This topic covers functions formed by dividing polynomials and more general expressions, their key features, and how to analyse and graph them.
A rational function has the form $f(x) = \dfrac{P(x)}{Q(x)}$ where $P$ and $Q$ are polynomials and $Q \not\equiv 0$.
More general quotient functions include expressions like $\dfrac{\sqrt{x}}{x+1}$ or $\dfrac{e^x}{x^2+1}$.
| Feature | How to Find |
|---|---|
| Domain | Exclude values making $Q(x) = 0$ |
| Vertical asymptote | Values where $Q(x) = 0$ and $P(x) \neq 0$ |
| Hole (removable discontinuity) | Common factor of $P$ and $Q$ |
| Horizontal asymptote | Compare degrees of $P$ and $Q$ |
| Oblique asymptote | When $\deg P = \deg Q + 1$; perform polynomial division |
| $x$-intercepts | Solve $P(x) = 0$ within the domain |
| $y$-intercept | $f(0)$ if $0$ is in the domain |
Let $f(x) = \dfrac{a_n x^n + \cdots}{b_m x^m + \cdots}$.
| Degree comparison | Horizontal asymptote |
|---|---|
| $n < m$ | $y = 0$ |
| $n = m$ | $y = a_n/b_m$ |
| $n > m$ | None (oblique or higher) |
A sign diagram tracks where $f(x) > 0$ or $f(x) < 0$ by testing intervals separated by zeros and vertical asymptotes.
Example: $f(x) = \dfrac{x-1}{(x+2)(x-3)}$.
Key $x$-values: $-2$ (VA), $1$ (zero), $3$ (VA).
| Interval | Sign of $f$ |
|---|---|
| $x < -2$ | $-/((-)(-)) = -/+ = -$ |
| $-2 < x < 1$ | $-/((+)(-)) = -/- = +$ |
| \$1 < x < 3$ | $+/((+)(-)) = +/- = -$ |
| $x > 3$ | $+/((+)(+)) = +$ |
For any rational function, as $x \to \pm\infty$, keep only the leading terms:
$$\frac{3x^2 - 2x + 1}{x^2 + 5} \approx \frac{3x^2}{x^2} = 3 \quad \text{as } x \to \pm\infty$$
So the horizontal asymptote is $y = 3$.
When $\deg P = \deg Q + 1$, perform polynomial long division:
$$\frac{x^2 + 1}{x - 2} = x + 2 + \frac{5}{x-2}$$
As $x \to \pm\infty$, $\dfrac{5}{x-2} \to 0$, so the oblique asymptote is $y = x + 2$.
KEY TAKEAWAY: Analysing a rational function requires systematically finding its domain, intercepts, asymptotes, and sign. Each feature informs a different aspect of the graph.
STUDY HINT: Practise polynomial long division for oblique asymptotes. It also feeds directly into partial fraction decomposition.
VCAA FOCUS: Sketching rational function graphs with full labelling of asymptotes and intercepts is a core task in both paper 1 and 2.
