Transformations

What you will learn

• Name and describe translations, reflections, rotations and enlargements.
• Find the image of a shape on a coordinate grid.
• Say which properties each transformation keeps.

Before you start: you should be able to plot points on a coordinate grid, including negative numbers. See Line and Rotational Symmetry.

The idea

A transformation moves or changes a shape according to a rule. The starting shape is the object; the result is the image. There are four main transformations:

1. Translation: slide the shape without turning it.
2. Reflection: flip it over a mirror line.
3. Rotation: turn it about a fixed centre.
4. Enlargement: resize it from a centre by a scale factor.

Translations, reflections and rotations are isometries: they keep lengths and angles, so the image is congruent to the object. An enlargement keeps angles but changes lengths, so the image is similar to the object.

Reflection

Translation

Rotation

Enlargement

Describing a transformation fully

Each transformation needs specific information:

• Translation: a column vector, such as $\begin{pmatrix} 3 \\ -2 \end{pmatrix}$ for “3 right, 2 down”.
• Reflection: the equation of the mirror line, such as $y = x$.
• Rotation: the centre, the angle and the direction (clockwise or anticlockwise).
• Enlargement: the centre and the scale factor.

Worked examples

Example 1: Translation

Triangle $T$ has vertices $(1, 2)$, $(4, 2)$ and $(1, 5)$. Translate it by $\begin{pmatrix} -3 \\ 1 \end{pmatrix}$.

1. Subtract $3$ from every $x$-coordinate and add $1$ to every $y$-coordinate.
2. $(1, 2) \to (-2, 3)$, $(4, 2) \to (1, 3)$, $(1, 5) \to (-2, 6)$.

Answer: $(-2, 3)$, $(1, 3)$, $(-2, 6)$

Example 2: Reflection in y = x

Reflect the point $P(5, 2)$ in the line $y = x$.

1. Reflecting in $y = x$ swaps the coordinates: $(x, y) \mapsto (y, x)$.
2. So $P(5, 2) \to P'(2, 5)$.
3. Check: the midpoint of $PP'$ is $(3.5, 3.5)$, which lies on $y = x$.

Answer: $P'(2, 5)$

Example 3: Enlargement

A rectangle $2$ cm by $3$ cm is enlarged by scale factor $4$. Find the new dimensions and compare the areas.

1. Every length multiplies by $4$: the image is $8$ cm by $12$ cm.
2. Original area: $6 \text{ cm}^2$. New area: $96 \text{ cm}^2$.
3. The ratio is $96 \div 6 = 16 = 4^2$.

Answer: $8 \text{ cm} \times 12 \text{ cm}$; the area is $16$ times larger.

Tip: a scale factor between $0$ and $1$ makes the shape smaller, but it is still called an enlargement. A negative scale factor flips the image to the other side of the centre.

Why it matters

Computer graphics are built on transformations: every time a game character walks, turns or zooms toward the camera, the software applies translations, rotations and enlargements to thousands of coordinates. Artists, architects and map makers use the same ideas to move, flip and resize drawings.