18.1 Reflection, rotation and translation — Learn

Prerequisites: coordinates, positive and negative numbers, and reading a number line.

Objectives: reflect, rotate and translate points and shapes on a coordinate grid; describe a transformation using its defining information.

Higher only: describe changes and invariance after combining transformations.

A transformation moves or changes a shape according to a rule. Reflection, rotation and translation keep lengths and angles unchanged. Such transformations preserve size and shape.

Reflection

A reflection flips a shape across a mirror line. Each point and its image lie the same perpendicular distance from that line.

Example. Reflect $(3,2)$ in the $y$-axis.

The $y$-coordinate stays $2$ because the point remains level. The $x$-coordinate changes sign because the point crosses the vertical axis. The image is $(-3,2)$.

Rotation

A rotation turns a shape around a fixed point called the centre of rotation. State the centre, angle and direction.

For a $90^\circ$ anticlockwise turn about the origin, $(x,y)$ maps to $(-y,x)$. A clockwise turn maps $(x,y)$ to $(y,-x)$. A half-turn maps $(x,y)$ to $(-x,-y)$.

Example. Rotate $(2,1)$ by $90^\circ$ anticlockwise about the origin.

Use $(-y,x)$ for a quarter-turn anticlockwise. Substitute $x=2$ and $y=1$: $(-1,2)$.

Translation

A translation slides every point by the same vector, a column showing horizontal then vertical movement.

Example. Translate $(2,-1)$ by $\begin{pmatrix}-3\\4\end{pmatrix}$.

Add $-3$ to the $x$-coordinate: $2+(-3)=-1$. Add $4$ to the $y$-coordinate: $-1+4=3$. The image is $(-1,3)$.

Higher only: combinations of transformations

Apply each transformation in the order given. Keep the image from the first move as the starting point for the second.

Example. Translate the triangle with vertices $(1,1)$, $(3,1)$ and $(1,2)$ by $\begin{pmatrix}1\\1\end{pmatrix}$. Then reflect it in the $y$-axis and rotate it $90^\circ$ clockwise about the origin.

After the translation, the vertices are $(2,2)$, $(4,2)$ and $(2,3)$.

After reflection, they are $(-2,2)$, $(-4,2)$ and $(-2,3)$.

The clockwise quarter-turn maps $(x,y)$ to $(y,-x)$.

The final vertices are $(2,2)$, $(2,4)$ and $(3,2)$.

Each move preserves side lengths and angles, so these remain unchanged.

Watch out. A reflection in the $y$-axis changes the sign of $x$, not $y$. Check that the point and image sit opposite each other across that axis.

Progress check

The mural tile has a corner at $(3,1)$. Work out its image after a half-turn about the origin, then its image after a reflection in the $y$-axis. Explain how the signs change in each case.

You should now be able to

• reflect points and shapes in horizontal, vertical and diagonal mirror lines;
• rotate a shape when the centre, angle and direction are given;
• translate a shape using a vector;
• describe a transformation with enough detail to identify it.

• combine transformations in order and state what stays unchanged.