AC9M7SP03 · Year 7 · Space

Transformations in the Cartesian plane

ACARA v9 CONTENT DESCRIPTION “describe transformations of a set of points using coordinates in the Cartesian plane, translations and reflections on an axis, and rotations about a given point”

Shapes do not have to stay still. Moving them around, sliding, flipping and turning, is the basis of everything from tiling patterns and logos to computer graphics and animation. This year you learn to describe these movements precisely using coordinates on the Cartesian plane. The three basic transformations are translation, reflection and rotation, and each can be described exactly by what it does to the points of a shape.

What unites these three transformations is that none of them changes the size or shape of the figure. They are rigid motions: the object that comes out is identical to the one that went in, just in a new position or orientation. The Cartesian plane lets us pin down exactly where it ends up.

Translation: sliding

A translation slides a shape across the plane without turning or flipping it. Every single point moves by the same amount in the same direction, described by a horizontal and a vertical shift. If a shape is translated 5 to the right and 2 up, then each point with coordinates (x, y) moves to (x plus 5, y plus 2). Because every point shifts identically, the shape arrives unchanged in size and orientation, simply relocated.

Translation: sliding
A translation slides every point by the same horizontal and vertical amount.
A translation slides a shape without turning or flipping it. Every point moves by the same amount, here 5 to the right and 2 up, so each coordinate (x, y) becomes (x plus 5, y plus 2). The shape keeps its size and orientation.

This is why translations are so easy to describe with coordinates: you only need the two numbers that say how far across and how far up or down. Applying them to each vertex of a shape gives the translated shape exactly. The same rule, applied to every point, captures the whole movement in a compact form.

Reflection and rotation

A reflection flips a shape across a line, called the mirror line or axis. The reflected shape is a mirror image, with every point the same distance from the line as before but on the opposite side. Reflecting across the y-axis, for instance, keeps the height of each point the same but reverses its left-right position. The result looks like the original seen in a mirror.

Reflection and rotation
A reflection flips across a line; a rotation turns about a fixed point.
A reflection flips a shape across a line, the mirror line, producing a mirror image the same distance from the line on the other side. A rotation turns a shape about a fixed point by a given angle. Both keep the shape the same size, changing only its position or orientation.

A rotation turns a shape about a fixed point, the centre of rotation, by a given angle such as 90 or 180 degrees. Every point swings around the centre, staying the same distance from it, so the shape turns as a whole without changing size. Translation, reflection and rotation together give a complete toolkit for moving shapes around the plane, and describing each one through its effect on coordinates turns these everyday movements into precise mathematics, the same mathematics that powers patterns, design and the graphics on every screen.

Teaching tip: tracing paper makes transformations physical. Draw a shape, trace it, then slide the tracing for a translation, flip it over for a reflection, or pin one point and turn it for a rotation. Watching the traced copy move while the original stays put makes each transformation clear and memorable.

For coordinates, work through what happens to a single point first, then apply the same rule to every vertex. Seeing that a translation adds the same numbers to every point, or that a reflection across the y-axis flips the sign of each x, turns the visual idea into a reliable method.

Builds on: Tables of values and the Cartesian plane (AC9M7A05). That unit plotted points on the Cartesian plane; this unit moves sets of points by transformations.
Quick self-check
1. A translation moves a shape by
2. Under a translation of (+3, -1), the point (2, 5) moves to
3. A reflection produces
4. Which transformation turns a shape about a fixed point?
5. What do translation, reflection and rotation all have in common?