Rational and simple quotient functions: VCE Specialist Mathematics Units 3 & 4

1. A quotient is a rule together with a domain

A rational function is a quotient of polynomials, f(x) = P(x)/Q(x), with Q(x) ≠ 0. Its behaviour comes from both the algebraic expression and the excluded inputs. Factoring, cancelling, dividing and decomposing into partial fractions can reveal different features of the same graph. None of those operations silently restores an input excluded by the original rule.

Begin a graph investigation by writing the original domain. Next factor numerator and denominator where practical. An uncancelled denominator zero usually produces a vertical asymptote for the low-degree rational functions considered here; a cancelled factor can instead leave a removable hole. The decisive question is what the function approaches near the excluded input, rather than whether a denominator appeared in the first line.

An x-intercept requires the numerator to be zero at an allowed input. A y-intercept is f(0) only if zero belongs to the domain. A point that would lie on a simplified graph may still be missing from the actual function. Use an open circle for a hole and state its coordinates; an empty point is not an asymptote.

2. Division explains end behaviour

Polynomial division expresses an improper rational function as a polynomial plus a proper fraction. If the remaining fraction tends to zero as x becomes large in magnitude, the polynomial identifies the graph's end behaviour. In this guide, the important cases produce a constant horizontal asymptote or a linear oblique asymptote. Give an asymptote as an equation such as y = 2x − 1, rather than listing a slope or intercept separately.

When the numerator degree is lower than the denominator degree, the horizontal asymptote is y = 0. When their degrees are equal, it is the ratio of their leading coefficients. When the numerator degree is one higher, divide to obtain the oblique line. These shortcuts describe large-|x| behaviour; they do not determine vertical asymptotes, holes or turning points.

A graph may cross a horizontal or oblique asymptote. An asymptote constrains a limiting difference, not all finite points. For y = x/(x² + 1), the graph crosses y = 0 at the origin while approaching that same line at both ends. Conversely, y = x + 4/x never crosses y = x because their difference 4/x cannot equal zero. Use the actual difference to settle a crossing question.

One-sided behaviour near a vertical asymptote needs sign analysis. For 3/(x − 2), a small positive denominator gives a large positive value, while a small negative denominator gives a large negative value. For 3/(x − 2)², both sides approach positive infinity. Multiplicity and the other factors determine the direction; a dotted vertical line alone is not a complete graph.

3. Partial fractions expose the separate components

A proper rational expression with factorised denominator can often be written as a sum of simpler quotients. For example, (3x + 5)/[(x + 1)(x + 2)] = A/(x + 1) + B/(x + 2). Multiplying by the denominator gives the polynomial identity 3x + 5 = A(x + 2) + B(x + 1). Equating coefficients or choosing x = −1 and −2 gives A = 2 and B = 1.

The coefficient identity can be evaluated at −1 and −2 because it is an identity between polynomials. The original fractions remain undefined at those inputs. This distinction explains why substitution into the cleared identity is legitimate without claiming the original function has acquired values at its poles.

A repeated linear factor needs each power: a denominator (x − a)² uses terms A/(x − a) + B/(x − a)². An irreducible quadratic may require a linear numerator. Divide first if the expression is improper. For graph analysis, these forms expose shifts and local signs; for integration using partial fractions, they expose standard antiderivatives. Always recombine your proposed fractions as an algebraic check.

4. Join algebra to calculus

Stationary points satisfy f′(x) = 0 at allowed inputs. Solve that equation, obtain the corresponding coordinates and use the sign of f′ to determine increasing and decreasing intervals. A derivative zero does not by itself distinguish a maximum, minimum or stationary inflection. A point where the function is undefined cannot become stationary simply because a denominator was cleared.

The second derivative describes concavity. Positive f″ means increasing gradient, or concave up; negative f″ means decreasing gradient, or concave down. An inflection point requires a change of concavity at a point on the curve. A vertical asymptote can separate regions with opposite concavity without creating an inflection point, because there is no point of the function there.

Check symmetry from the rule and domain. An even function satisfies f(−x) = f(x) on a domain symmetric about zero. An odd function satisfies f(−x) = −f(x). A translated reciprocal graph may instead have rotational symmetry about the intersection of its asymptotes. Symmetry is useful evidence, but it cannot replace labelling intercepts, asymptotes and critical points precisely.

5. Simple non-rational quotients need the same discipline

A quotient containing a logarithm, exponential or circular function need not be rational. The graphing workflow still works, but numerator-degree shortcuts do not. First intersect every domain condition: a logarithm needs a positive argument, an inverse circular function has its own permitted inputs, and a denominator cannot be zero. Then use appropriate limiting or numerical reasoning for the ends.

For instance, ln x/x exists only for x > 0. The positive denominator never vanishes inside that domain, but it approaches zero at the boundary. A graphing window that starts at x = 0.1 can hide the rapid fall near zero. A sensible sketch records the boundary behaviour explicitly and does not draw an unsupported left-hand branch.

6. Investigation and technology checks

Consider the family x + a/x. For a > 0, stationary inputs are ±√a, and the two branches have separated ranges. For a < 0, the derivative 1 − a/x² is always positive and there are no stationary points. At a = 0, the original expression still excludes zero unless the function is deliberately redefined. A parameter can change graph structure as well as numerical coordinates.

With technology, store the original rule and domain before simplifying. Request exact roots and derivatives, then test the sign on each interval cut by critical and excluded inputs. Plot at more than one scale. A missing hole, a false vertical join or a turning point outside the window is a display limitation, not a new mathematical property.

For a technology-free response, a feature list and sign chart provide a robust route to a smooth sketch. Label the requested equations and coordinates, draw asymptotes distinctly and show a hole clearly. State exact values such as e or √a unless approximations are requested. The graph and the written reasoning should tell the same story.