Apply translations, dilations and reflections to the graph of a function, including the form $y = a f(b(x - h)) + k$ and the effect of each parameter

What this dot point is asking

QCAA wants you to identify and apply translations, dilations and reflections to function graphs, working with the form $y = a f(b(x - h)) + k$.

The general form

$y = a f(b(x - h)) + k$.

Vertical dilation

$y = a f(x)$ scales $y$-values by $|a|$. $x$-axis fixed.

Horizontal dilation

$y = f(bx)$ compresses horizontally by factor $b$ (equivalently dilates by $1/b$). $y$-axis fixed.

A common slip: $y = f(2x)$ is a compression, not a stretch.

Translations

$y = f(x - h)$ shifts the graph right by $h$ units (the bracket sign is opposite to the shift direction).

$y = f(x) + k$ shifts the graph up by $k$ units.

Reflections

$y = -f(x)$ reflects in the $x$-axis. $y = f(-x)$ reflects in the $y$-axis. $y = f^{-1}(x)$ reflects in the line $y = x$ (the inverse function).

Order of transformations

Apply horizontal dilation/reflection (with $b$) first inside the bracket, then horizontal translation ($h$), then vertical dilation/reflection ($a$), then vertical translation ($k$).

Worked example

Sketch $y = -2\sqrt{x - 3} + 1$ from $y = \sqrt x$.

$a = -2$, $h = 3$, $k = 1$.

Domain: $x \ge 3$. Range: $y \le 1$. Starting point of the curve: $(3, 1)$. Reflection flips downward; dilation by $2$ steepens.

Common traps

Direction of horizontal translation. $(x - 3)$ shifts right $3$, not left.

Reciprocal scale factor for horizontal. $y = f(2x)$ halves $x$-values.

Forgetting reflection sign. $y = -f(x)$ flips vertically; $y = f(-x)$ flips horizontally.

In one sentence

For $y = af(b(x - h)) + k$: $a$ dilates/reflects vertically, $b$ dilates/reflects horizontally with reciprocal factor, $h$ shifts horizontally (right if positive), $k$ shifts vertically; the standard application order is horizontal dilation, then horizontal translation, then vertical dilation, then vertical translation.