Find vertical asymptotes of rational functions by solving denominator = 0 after canceling common factors — distinguishes asymptotes from holes (removable discontinuities).
Set remaining denominator = 0 after simplifying
Vertical asymptotes occur where the denominator equals zero and the factor did not cancel with the numerator. The function approaches ±∞ at these x-values.
Process: (1) Factor both numerator and denominator. (2) Cancel any common factors — these create holes, not asymptotes. (3) Set the remaining denominator factors equal to zero. (4) Solve for x — those are your vertical asymptotes.
Always simplify the rational expression first. Trying to find VAs without cancelling first will give wrong answers when common factors exist.
Cancelled factor → hole, not an asymptote
When a factor cancels from both numerator and denominator, the function has a hole (removable discontinuity) at that x-value — a single missing point, not a vertical asymptote.
To find the hole's y-coordinate: substitute the x-value into the simplified form of the function.
One-on-one Algebra 2 tutoring makes holes, asymptotes, and graphing rational functions click — we work through your actual problems and build lasting intuition for tests and beyond.