Rational functions and asymptotes (WACE Year 12 Mathematics Specialist Unit 3)

What this dot point is asking

SCSA wants a structured sketch of a rational function: intercepts, every asymptote, behaviour near each asymptote, and stationary points where relevant.

Vertical asymptotes and holes

Factor numerator and denominator. A zero of the denominator that is not cancelled by the numerator gives a vertical asymptote, where the curve shoots to $\pm\infty$. A factor common to both gives a hole (removable discontinuity) at that $x$-value, not an asymptote. Near a vertical asymptote, check the sign of $y$ on each side to decide whether the branch goes up or down.

End behaviour: horizontal and oblique asymptotes

Compare degrees of numerator and denominator:

For an oblique asymptote, divide $P$ by $Q$; the quotient (a linear expression) is the asymptote and the remainder term vanishes as $x \to \pm\infty$.

Reading degree differences quickly

The degree comparison can be made mechanical. Divide the numerator and denominator by the highest power of $x$ in the denominator and let $x \to \pm\infty$. If the numerator degree is smaller, every surviving term has a positive power of $x$ in the denominator and the quotient tends to $0$, giving the asymptote $y = 0$. If the degrees are equal, only the ratio of leading coefficients survives, giving a non-zero horizontal asymptote. If the numerator degree is exactly one greater, the polynomial division leaves a linear quotient plus a remainder that vanishes, giving the oblique asymptote. A numerator two or more degrees higher has no linear asymptote and the curve grows without bound, which is beyond the standard SCSA cases but worth recognising.

Intercepts and sign analysis

The $y$-intercept is the value at $x = 0$ (if defined). The $x$-intercepts are the zeros of the numerator that are not also zeros of the denominator. A sign table across the intercepts and vertical asymptotes shows which regions sit above or below the $x$-axis, fixing the shape of each branch.

Crossing a horizontal or oblique asymptote

A frequent misconception is that a curve can never touch its asymptote. This is true only for vertical asymptotes. A horizontal or oblique asymptote describes end behaviour as $x \to \pm\infty$, so the curve is free to cross it in the middle of the domain. To find where $y = \dfrac{P(x)}{Q(x)}$ meets its horizontal asymptote $y = c$, solve $\dfrac{P(x)}{Q(x)} = c$; a solution is a genuine crossing point that should be marked on the sketch.

Behaviour near a vertical asymptote

The sign of the function immediately on each side of a vertical asymptote determines whether the branch rises to $+\infty$ or falls to $-\infty$. The quickest method is to substitute a value just above and just below the asymptote into the factored form and read the sign, or to use a full sign table. An odd-power factor in the denominator gives opposite signs on the two sides (the branches go opposite ways), while an even-power factor gives the same sign on both sides (both branches go the same way).

Assembling the sketch

Draw the asymptotes as dashed lines first, mark the intercepts, then connect each branch consistently with the sign analysis and end behaviour. Stationary points can be located with calculus if the question requires them.