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Vertical Asymptotes

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

Vertical asymptotes mark x-values where the simplified denominator is zero. 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 vertical asymptote x-value for r(x) = 3/(x - (-2)).

  1. Worked Example: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (-2)).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Set the denominator equal to 0.
  4. x - (-2) = 0 gives x = -2.
  5. Because no factor cancels, this is a vertical asymptote.
  6. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

Practice problems

1. Find the vertical asymptote x-value for r(x) = 3/(x - (-2)).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (-2)).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Set the denominator equal to 0.
  4. x - (-2) = 0 gives x = -2.
  5. Because no factor cancels, this is a vertical asymptote.
  6. Check the result by substituting or estimating: the response should match -2 and make sense in the original problem.

Answer: -2

2. Find the vertical asymptote x-value for r(x) = 3/(x - (-1)).

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (-1)).
  2. For signed numbers, track both distance from zero and direction so the sign of the answer makes sense.
  3. Set the denominator equal to 0.
  4. x - (-1) = 0 gives x = -1.
  5. Because no factor cancels, this is a vertical asymptote.
  6. Check the result by substituting or estimating: the response should match -1 and make sense in the original problem.

Answer: -1

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

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (4)).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set the denominator equal to 0.
  4. x - (4) = 0 gives x = 4.
  5. Because no factor cancels, this is a vertical asymptote.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. Find the vertical asymptote x-value for r(x) = 3/(x - (1)).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (1)).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set the denominator equal to 0.
  4. x - (1) = 0 gives x = 1.
  5. Because no factor cancels, this is a vertical asymptote.
  6. Check the result by substituting or estimating: the response should match 1 and make sense in the original problem.

Answer: 1

5. Find the vertical asymptote x-value for r(x) = 3/(x - (2)).

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the vertical asymptote x-value for r(x) = 3/(x - (2)).
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. Set the denominator equal to 0.
  4. x - (2) = 0 gives x = 2.
  5. Because no factor cancels, this is a vertical asymptote.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

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