Graphing Rational Functions: Asymptotes, Holes, and Intercepts

Rational functions $f(x) = \frac{P(x)}{Q(x)}$ produce some of the most distinctive graphs in algebra — branches diverging to infinity, holes you cannot see at first, and asymptotes that the curve hugs forever without crossing. This guide gives you a checklist to graph any rational function.

The 5-step workflow

1. Factor numerator and denominator completely.
2. Identify holes at common factors (cancel them, but mark the x-values as holes).
3. Vertical asymptotes at remaining zeros of the denominator.
4. Horizontal or slant asymptote from the degree comparison.
5. Intercepts: y-intercept at $f(0)$ if defined; x-intercepts at zeros of the simplified numerator.

Step-by-step on $f(x) = \frac{x^2 - 1}{x^2 - x - 6}$

Factor

$f(x) = \frac{(x-1)(x+1)}{(x-3)(x+2)}$

No common factors → no holes.

Vertical asymptotes

Denominator zeros are $x = 3$ and $x = -2$. Two vertical asymptotes.

Horizontal asymptote

Degree of numerator (2) = degree of denominator (2). The horizontal asymptote is the ratio of leading coefficients: $y = 1/1 = 1$.

Intercepts

• $f(0) = (-1)(1)/((-3)(2)) = -1 / -6 = 1/6$. y-intercept: $(0, 1/6)$.
• Numerator zeros: $x = 1$ and $x = -1$. x-intercepts at those.

Sketch

Two vertical asymptotes split the x-axis into three regions. In each, test a sample point to see if $f$ is positive or negative. The graph approaches $y = 1$ as $x \to \pm\infty$ and crosses through the intercepts found above.

The asymptote rules in one table

Compare degrees	Asymptote type
deg(P) < deg(Q)	$y = 0$ horizontal
deg(P) = deg(Q)	$y = a/b$ horizontal (ratio of leading coeffs)
deg(P) = deg(Q) + 1	slant asymptote (do polynomial long division)
deg(P) ≥ deg(Q) + 2	no horizontal/slant; ends fly off polynomially

Worked example: a hole

$g(x) = \frac{x^2 - 4}{x - 2} = \frac{(x-2)(x+2)}{x-2}$

Cancel: $g(x) = x + 2$ for $x \ne 2$. Graph the line $y = x + 2$ with an open circle at $(2, 4)$ — that is the hole.

Common mistakes

• Forgetting holes — cancelling factors removes vertical asymptotes but leaves holes.
• Mis-applying the horizontal asymptote rule when degrees differ.
• Assuming graphs never cross horizontal asymptotes — they often do, just never as $x \to \pm\infty$.

Try with the AI Equation Solver

Plug your rational function into the Equation Solver to factor it and identify zeros / poles automatically.

Related references:

• Polynomial Calculator — for the long-division step in slant cases
• Factor Calculator — the foundation of step 1
• Limit Calculator — asymptotes are limits at infinity