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Graphs of Rational Functions

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

A rational function is a ratio of polynomials. VERTICAL ASYMPTOTES happen where the denominator is zero AND the factor does not cancel with the numerator. HOLES happen where a factor cancels in both. HORIZONTAL ASYMPTOTES depend on degrees: if degree of numerator is less than degree of denominator, y = 0; if equal, y = leading-coefficient ratio; if greater, no horizontal asymptote.

What you'll learn

Why it matters: Drug concentrations over time, economic supply/demand limits, gear ratios, and physics relationships involving 1/x all show rational-function behavior including asymptotes.

Worked example

Problem. For f(x) = (x + 3) / ((x - 2)(x + 3)), find the vertical asymptote and the hole.

  1. The factor (x + 3) cancels between numerator and denominator -> hole at x = -3.
  2. Remaining denominator factor (x - 2) gives the vertical asymptote at x = 2.

Answer: VA: x = 2. Hole: x = -3.

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. f(x) = 1 / (x - 3). Enter the x-value of the vertical asymptote.

Show solution
  1. Denominator = 0 at x = 3.

Answer: 3

2. f(x) = 1 / (x + 5). Enter the x-value of the vertical asymptote.

Show solution
  1. x + 5 = 0 at x = -5.

Answer: -5

3. f(x) = (x - 2) / (x^2 - 4). Enter the x-value of the hole.

Show solution
  1. x^2 - 4 = (x - 2)(x + 2).
  2. The (x - 2) cancels with the numerator -> hole at x = 2.

Answer: 2

4. Same f(x) = (x - 2) / (x^2 - 4). Enter the x-value of the vertical asymptote.

Show solution
  1. After cancelling, the remaining denominator factor is (x + 2).

Answer: -2

5. f(x) = 3x / x^2. Enter the y-value of the horizontal asymptote.

Show solution
  1. Numerator degree 1 < denominator degree 2 -> HA at y = 0.

Answer: 0

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