A transformation moves or changes a shape. A translation slides it using a column vector, a reflection flips it in a mirror line, a rotation turns it about a centre, and an enlargement changes its size by a scale factor from a centre. Translations, reflections and rotations keep the shape the same size; enlargements change the size but keep the angles.
Transformation rules
- Translation by (a, b): (x, y) → (x + a, y + b)
- Reflection in the y-axis: (x, y) → (−x, y); in the x-axis: (x, y) → (x, −y)
- Reflection in y = x: (x, y) → (y, x)
- Rotation 90° clockwise about O: (x, y) → (y, −x); 180°: (x, y) → (−x, −y)
- Enlargement scale factor k, centre O: (x, y) → (kx, ky)
Translations
A translation slides every point the same distance in the same direction, without turning or resizing. It is described by a column vector: the top number is the movement right (negative means left) and the bottom number is the movement up (negative means down).
Reflections
A reflection flips a shape in a mirror line. Every image point is the same distance from the line as the original, on the opposite side. Learn the rules for the axes and the lines y = x and y = −x. For other lines, such as x = 2, count squares perpendicular to the mirror line.
Rotations
A rotation turns a shape about a fixed centre through an angle, clockwise or anticlockwise. About the origin, 180° sends (x, y) to (−x, −y), and 90° clockwise sends (x, y) to (y, −x). Tracing paper is the most reliable way to rotate about any other centre.
Enlargements
An enlargement changes the size of a shape by a scale factor from a centre of enlargement. With centre the origin, multiply each coordinate by the scale factor: (3, 1) with scale factor 3 becomes (9, 3). A scale factor between 0 and 1 makes the shape smaller, and lengths are multiplied by the scale factor.
Describing a transformation
To “describe fully” a single transformation, give every required detail. Translation: the column vector. Reflection: the equation of the mirror line. Rotation: the angle, direction and centre. Enlargement: the scale factor and centre. Missing one detail loses marks.
Worked examples
Example 1. Translate the point (2, 5) by the vector with top number 3 and bottom number −4.
- Add 3 to the x-coordinate and −4 to the y-coordinate.
Answer: (5, 1).
Example 2. Reflect the point (4, −3) in the x-axis, and the point (2, 7) in the line y = x.
- In the x-axis, the y-coordinate changes sign: (4, 3).
- In y = x, the coordinates swap: (7, 2).
Answer: (4, 3) and (7, 2).
Example 3. Rotate the point (5, 2) by 90° clockwise about the origin.
- 90° clockwise: (x, y) → (y, −x).
Answer: (2, −5).
Example 4. Enlarge the point (3, 1) by scale factor 3, centre the origin.
- Multiply each coordinate by 3.
Answer: (9, 3).
Example 5. Shape A has vertices (1, 1), (3, 1) and (1, 2). Shape B has vertices (−1, −1), (−3, −1) and (−1, −2). Describe fully the single transformation that maps A onto B.
- Every point (x, y) has become (−x, −y).
Answer: A rotation of 180° about the origin.
Common mistakes
- Writing a column vector the wrong way up. Top number: horizontal; bottom number: vertical.
- Confusing reflection in x = a with y = a. x = 2 is a vertical line; y = 2 is horizontal.
- Leaving out the centre of a rotation or enlargement. “Describe fully” needs the centre.
- Giving two transformations. The question asks for a single transformation.
- Not stating the direction of rotation. Say clockwise or anticlockwise (not needed for 180°).
How transformations are examined
Every Edexcel GCSE and iGCSE tier includes transformations on a grid, and Higher papers add negative and fractional scale factors. Questions ask you to:
- translate, reflect, rotate or enlarge a shape on a grid
- describe fully a single transformation
- find the coordinates of an image point
- combine two transformations and describe the result
The transformations worksheet has questions with answers, and the GCSE geometry revision engine gives unlimited practice.
Enlargement links to similar shapes, and column vectors lead to vectors.
Frequently asked questions
What details do you need to describe a rotation?
The angle, the direction (clockwise or anticlockwise) and the centre of rotation.
What happens to a shape in an enlargement?
Its lengths are multiplied by the scale factor, but its angles stay the same.
How do you reflect a point in the line y = x?
Swap its coordinates: (a, b) becomes (b, a).