Sketch rational functions, reciprocal and modulus graphs, and use transformations and asymptotic behaviour

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.