Cambridge IGCSE Mathematics · Syllabus 0580 · Transformations and Vectors
Transformations
What is Transformations?
A transformation is a rule that assigns to every point of the plane a single image point, so that a whole shape is carried to an image shape. Cambridge IGCSE Mathematics 0580 examines four: reflection, rotation, enlargement and translation. On the Core route, statement C7.1 restricts them: a reflection is in a vertical or horizontal line, a rotation is about the origin, a vertex of the shape or the midpoint of one of its edges through a multiple of 90 degrees, an enlargement uses a positive or fractional scale factor, and a translation is by a column vector; Core questions do not involve combinations of transformations. On the Extended route, statement E7.1 removes those restrictions: the mirror line may be any straight line, the centre of rotation may be any point, the scale factor may also be negative, and questions may involve combinations of transformations. A ruler must be used for all straight edges on both routes. Reflection, rotation and translation preserve every length and angle, so the image is congruent to the object; reflection alone reverses orientation. Enlargement preserves angles and shape but multiplies every length by the modulus of the scale factor and every area by its square, so the image is similar to the object. Describing a transformation completely means stating its name together with every parameter it requires: the mirror line as an equation for a reflection, the angle, direction and centre for a rotation, the centre and scale factor for an enlargement, and the column vector for a translation.
This definition is part of the Transformations and Vectors chapter in Cambridge IGCSE Mathematics.
Transformations in context
“Enlargement, centre \((1,1)\), scale factor 2” for an image on the far side of the centre. The sign is missing. If the image is on the opposite ray, the scale factor is \(-2\), and \(2\) describes a different image altogether. “Reflection in the line \(y=x\), then translation by \(\begin{pmatrix}0\\2\end{pmatrix}\).” Two transformations offered where one was asked for. Find the single equivalent transformation instead.
Common mistakes with Transformations
- A seventh, quieter one Extended E7.1. Order matters when transformations are combined. Two translations are the safe exception: they always give the same result in either order, because adding column vectors is commutative. A handful of other pairs commute too — two rotations about the same centre, two enlargements from the same centre — but every one of those shares a fixed point. Where no fixed point is shared, assume reversing the order changes the image. Core questions do not combine transformations, so this danger belongs to the Extended route alone.