large in either direction.
| Degrees of p and q | Horizontal asymptote |
|---|---|
| Degree of p < degree of q | y = 0 |
| Degree of p = degree of q | y = (leading coefficient of p) / (leading coefficient of q) |
| Degree of p > degree of q | None |
How do you graph y = a/(x − h) + k?
This form is the parent function y = 1x stretched by a, shifted right h and up k. The vertical asymptote is x = h and the horizontal asymptote is y = k. If a > 0 the two branches lie in the upper right and lower left of the asymptote cross; if a < 0 they lie in the other two regions.
- Factor the numerator and denominator and cancel common factors; each canceled factor is a hole.
- Draw the vertical asymptotes at the remaining zeros of the denominator.
- Find the horizontal asymptote from the degrees.
- Plot the intercepts and a few points, then sketch each branch.
f(x) = x2 + x − 12x − 3 = (x + 4)(x − 3)x − 3. The factor x − 3 cancels, so f(x) = x + 4 with a hole at x = 3. The hole’s height is 3 + 4 = 7, so the hole is at (3, 7).
What is inverse variation?
Two quantities vary inversely when their product stays constant: xy = k, or y = kx. When one quantity doubles, the other halves. The graph is a rational function with asymptotes at the axes.
Do not confuse a hole with an asymptote: if a factor cancels, there is a hole, not an asymptote. Also, a graph can cross a horizontal asymptote; only vertical asymptotes are never crossed.
Cancel first, ask questions later. Everything that cancels is a hole; everything left in the denominator is an asymptote.
- The domain excludes every zero of the denominator.
- A canceled factor creates a hole; a remaining denominator factor creates a vertical asymptote.
- The horizontal asymptote depends on the degrees: smaller top gives y = 0, equal degrees give the ratio of leading coefficients.
- y = a/(x − h) + k has asymptotes x = h and y = k.
- Inverse variation means xy = k, so the graph is y = k/x.
Rational Functions: Asymptotes, Holes, and Graphs: frequently asked questions
What is the difference between a hole and a vertical asymptote?
Both come from a zero of the denominator. If the factor also cancels with the numerator, the graph has a hole at that x. If the factor does not cancel, the graph has a vertical asymptote there.
How do you find a horizontal asymptote?
Compare the degrees of the numerator and denominator. If the numerator has the smaller degree, the asymptote is y = 0. If the degrees are equal, it is the ratio of the leading coefficients. If the numerator has the larger degree, there is none.
What is the domain and range of y = a/(x − h) + k?
The domain is all real numbers except x = h. The range is all real numbers except y = k. Both excluded values are the equations of the asymptotes.
How do you recognize inverse variation?
The product of the two quantities is always the same number. A table in which x and y change in opposite directions with a constant product describes inverse variation.
Can a graph cross its horizontal asymptote?
Yes, for some rational functions the graph crosses a horizontal asymptote at a finite point. The asymptote describes the behavior for very large positive or negative x, so a crossing in the middle is allowed.
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