Transformations of functions (QCE Math Methods Unit 1)

Syllabus dot point

“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”

7 min readUpdated 7 Oct 2026

Quick answer

In $y = af(b(x-h)) + k$ : $a$ scales/reflects vertically, $b$ scales/reflects horizontally by reciprocal factor $1/|b|$ , $h$ shifts right and $k$ shifts up. Apply horizontal dilation, then horizontal shift, then vertical dilation, then vertical shift.

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What this dot point is asking
The general form
Vertical dilation
Horizontal dilation
Translations
Reflections
Order of transformations
How transformations act on points
In one sentence
Exam-style questions

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

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

each control one transformation.

Parameter	Effect
$a$	Vertical dilation by factor $\|a\|$ ; reflection in $x$ -axis if $a < 0$
$b$	Horizontal dilation by factor $1/\|b\|$ ; reflection in $y$ -axis if $b < 0$
$h$	Horizontal translation $h$ units right
$k$	Vertical translation $k$ units up

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.

Sketching a transformed root function

Sketch $y = -2\sqrt{x - 3} + 1$ by applying transformations to the parent $y = \sqrt{x}$ .

Step 1: Identify each parameter and its effect

Write the function in the form $y = a f(x - h) + k$ to read off the parameters: $a = -2$ , $h = 3$ , $k = 1$ . The parent $y = \sqrt{x}$ starts at $(0, 0)$ , has domain $x \geq 0$ , and increases to the right.

Step 2: Apply the horizontal translation inside the bracket

The term $(x - 3)$ shifts the entire graph $3$ units to the right, because we now need $x - 3 \geq 0$ , i.e. $x \geq 3$ . The start point moves from $(0, 0)$ to $(3, 0)$ .

Step 3: Apply the vertical dilation and reflection outside

The factor $a = -2$ does two things at once: the magnitude $2$ stretches the graph vertically by a factor of $2$ (making it steeper), and the negative sign reflects it in the $x$ -axis, so the curve now falls to the right instead of rising.

Step 4: Apply the vertical translation

Adding $k = 1$ shifts the whole graph up by $1$ unit. The start point moves from $(3, 0)$ to $(3, 1)$ .

Step 5: State domain, range and shape

Domain: $x \geq 3$ (argument of the square root must be non-negative). Range: $y \leq 1$ (the curve starts at the maximum value $y = 1$ and decreases). The curve begins at $(3, 1)$ , falls to the right, and flattens gradually.

Final answer: Start point $(3, 1)$ ; domain $x \geq 3$ ; range $y \leq 1$ ; curve falls and flattens to the right.

Common traps

Direction of horizontal translation: $(x - 3)$ shifts the graph right by $3$ , not left; the bracket sign is opposite to the shift.
Reciprocal horizontal scale: $y = f(2x)$ compresses, halving the $x$ -values, rather than stretching.
Reflection signs: $y = -f(x)$ reflects in the $x$ -axis (vertical flip); $y = f(-x)$ reflects in the $y$ -axis (horizontal flip).
Wrong order: Apply the operations inside the bracket (on $x$ ) before those outside (on $y$ ).

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.

Exam-style questions

Questions in the style of QCAA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

Practice3 marks

Paper 1 (technique). Describe the sequence of transformations that maps

y = x^2

onto

y = -3(x - 1)^2 + 4

, and state the new vertex.

Show worked answer →

Parameters $a = -3$ , $h = 1$ , $k = 4$ .

Sequence: a vertical dilation by factor $3$ , a reflection in the $x$ -axis (since $a < 0$ ), then a translation $1$ right and $4$ up.

The vertex moves from $(0, 0)$ to $(1, 4)$ and the parabola opens downward.

Markers reward naming each transformation, the direction of each, and tracking the vertex.

Practice4 marks

Paper 2 (complex familiar). The point

(2, 5)

lies on

y = f(x)

. Determine the coordinates of the image of this point under (a)

y = f(x - 3) + 1

and (b)

y = 2f(x)

, and (c) state the equation of the image of the line

y = f(x)

after a reflection in the

y

-axis.

Show worked answer →

(a) $f(x - 3)$ shifts right $3$ and $+1$ shifts up $1$ , so $(2, 5) \to (5, 6)$ .

(b) A vertical dilation by factor $2$ multiplies the $y$ -coordinate: $(2, 5) \to (2, 10)$ .

Markers reward applying each transformation to the coordinates and the reflected equation.

Practise this

Sources & how we know this

Mathematical Methods General senior syllabus 2025 — QCAA