Functions and sketching graphs (WACE Year 12 Mathematics Specialist Unit 3)

What this dot point is asking

SCSA expects you to analyse and sketch graphs of rational functions $y = \tfrac{P(x)}{Q(x)}$, graphs of $y = \tfrac{1}{f(x)}$ from the graph of $y = f(x)$, and modulus graphs $y = |f(x)|$ and $y = f(|x|)$, using asymptotes, intercepts and transformations rather than plotting points.

Rational functions and asymptotes

A vertical asymptote occurs where $Q(x) = 0$ but $P(x) \neq 0$. Near it the graph shoots to $\pm\infty$; the sign on each side is found from a sign table.

For end behaviour, compare degrees of $P$ and $Q$:

• $\deg P < \deg Q$: horizontal asymptote $y = 0$.
• $\deg P = \deg Q$: horizontal asymptote $y = \tfrac{\text{leading coeff } P}{\text{leading coeff } Q}$.
• $\deg P = \deg Q + 1$: an oblique (slant) asymptote, found by polynomial division.

The reciprocal of a graph

To sketch $y = \tfrac{1}{f(x)}$ from $y = f(x)$:

• Zeros of $f$ become vertical asymptotes of $\tfrac{1}{f}$.
• Where $f \to \pm\infty$, $\tfrac{1}{f} \to 0$.
• Maxima of $f$ (above the axis) become minima of $\tfrac{1}{f}$ and vice versa.
• Points where $f = \pm 1$ are fixed, since $\tfrac{1}{\pm 1} = \pm 1$.
• The sign of $\tfrac{1}{f}$ matches the sign of $f$.

Modulus functions

For $y = |f(x)|$, reflect any part of $y = f(x)$ that lies below the $x$-axis up across the axis. For $y = f(|x|)$, take the part of $y = f(x)$ for $x \ge 0$ and reflect it in the $y$-axis to give an even function.

A systematic sketching checklist

SCSA rewards a methodical approach, not point-plotting. Work through the same list every time:

1. Domain. State any values excluded by a zero denominator or even root.
2. Intercepts. Set $y = 0$ for $x$-intercepts and $x = 0$ for the $y$-intercept.
3. Vertical asymptotes. Solve $Q(x) = 0$ where $P(x) \neq 0$, then a sign table on each side.
4. End behaviour. Compare the degrees of numerator and denominator for a horizontal or oblique asymptote.
5. Turning points. Differentiate and solve $\tfrac{dy}{dx} = 0$ where the question needs them.
6. Join the pieces respecting the sign of the function between consecutive features.

Sign tables make the branches unambiguous

The single most reliable tool for a rational sketch is the sign table. List every $x$-value where the numerator or denominator is zero, in order; these split the number line into intervals. On each interval the function keeps a constant sign, found by testing one value or counting the negative factors. The sign table tells you whether each branch approaches a vertical asymptote from $+\infty$ or $-\infty$ and on which side of the $x$-axis each branch lives, which is exactly the information a hand sketch needs.

Transformations of standard graphs

Many Specialist graphs are transformations of a known parent curve. The graph of $y = a\,f(b(x - h)) + k$ takes $y = f(x)$, stretches it by factor $a$ vertically and $\tfrac{1}{b}$ horizontally, then translates it by $h$ right and $k$ up. Recognising a rational function as a translated reciprocal, for instance $y = \dfrac{1}{x - 2} + 3$ as the curve $y = \dfrac{1}{x}$ shifted right $2$ and up $3$, gives the asymptotes $x = 2$ and $y = 3$ instantly without further work.