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 combined form $y = a f(b(x - h)) + k$ and understanding the effect of each of the four parameters. Transformations let you build any member of a function family from its parent graph, which is far quicker than plotting points, and they reappear throughout the course wherever a standard curve is shifted or scaled.

The general form

The four parameters in

each control one transformation.

Vertical dilation

$y = a f(x)$ multiplies every output by $a$, stretching the graph vertically by factor $|a|$ while leaving the $x$-axis (where $y = 0$) fixed. Points farther from the $x$-axis move farther, so intercepts on the $x$-axis stay put and the overall height scales.

Horizontal dilation

$y = f(bx)$ acts on the input, so a point that was at $x$ now occurs at $\tfrac{x}{b}$, compressing the graph horizontally by factor $b$ (equivalently dilating by $\tfrac{1}{b}$) while fixing the $y$-axis. A common slip is to read $y = f(2x)$ as a stretch; it is a compression, because the curve reaches each output value at half the previous $x$-value.

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

A reflection is the special case of a dilation with a negative factor. $y = -f(x)$ reflects in the $x$-axis (every $y$-value changes sign), $y = f(-x)$ reflects in the $y$-axis (every $x$-value changes sign), and the inverse $y = f^{-1}(x)$ reflects the graph in the line $y = x$, swapping the roles of input and output. Recognising a reflection as a sign on $a$ or $b$ keeps it inside the same general framework rather than treating it as a separate idea.

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 ($k$).

How transformations act on points

Each transformation acts on the coordinates of every point. A vertical dilation by $a$ sends $(x, y)$ to $(x, ay)$; a horizontal dilation by factor $\tfrac{1}{b}$ sends $(x, y)$ to $(\tfrac{x}{b}, y)$; a translation sends $(x, y)$ to $(x + h, y + k)$. Tracking a single known point (such as a vertex, endpoint or intercept) through the sequence is the fastest way to locate the transformed graph, and it is exactly what extended-response questions reward.

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.