CMClearMathAcademy

Horizontal and Slant Asymptotes

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

Horizontal and slant asymptotes describe what a rational graph approaches far left and far right. 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. For r(x) = (3x^2 + 1)/(x^2 - 4), what is the horizontal asymptote?

  1. Worked Example: First identify exactly what the question is asking: For r(x) = (3x^2 + 1)/(x^2 - 4), what is the horizontal asymptote?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The numerator and denominator have equal degree.
  4. Use the ratio of leading coefficients.
  5. That ratio is 3/1, so y = 3.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y = 3

Practice problems

1. For r(x) = (3x^2 + 1)/(x^2 - 4), what is the horizontal asymptote?

Choices: y = 3 · y = 0 · x = 2 · y = 1/3

Show solution
  1. Warm-up: First identify exactly what the question is asking: For r(x) = (3x^2 + 1)/(x^2 - 4), what is the horizontal asymptote?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The numerator and denominator have equal degree.
  4. Use the ratio of leading coefficients.
  5. That ratio is 3/1, so y = 3.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y = 3

2. For r(x) = (4x^2 + 1)/(x^2 - 4), what is the horizontal asymptote?

Choices: y = 4 · y = 0 · x = 2 · y = 1/4

Show solution
  1. Warm-up: First identify exactly what the question is asking: For r(x) = (4x^2 + 1)/(x^2 - 4), what is the horizontal asymptote?
  2. Use inverse operations to isolate the unknown, and keep both sides balanced at every step.
  3. The numerator and denominator have equal degree.
  4. Use the ratio of leading coefficients.
  5. That ratio is 4/1, so y = 4.
  6. Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.

Answer: y = 4

3. For r(x) = (5x + 1)/(x^2 - 4), what is the horizontal asymptote?

Choices: y = 0 · y = 5 · no horizontal asymptote · x = 2

Show solution
  1. The numerator degree 1 is less than the denominator degree 2.
  2. When the top degree is smaller, outputs shrink toward 0 far from the origin.
  3. So the horizontal asymptote is y = 0.

Answer: y = 0

4. Find the slant asymptote of r(x) = (x^2 + 2)/(x - 1) by long division. Enter it as y = mx + b.

Show solution
  1. The numerator degree is one more than the denominator, so expect a slant asymptote.
  2. Long division of x^2 + 2 by x - 1 gives quotient x + 1 (remainder 3).
  3. The slant asymptote is the quotient line y = x + 1.

Answer: y = x + 1

5. A slant asymptote usually appears when:

Choices: the numerator degree is exactly one more than the denominator degree · the denominator degree is larger · both degrees are equal · there is no denominator

Show solution
  1. A one-degree difference creates a linear quotient.
  2. Polynomial division reveals the slant asymptote.
  3. Equal degrees give a horizontal asymptote.

Answer: the numerator degree is exactly one more than the denominator degree

Practice this interactively with instant feedback and an AI tutor.

Practice Horizontal and Slant Asymptotes Take the free placement check

More Precalculus lessons