Rational Function Grapher

About the Rational Function Grapher

Plot rational functions y = p(x)/q(x) and visualize vertical and horizontal asymptotes.

How to use this calculator

1. Enter the numerator polynomial \(p(x)\).
2. Enter the denominator polynomial \(q(x)\).
3. Click graph to plot the function and display asymptotes.
4. Check the graph for holes, vertical asymptotes, and any horizontal asymptote.

The formula explained

This tool graphs \(y=\frac{p(x)}{q(x)}\), a ratio of two polynomials. It shows how the function behaves where the denominator is zero and how it acts far to the left and right.

• x = the input variable on the horizontal axis
• p(x) = the polynomial in the numerator
• q(x) = the polynomial in the denominator
• y = the output value of the rational function

Step by step method

1. Factor the numerator and denominator if possible.
2. Find values of \(x\) that make \(q(x)=0\), these are excluded from the domain and may create vertical asymptotes or holes.
3. Compare the degrees of \(p(x)\) and \(q(x)\) to estimate the horizontal asymptote.
4. Plot the function to confirm the intercepts and asymptotes.

Worked example

Problem. Graph \(y=\frac{x+2}{x-1}\) and identify its asymptotes.

1. The denominator is zero at \(x=1\), so the function is undefined there.
2. Since the numerator and denominator have the same degree, the horizontal asymptote is \(y=\frac{1}{1}=1\).
3. So the graph has a vertical asymptote at \(x=1\) and a horizontal asymptote at \(y=1\).

Tips and common mistakes

• If a factor cancels between the numerator and denominator, that point is a hole, not a vertical asymptote.
• A horizontal asymptote describes end behavior, so it tells you what happens as \(x\to\infty\) or \(x\to-\infty\).

Frequently asked questions