CMClearMathAcademy

Graphing Rational Functions

A free Precalculus lesson from the “Rational Functions” unit, with a worked example and practice problems including step-by-step solutions.

A rational graph sketch is built from restrictions, intercepts, asymptotes, and interval signs. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Rational functions model rates, constraints, efficiency, and quantities that change sharply near restricted inputs.

Worked example

Problem. Find the x-intercept of r(x) = (x - 3)/(x - 8).

  1. An x-intercept occurs where the numerator is 0 (and the denominator is not).
  2. Set x - 3 = 0, so x = 3. The denominator there is 3 - 8 = -5, nonzero.
  3. So the x-intercept is at x = 3.

Answer: 3

Practice problems

1. Find the x-intercept of r(x) = (x - 3)/(x - 8).

Show solution
  1. An x-intercept occurs where the numerator is 0 (and the denominator is not).
  2. Set x - 3 = 0, so x = 3. The denominator there is 3 - 8 = -5, nonzero.
  3. So the x-intercept is at x = 3.

Answer: 3

2. Find the x-intercept of r(x) = (x + 3)/(x - 4).

Show solution
  1. Set the numerator equal to 0.
  2. x + 3 = 0 gives x = -3; the denominator -3 - 4 = -7 is nonzero.
  3. So the x-intercept is at x = -3.

Answer: -3

3. Find the y-intercept value r(0) for r(x) = (2x + 5)/(x + 1).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the y-intercept value r(0) for r(x) = (2x + 5)/(x + 1).
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. The y-intercept is r(0): substitute x = 0.
  4. r(0) = (2*0 + 5)/(0 + 1) = 5/1.
  5. So the y-intercept value is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

4. Find the y-intercept value r(0) for r(x) = (x - 6)/(x - 3).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the y-intercept value r(0) for r(x) = (x - 6)/(x - 3).
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Substitute x = 0 into r(x).
  4. r(0) = (0 - 6)/(0 - 3) = -6/-3.
  5. That simplifies to 2, so the y-intercept value is 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

5. Find the vertical asymptote x-value for r(x) = 4/(x - 3).

Show solution
  1. A vertical asymptote occurs where the uncanceled denominator is 0.
  2. Set x - 3 = 0, so x = 3; the numerator 4 does not cancel it.
  3. So the vertical asymptote is x = 3.

Answer: 3

Practice this interactively with instant feedback and an AI tutor.

Practice Graphing Rational Functions Take the free placement check

More Precalculus lessons