CMClearMathAcademy

Rational Functions and Asymptotes

A free College Algebra lesson from the “Advanced Function Types” unit, with a worked example and practice problems including step-by-step solutions.

A rational function is a ratio of polynomials. Its graph can have vertical asymptotes where the denominator is zero and horizontal or slant asymptotes based on end behavior. These features matter because they describe where the function is undefined and how the graph behaves far from the origin. When practicing, factor the numerator and denominator first, identify values that make the denominator zero, and check whether any factors cancel to create holes instead of asymptotes. A common mistake is calling every excluded value an asymptote without checking for cancellation.

What you'll learn

Why it matters: Reciprocal models, rate limits, concentration formulas, and average-cost curves can approach values they never reach or break at restricted inputs. Asymptotes make that behavior visible instead of treating it as a calculator error.

Worked example

Problem. For f(x) = 1/(x - 6), what is the vertical asymptote?

  1. Set the denominator equal to zero.
  2. x - 6 = 0.
  3. The vertical asymptote is x = 6.

Answer: x = 6

Lesson preview: this page shows 5 practice items. The interactive activity contains 28 practice items and 15 quiz items (typed response and multiple choice). Counts include repeated prompts and variants.

Practice problems

1. For f(x) = 1/(x + 3), what x-value is the vertical asymptote?

Show solution
  1. x + 3 = 0.

Answer: -3

2. For f(x) = 5/(x - 9), what x-value is the vertical asymptote?

Show solution
  1. x - 9 = 0.

Answer: 9

3. A vertical asymptote describes behavior near a restricted...

Choices: x-value · y-value only · slope · degree

Show solution
  1. It is a vertical line x = a.

Answer: x-value

4. What is the vertical asymptote of f(x) = 1/(x - 4)? Enter the x-value.

Show solution
  1. The denominator is zero at x = 4.

Answer: 4

5. What is the horizontal asymptote of f(x) = 3/(x + 1)? Enter the y-value.

Show solution
  1. The denominator degree is larger than the numerator degree.

Answer: 0

Practice this interactively with instant feedback and an AI tutor.

Practice Rational Functions and Asymptotes Take the free placement check

More College Algebra lessons