Year 12 Functions and relations: Function transformations

Function transformations describe how one graph is built from another. In Year 12, this includes translations, reflections, dilations, transformed polynomials, and special shapes such as the absolute value function. The key difficulty is that horizontal changes and vertical changes behave differently. Students often know the words but still apply the movement in the wrong direction. A transformation question becomes easier when you identify whether the change happens outside the function or inside its input.

Changes outside the function affect outputs vertically. Changes inside the bracket affect inputs horizontally.

Main transformation types

$y = f (x) + a$ shifts the graph up or down.
$y = f (x - a)$ shifts the graph right or left.
$y = - f (x)$ reflects the graph in the x-axis.
$y = f (- x)$ reflects the graph in the y-axis.
$y = | x |$ and related absolute-value graphs introduce a sharp turning shape.

Worked example 1

Problem: Describe how the graph of

y = {(x - 2)}^{2} + 3

is obtained from

y = x^{2}

The expression $x - 2$ inside the square shifts the graph right by 2.
The $+ 3$ outside the function shifts the graph up by 3.
The vertex therefore moves from $(0, 0)$ to $(2, 3)$ .

Answer: shift the parabola 2 units right and 3 units up.

Worked example 2

Problem: Sketch

y = | x - 1 | - 2

Start from the base graph $y = | x |$ , a V-shape with vertex at the origin.
The term $x - 1$ shifts the graph right by 1.
The $- 2$ shifts the graph down by 2.
The new vertex is at $(1, - 2)$ .
The graph still opens upward in a V-shape, with slopes -1 and 1 on either side.

Answer: the graph is an absolute-value V-shape with vertex at

(1, - 2)

What this page covers

This tutorial also covers identifying transformed polynomials, because the same transformation logic is used there. A transformed cubic, for example, is still recognised by its shifted turning behaviour and overall shape. The same idea applies when reading an absolute-value graph. The base graph matters, and the transformation tells you how that base graph has moved or reflected.

Worked example 3

Problem: Describe the transformations from

y = x^{3}

y = - 2 {(x + 1)}^{3}

The $x + 1$ inside the bracket shifts the cubic left by 1.
The factor $2$ outside the function creates a vertical stretch by factor 2.
The negative sign reflects the graph in the x-axis.
The point of inflection, originally at the origin, moves to $(- 1, 0)$ .

Answer: shift left 1, stretch vertically by 2, then reflect in the x-axis.

Common traps

Thinking $f (x - 2)$ shifts left when it actually shifts right.
Mixing vertical reflection with horizontal reflection.
Sketching an absolute-value graph smoothly instead of with a corner.

Revision focus

The best transformation habit is to say the move in words before drawing it. If you cannot say what the graph is doing, you are more likely to move it in the wrong direction. Naming the base graph first also helps: parabola, cubic, power graph, or absolute-value graph. Then the transformation becomes a change applied to something already familiar rather than a graph built from scratch.

Another useful check is to track one key point, such as a vertex or intercept, through the transformation. If the transformed graph is meant to pass through a certain shifted location, your sketch should show that explicitly.