Transformations: O-Level / IGCSE Maths (0580)

A transformation changes the position, orientation, or size of a shape on the coordinate grid. In this topic you work with the four transformations on the 0580 syllabus: reflections, rotations, translations, and enlargements. For each one you need to do two things: perform the transformation to draw the image of a given object, and describe a transformation fully when you are shown both an object and its image. Reflections, rotations and translations are congruent transformations: the image is the same shape and size as the object. An enlargement is generally not congruent, because it scales the lengths.

The key is knowing exactly what defines each one. A reflection needs a mirror line, often given as an equation such as $x = 2$, $y = -1$, $y = x$, or $y = -x$. A rotation needs a centre, an angle, and a direction (clockwise or anticlockwise): for example, a rotation of $90^\circ$ anticlockwise about the origin. A translation is given by a column vector $\begin{pmatrix} a \\ b \end{pmatrix}$, where $a$ is the movement in the $x$-direction and $b$ the movement in the $y$-direction. An enlargement needs a centre of enlargement and a scale factor $k$; a point at distance $d$ from the centre moves to distance $kd$, and a negative or fractional $k$ is also possible. When asked to identify a single transformation, name the type and then state every property it requires, since a description that omits the centre, the vector, or the mirror line is incomplete.

A reliable method is to track what happens to the vertices: find the image of each corner using the rule for that transformation, then join them in order. To describe an unknown transformation, first check whether the size has changed: if it has, it is an enlargement; if not, decide between reflection, rotation, or translation by checking the orientation and how corresponding points correspond. The worked examples below are original, written from the syllabus objective to show these methods in action.