Transformations: reflection, rotation, translation and enlargement

• Reflect in the y-axis: $(-2, 3)$.
• Reflect in the line $y = x$: swap the coordinates, $(3, 2)$.
• Rotate 90° clockwise about the origin: $(3, -2)$.
• Translate by $\begin{pmatrix} 4 \\ -1 \end{pmatrix}$: add 4 to $x$ and $-1$ to $y$: $(6, 2)$.
• Enlarge by scale factor 2, centre the origin: $(4, 6)$.
• Enlarge by scale factor $-1$, centre the origin: $(-2, -3)$, the same as a 180° rotation.

Exam questions often give two shapes and ask you to describe the single transformation. Full marks need every detail:

• Reflection in the line $x = 1$.
• Rotation 90° anticlockwise about $(0, 2)$.
• Translation by the vector $\begin{pmatrix} -3 \\ 5 \end{pmatrix}$.
• Enlargement by scale factor 3, centre $(1, 1)$.

Naming the transformation alone usually earns only one mark.

• Rotation: use tracing paper, try possible centres, and check which turns the shape onto its image.
• Enlargement: draw straight lines through matching corners of the shape and its image; they meet at the centre.
• Scale factor: divide a length on the image by the matching length on the original.

A scale factor between 0 and 1 makes the shape smaller (it's still called an enlargement). A negative scale factor puts the image on the opposite side of the centre, upside down. Both are Higher tier at GCSE.

Doing one transformation after another can equal a single one: two reflections in parallel lines make a translation, and two reflections in lines that cross make a rotation. Questions ask you to carry out both, then describe the single transformation that has the same effect.

• Missing the direction of a rotation.
• Using coordinates instead of a column vector for a translation.
• Measuring an enlargement from the wrong centre.
• Reflecting in $y = 2$ when the question says $x = 2$ (horizontal vs vertical lines).

See similar and congruent shapes for what each transformation preserves.

An invariant point doesn't move under a transformation. Points on a mirror line are invariant under that reflection; the centre of a rotation or enlargement is invariant too. A translation (other than by the zero vector) has no invariant points. Questions sometimes ask you to identify them, which is a quick way to check you've used the right line or centre.

Use the point $(4, 1)$.

1. Reflect it in the x-axis.
2. Rotate it 180° about the origin.
3. Translate it by $\begin{pmatrix} -5 \\ 2 \end{pmatrix}$.
4. Enlarge it by scale factor 3, centre $(1, 1)$.

Answers: 1. $(4, -1)$. 2. $(-4, -1)$. 3. $(-1, 3)$. 4. $(10, 1)$: it is 3 units right of the centre, so it moves to 9 units right.

• Use tracing paper for rotations: it's allowed in the exam and makes the answer easy to check.
• Count squares carefully for translations and write the vector with the horizontal move on top.
• After any transformation, check one corner of the image against the original before drawing the rest.
• If a question says "single transformation", don't describe two.