Transformations and vectors
Maths · GCSE · The School
Four moves, four descriptions
Translation: give the vector. Reflection: give the mirror LINE (an equation, e.g. y = x). Rotation: give angle, direction and CENTRE. Enlargement: give scale factor and CENTRE. Marks are lost for incomplete descriptions more than for wrong ones — "rotation 90°" without a centre is not an answer.
Vectors as instructions
The column vector (3, −2) means 3 right, 2 down. Adding vectors means doing one move then the other: (3, −2) + (1, 5) = (4, 3). Multiplying by a scalar stretches without turning: 2(3, −2) = (6, −4), the same direction, twice as far. Negative flips the direction entirely.
Enlargement with fractions and negatives
A scale factor between 0 and 1 makes the shape SMALLER while still being called an enlargement — the word is technical, not descriptive. A negative scale factor puts the image on the opposite side of the centre, upside down. Both appear in exams precisely because they defeat guessing from the picture.
Finding a centre you have not been given
Exams ask you to DESCRIBE a transformation from two drawn shapes, which means finding the centre yourself. For a rotation: join each point to its image, construct the perpendicular bisector of two such lines, and the centre is where the bisectors cross — every point of the shape has turned about it, so it is the one point equidistant from each pair. For an enlargement it is easier: draw a ray through each point and its image and extend them backwards; they all meet at the centre. Worked: triangle A(1, 1), B(3, 1), C(1, 2) maps to A′(3, 3), B′(7, 3), C′(3, 5). Side AB was 2 units and A′B′ is 4, so the scale factor is 2. The ray through A(1, 1) and A′(3, 3) and the ray through B(3, 1) and B′(7, 3) meet at (−1, −1), and that is the centre. State all of it: enlargement, scale factor 2, centre (−1, −1).