Graphing Rational Functions Calculator

What Is a Rational Function?

A rational function is a fraction made of two polynomials. One polynomial sits on top, the other on the bottom. A simple example is f(x) = 1 / x. Another is f(x) = (x + 1) / (x - 2).

The top part is the numerator. The bottom part is the denominator. The most important fact about rational functions is this: you can never divide by zero. Wherever the denominator is zero, the function has no value — and the graph breaks.

How to Find Asymptotes

An asymptote is a line that the graph gets closer and closer to but never touches. Rational functions have two main kinds.

Vertical asymptotes

Set the denominator equal to zero and solve for x. Each answer is a vertical asymptote, unless that factor also cancels with the numerator. For f(x) = 1 / (x - 2), the denominator is zero at x = 2, so the graph has a vertical asymptote at x = 2. Near that line, the y values shoot up to positive infinity on one side and down to negative infinity on the other.

Horizontal asymptotes

Compare the degree of the numerator with the degree of the denominator:

• Top degree smaller: the horizontal asymptote is y = 0.
• Degrees equal: the horizontal asymptote is y = the ratio of the leading coefficients. For (3x + 1) / (2x - 5), it is y = 3/2.
• Top degree larger by one: there is no horizontal asymptote. Instead, there is a slant asymptote — a diagonal line. Divide the polynomials to find it.

The function graphing calculator draws the asymptotes as part of the graph, so you can see at once where the curve flattens out.

Holes and Intercepts

Not every break in a rational graph is an asymptote. Sometimes the graph has a hole — a single missing point. A hole appears when the same factor is in both the numerator and the denominator and cancels out. For f(x) = (x - 1)(x + 2) / (x - 1), the factor (x - 1) cancels, so the graph looks like the line y = x + 2 with a hole at x = 1.

Intercepts are easier. The x-intercepts are the x values where the numerator is zero (and the denominator is not zero at the same time). The y-intercept is the value f(0) — just put 0 in for x. For f(x) = (x + 1) / (x - 2), the x-intercept is (-1, 0) and the y-intercept is (0, -1/2).

How to Graph a Rational Function

Follow these steps in order, and any rational function becomes easy to graph:

1. Factor the numerator and the denominator.
2. Find holes from factors that cancel.
3. Find vertical asymptotes from the remaining denominator zeros.
4. Find the horizontal or slant asymptote by comparing degrees.
5. Find the intercepts. Numerator zero gives x-intercepts; f(0) gives the y-intercept.
6. Check the sign on each side of the asymptotes, then sketch the curve.

Try it with f(x) = 1 / (x - 2). The denominator is zero at x = 2, so there is a vertical asymptote at x = 2. The top degree is smaller, so the horizontal asymptote is y = 0. The graph has two branches: one in the top right and one in the bottom left. Type the same function into the calculator above and compare.

Domain of a Rational Function

The domain is all x values except the ones that make the denominator zero. For f(x) = 1 / (x - 2), the domain is all real numbers except x = 2. We write that as (-∞, 2) ∪ (2, ∞).

Remember: a cancelled factor still counts. In f(x) = (x - 1)(x + 2) / (x - 1), x = 1 is not in the domain even though the factor cancels — the graph has a hole, not a value.

Practice With Our Free Tool

Use the graphing rational functions calculator above to check your homework. Enter the function, look at the asymptotes and holes, and compare the graph with your sketch. Everything is free, with no download and no sign-up.

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