Graphing Rational Functions

A rational function is a ratio of polynomials, and its graph is shaped by asymptotes — lines the graph approaches but never touches. Vertical and horizontal asymptotes are the main features.

Vertical asymptotes come from denominator zeros; horizontal asymptotes come from comparing the degrees of top and bottom. Find both and the graph's skeleton appears.

Vertical asymptotes

Set the denominator equal to zero and solve. Each solution that does not also cancel from the numerator is a vertical asymptote.

For $f(x) = \dfrac{2x + 4}{x - 1}$ , the denominator is zero at $x = 1$ , so $x = 1$ is a vertical asymptote.

Horizontal asymptotes from degrees

Compare the degrees of numerator and denominator. Equal degrees give a horizontal asymptote at the ratio of leading coefficients.

For $\dfrac{2x + 4}{x - 1}$ , both are degree 1, so the horizontal asymptote is $y = \dfrac{2}{1} = 2$ . A smaller top degree gives $y = 0$ ; a larger top has none.

Worked examples

Example 1: both asymptotes

Find the vertical and horizontal asymptotes of $f(x) = \dfrac{2x + 4}{x - 1}$ .

Denominator zero for vertical

x - 1 = 0 \Rightarrow x = 1

Equal degrees for horizontal

y = \dfrac{2}{1} = 2

Answer: Vertical $x = 1$ , horizontal $y = 2$

Example 2: horizontal at zero

What is the horizontal asymptote of $\dfrac{5}{x + 2}$ ?

Top degree smaller than bottom

0 < 1

So the asymptote is y = 0

y = 0

Answer: $y = 0$

Example 3: a factor that cancels

Does $f(x) = \dfrac{x^2 - 4}{x - 2}$ have a vertical asymptote at $x = 2$ ?

Denominator zero

x - 2 = 0 \Rightarrow x = 2

Factor the numerator

\dfrac{(x + 2)(x - 2)}{x - 2}

The factor cancels, so it is a hole

x + 2, \; x \neq 2

Answer: No — a hole at $x = 2$ , not a vertical asymptote

Try one yourself

Common questions

How do I find vertical asymptotes?

Set the denominator equal to zero and solve. Each non-canceling zero is a vertical asymptote.

How do I find the horizontal asymptote?

Compare degrees: equal degrees give the ratio of leading coefficients; smaller top gives $y = 0$ ; larger top gives none.

What if a factor cancels?

A canceling factor makes a hole, not a vertical asymptote, at that x-value.

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Graphing Rational Functions

Vertical asymptotes

Horizontal asymptotes from degrees

Worked examples

Example 1: both asymptotes

Example 2: horizontal at zero

Example 3: a factor that cancels

Try one yourself

Common questions

Keep learning

Adding & Subtracting Rational Expressions

Solving Rational Equations & Inequalities

Functions & Continuity