How to Find a Hole on a Graph: Rational Functions Guide

A hole on a graph is a single point where a rational function is undefined even though the surrounding curve is continuous — the graph looks like an unbroken line or curve with one missing dot. To find it, factor the numerator and denominator, cancel any common factor, and note the x-value that cancellation removes. That x-value is the hole’s location; substitute it into the simplified expression to get the y-coordinate.

Why Holes Happen: Common Factors

A rational function is a fraction of two polynomials, written as f(x) = P(x) / Q(x). It is undefined wherever Q(x) = 0, because division by zero is not allowed. Those values normally produce vertical asymptotes — but not always. When the same factor appears in both P(x) and Q(x), the zero of that factor cancels out of the fraction. The function still can’t be evaluated at that exact x-value, but the curve doesn’t blow up to infinity there. Instead, the graph approaches a finite point and simply leaves it out. That missing point is the hole.

The distinction matters: a factor that cancels causes a hole, while a factor that remains in the denominator after simplifying causes a vertical asymptote. So the first real step in any problem is factoring, and the second is canceling — carefully keeping track of what you canceled.

Step-by-Step Method

Step 1: Factor both polynomials completely

Write the numerator and denominator as products of factors. For example:

f(x) = (x² − 4) / (x² − x − 6) = (x − 2)(x + 2) / (x − 3)(x + 2)

Step 2: Cancel common factors and record them

The factor (x + 2) appears in both parts, so it cancels. Write down its zero: x = −2. This is the x-coordinate of the hole. Never simply erase a canceled factor from your scratch work — the canceled factor is exactly the information you need.

Step 3: Simplify the expression

Removing the shared factor leaves:

f_simplified(x) = (x − 2) / (x − 3)

This simplified version describes every point on the graph except x = −2. It also reveals the vertical asymptote at x = 3.

Step 4: Find the y-coordinate

Plug the hole’s x-value into the simplified expression:

f_simplified(−2) = (−2 − 2) / (−2 − 3) = (−4) / (−5) = 4/5

So the hole is at the point (−2, 4/5). If you plug x = −2 into the original function instead, you get 0/0, which is undefined — that’s the signature of a hole rather than an asymptote.

Reading Holes From a Graph

Sometimes you’re handed a picture rather than an equation. On a graph, a hole shows up as a small open circle sitting on an otherwise smooth curve. It looks like a break in an otherwise continuous path, but the curve clearly continues on both sides — the y-values approach the removed point from the left and right without shooting toward infinity. That’s the visual cue that distinguishes it from a vertical asymptote, where the curve races off the top or bottom of the screen.

To locate the coordinates from a picture, read the open circle’s position directly off the axes: the x-value is where the circle sits horizontally, and the y-value is its height. If the grid isn’t detailed enough, trace the curve from either side and estimate where the missing point would fall.

Common Mistakes to Avoid

• Calling a canceled factor an asymptote. If the factor cancels, the discontinuity is removable (a hole), not an asymptote.
• Forgetting the y-coordinate. The hole isn’t just x = −2; it’s the point (−2, 4/5). A hole needs both coordinates.
• Plugging into the original function. The original gives 0/0 at the hole. Always substitute into the simplified form to get the y-value.
• Canceling terms instead of factors. You can cancel (x + 2) as a whole factor, but you can never cancel individual terms across a plus or minus sign.
• Assuming every zero of the denominator is a hole. Check whether the factor actually appears in the numerator. If it doesn’t, you have an asymptote.

The distinction between holes and asymptotes is a recurring theme in precalculus and calculus, and both Khan Academy and Paul’s Online Math Notes cover the topic with additional practice problems. For another angle on graphing these functions, Purplemath offers clear walkthroughs of rational function behavior.

Holes in Terms of Limits

For readers who have met calculus, a hole at x = a is precisely the case where the two-sided limit exists but does not equal the function’s value — usually because the function has no value there at all. Writing it formally,

lim(x→a) f(x) = L, but f(a) is undefined.

This is the textbook definition of a removable discontinuity, and it explains why the graph can be redrawn as a continuous curve by filling in the single point (a, L). A vertical asymptote, by contrast, has no finite limit: the one-sided limits diverge to positive or negative infinity. Recognizing this distinction early makes later topics — continuity tests, derivative definitions, integrals over an interval containing a hole — much easier to reason about.

Verifying a Hole Numerically

When you are unsure whether a discontinuity is a hole or an asymptote, evaluate the function at values just to the left and right of the suspect x-value. Take the earlier example around x = −2:

• f(−2.1) ≈ 0.804
• f(−1.9) ≈ 0.796

Both outputs hover near 0.8 = 4/5, confirming that the curve is settling toward a finite height rather than racing off toward infinity. Compare that with x = 3, where the outputs grow large in opposite directions — the unmistakable behavior of an asymptote.

Writing the Final Answer

Teachers and textbooks differ slightly in what they expect, so match the format requested. Common acceptable forms include the coordinate pair, e.g. (−2, 4/5), or a sentence such as “the graph has a hole at x = −2.” If the problem asks for all discontinuities, list the holes and the vertical asymptotes separately, since they are different types of behavior and graders usually award points for each.

Key Takeaways

• A hole is a single missing point where a rational function is undefined, caused by a common factor that cancels from numerator and denominator.
• Find it by factoring both polynomials, canceling common factors, recording the canceled factor’s zero as the hole’s x-value, and substituting that value into the simplified expression for the y-coordinate.
• At a hole the original function gives 0/0; at a vertical asymptote the denominator stays zero after simplifying and the curve shoots to infinity.
• On a graph, a hole appears as an open circle on an otherwise continuous curve — closer to a missing dot than a break.
• Always report the hole as a coordinate pair, and never cancel individual terms — only whole shared factors.