Carrying out and describing reflections, rotations, translations and enlargements, including negative and fractional scale factors at Higher tier.

What this dot point is asking

AQA wants you to carry out and fully describe the four transformations (reflection, rotation, translation, enlargement), including negative and fractional scale factors at Higher tier. The two skills are linked: performing a transformation on a grid, and describing a given transformation completely with all the defining information. A description that misses a detail (the mirror line, the centre, the angle) loses marks even if the type is named correctly.

Reflection

A reflection produces a mirror image across a mirror line. Each point and its image are the same perpendicular distance from the line, on opposite sides. Common mirror lines are the axes ($x = 0$, $y = 0$), vertical and horizontal lines ($x = 3$, $y = -1$), and the diagonals ($y = x$, $y = -x$). A full description states "reflection" and the equation of the mirror line. Reflection preserves size and shape; only orientation flips.

Rotation

A rotation turns a shape about a fixed point (the centre of rotation) through a given angle in a given direction. To describe a rotation fully you need three things: the angle (often $90^\circ$ or $180^\circ$), the direction (clockwise or anticlockwise, though $180^\circ$ needs no direction), and the centre. Tracing paper is the reliable method: trace the shape, pin the centre, and turn.

Translation

A translation slides every point the same distance in the same direction, described by a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ is the movement right (negative for left) and $b$ is the movement up (negative for down). A translation by $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ moves every point $3$ right and $2$ down. Size, shape and orientation are all unchanged.

Enlargement at Higher tier

An enlargement changes the size of a shape by a scale factor from a centre of enlargement. The distance from the centre to each point is multiplied by the scale factor.

A fractional scale factor such as $\tfrac{1}{2}$ shrinks the shape (the image is smaller but still on the same side of the centre). A negative scale factor such as $-2$ both resizes and inverts: each image point lands on the opposite side of the centre, so the image is upside down relative to the original.

Describing a transformation fully

Examiners stress the word "fully". A complete description gives the transformation type plus all its defining details: reflection needs the mirror line; rotation needs angle, direction and centre; translation needs the column vector; enlargement needs the scale factor and centre. Naming two transformations loses marks because a single transformation is required.

What each transformation preserves

A useful way to identify a transformation is to ask what stays the same. Reflections, rotations and translations are all congruences: the image is exactly the same size and shape as the original, so lengths and angles are unchanged (only position or orientation differs). An enlargement is the odd one out: it preserves angles and shape but changes size, so the image is similar to the original rather than congruent. Spotting that two shapes are the same size narrows the options to reflection, rotation or translation, while a size change points to enlargement.

Finding the centre of rotation or enlargement

To find a centre of rotation, join corresponding points on the object and image, construct the perpendicular bisector of each join, and the centre is where these bisectors meet. To find a centre of enlargement, draw straight lines through pairs of corresponding vertices and extend them; they all pass through the centre. The scale factor of an enlargement is then the ratio of an image length to the matching original length, taken as negative if the image is inverted through the centre. These "describe the transformation" tasks reward this systematic locating method.