Algebra 2 · Unit 4 · Topic 4.4
Holes, Vertical Asymptotes, and End Behavior
For a rational function, a canceled denominator factor may create a removable hole, while a remaining denominator zero may produce a vertical asymptote.
Definition and explanation
For a rational function, a canceled denominator factor may create a removable hole, while a remaining denominator zero may produce a vertical asymptote. Horizontal asymptotes describe end behavior, and a graph may cross them.
Worked example
For f(x) = (x² − 4)/(x² − x − 2), factor to obtain (x + 2)/(x + 1) for x ≠ 2, −1. There is a hole at (2, 4/3), a vertical asymptote x = −1, and a horizontal asymptote y = 1 because numerator and denominator have equal degree and equal leading coefficients.
Common mistake
Not every original denominator zero is a vertical asymptote; check cancellation first.
Key takeaway
For a rational function, a canceled denominator factor may create a removable hole, while a remaining denominator zero may produce a vertical asymptote.
Explain each step of the worked example, then check the units, signs, and conditions that apply.
Further reading
Topic coverage was checked against the openly available OpenStax Intermediate Algebra 2e, Algebra and Trigonometry 2e, and Introductory Statistics 2e textbooks. The explanations and examples on this site are independently written; textbook prose, exercises, and artwork are not reproduced.