U9 Symmetry and Transformation: Essential Formulas

Reflection in a Vertical Line

When a point $P(x, y)$ is reflected in the vertical line $x = a$, the image $P'(x', y')$ has coordinates given by the formula below. The line $x = a$ is the axis of symmetry.

Reflection in a Horizontal Line

When a point $P(x, y)$ is reflected in the horizontal line $y = b$, the image $P'(x', y')$ has coordinates given by the formula below.

Rotation about the Origin

For a rotation of a point $P(x, y)$ about the origin $O$ through an angle $\theta$ (anticlockwise is positive), the image $P'(x', y')$ is given by the rotation matrix. Common angles are shown below.

For specific angles: $90^\circ$ rotation: $(x, y) \to (-y, x)$; $180^\circ$ rotation: $(x, y) \to (-x, -y)$; $270^\circ$ rotation: $(x, y) \to (y, -x)$.

Translation by a Vector

A translation moves every point by a fixed vector $\begin{pmatrix} h \\ k \end{pmatrix}$. If $P(x, y)$ is translated, its image $P'(x', y')$ is given by the following simple addition.

Enlargement about the Origin

An enlargement with centre at the origin $O(0,0)$ and scale factor $k$ transforms a point $P(x, y)$ to its image $P'(x', y')$. If $|k| > 1$, the shape is enlarged; if $0 < |k| < 1$, it is reduced. A negative $k$ gives an image on the opposite side of the centre.

Enlargement about a Point $(a, b)$

For an enlargement with centre $C(a, b)$ and scale factor $k$, the transformation can be considered as a translation of the centre to the origin, followed by an enlargement, and then a translation back.