Finding the Coordinates of a Hole in a Graph

Rational equations can have what are called discontinuities. Nonremovable discontinuities are vertical asymptotes, invisible lines that the graph approaches but does not touch. Other discontinuities are called holes. Finding and graphing a hole often involves simplifying the equation. This leaves a literal “hole” in the line of the graph that is often represented by an open circle.

Step 1

Factor the numerator and denominator of the rational equation by using trinomial, greatest common factor, grouping or difference of squares factoring.

Step 2

Look for any factors on the top and bottom that are identical and cross both of them out. Then, rewrite the equation without them. Graph this simplified form — it might be a linear, quadratic or rational equation since there is still an x in the denominator.

Step 3

Set the the denominator equal to zero and solve for x. The result is the x-coordinate of the hole. Note that it is possible to have more than one asymptote if you have a complex denominator, such as “(x + 1)(x – 1).” In such a case, you would have two x-coordinates: -1 and 1

Step 4

Plug the answer from Step 3 into the simplified version of the equation and solve for y. This gives you the y-coordinate of the hole.

Step 5

Write the x-coordinate and the y-coordinate in parentheses, separated by a comma, for the final answer.