How Do You Graph a Rational Function With a Quadratic Numerator or Denominator?

Graph the function systematically: simplify it, find its domain and intercepts, identify asymptotes or holes, determine its end behaviour, and then establish the shape of each branch. This is AHL 2.13: Rational functions in IB Mathematics: Analysis and Approaches HL.

Method

For $f(x)=\frac{P(x)}{Q(x)}$, follow these steps:

1. Factor $P(x)$ and $Q(x)$. A cancelled factor creates a removable discontinuity, or hole; an uncancelled denominator zero creates a vertical asymptote.
2. Find the intercepts. Solve $P(x)=0$ for the $x$-intercepts, excluding cancelled or undefined values. Calculate for the -intercept, if defined.

When the denominator is quadratic, solve $Q(x)=0$ and classify each real root after cancellation. A repeated uncancelled factor can make the graph approach the same infinity on both sides; a non-repeated factor usually produces opposite signs. Complex roots create no real discontinuity.

Worked Example

Consider

$$f(x)=\frac{x^2-1}{x-2}.$$

The denominator gives the vertical asymptote $x=2$. The numerator factors as $(x-1)(x+1)$, so the $x$-intercepts are $(-1,0)$ and . Also, , giving the -intercept .

Polynomial division gives

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

so the oblique asymptote is $y=x+2$. Sketch the two branches approaching both asymptotes while passing through the intercepts.

A common misconception is that every denominator zero produces a vertical asymptote. If its factor cancels completely, it produces a hole instead.

Exam Technique

For a sketch question, label all intercepts, holes, and asymptotes and show the correct branch behaviour. On Paper 1, show the algebraic analysis; on Paper 2, use a GDC to verify the shape, but still state the mathematical features explicitly.