# The four transformations

## In a nutshell

There are four main ways to mathematically alter a shape: translation, rotation, reflection and enlargement.

## Translation

Translating a shape means to move its position while keeping the orientation and size of the shape intact. Translations are described using column vectors $\begin{pmatrix}x\\y\end{pmatrix}$, which means to move the shape $x$ spaces to the right and $y$ spaces up.

##### Example 1

*The square $ABCD$ was originally in the top-right quadrant. It was then transformed to the bottom-left quadrant as shown below. Describe the transformation that took place.*

*The size and orientation of the shape is unchanged, so the shape has been translated. *

*To find the corresponding column vector, focus on one vertex and see where it moved to.*

$A=(1,1)$ *gets mapped to $A=(-4,-4)$.*

*The corresponding column vector is therefore $\begin{pmatrix}-5\\-5\end{pmatrix}$ as $A$ moved $5$ steps to the left and $5$ steps down to end up at $(-4,-4)$.*

*The transformation is a **translation** by column vector *$\underline{\begin{pmatrix}-5\\-5\end{pmatrix}}$__.__

## Rotation

Rotating a shape has three details associated with it:

- The angle by which the shape has been rotated.
- The direction of rotation (clockwise or anticlockwise).
- The centre of rotation.

The *centre of rotation* is the point you rotate everything around.

##### Example 2

*Describe the transformation that maps the shape $ABCD$ to the shape $A'B'C'D'$ in the diagram below.*

*This is a **rotation** of *$\underline{180 \degree}$* with centre **O**.*

**Note: **In this case, there was no need to specify direction as a rotation of $180^\circ$ clockwise is the same as a rotation of $180^\circ$ anticlockwise.

## Reflection

Reflecting a shape means to draw a mirror line and reflect the shape through that mirror line. You may be asked to find the equation of a mirror line.

##### Example 3

*Describe the transformation that maps the shape $ABCD$ to the shape $A'B'C'D'$ in the diagram below.*

*The size of the shape is the same but the orientation has been slightly changed. Visually, it looks like a reflection, which can be verified by joining up the matching vertices.*

*The midpoints of all of these lines form the mirror line - which is the line $y=-x$.*

*Therefore, the transformation is a **reflection** in the line *$\underline{y=-x}$*.*

## Enlargement

Enlarging a shape means to make it bigger or smaller. There are always two details associated with an enlargement:

- The centre of enlargement.
- The scale factor.

### Enlarging a shape

To enlarge a shape by a given scale factor, follow this procedure.

#### procedure

1. | Measure the distance from the centre of enlargement to a vertex of the shape. |

2. | Multiply the distance by the scale factor - call this number $x$. |

3. | The corresponding enlarged vertex is found by moving in the same direction until the distance from the centre of enlargement is $x$. If the scale factor is negative, move in the *opposite* direction. |

4. | Repeat this for all the vertices of the shape and join the new vertices together with lines - this gives the enlarged shape. |

### Negative and fractional scale factors

Despite the name, enlarging a shape doesn't always result in the shape increasing in size.

- This is only true for scale factors larger than $1$.

- A scale factor between $0$ and $1$ results in the new shape being smaller than the old one and also closer to the centre of enlargement.

- A
*negative* scale factor means the enlarged shape is flipped and moved in the opposite direction.

##### Example 4

*Describe the transformations that map the shape $ABCD$ to the shapes $A'B'C'D'$ and $A''B''C''D''$ in the diagram below.*

$ABCD\rightarrow A'B'C'D'$:

*This is an enlargement with centre $O$. The scale factor is given by the ratio of two corresponding sides:*

$\frac{A'B'}{AB}=\frac{10\,\text{squares}}{4\,\text{squares}}=\frac{10}{4}=2.5$

*The scale factor is therefore $2.5$.*

*The transformatioin that maps ABCD to A'B'C'D'is an **enlargement** with scale factor *$\underline{2.5}$*, centre **O**.*

$ABCD\rightarrow A''B''C''D''$:

*This is an enlargement with centre $O$ that's in the opposite direction - so the scale factor will be negative. Comparing the side lengths:*

$\frac{A''B''}{AB}=\frac{2\,\text{squares}}{4\,\text{squares}}=\frac{2}{4}=0.5$

*The scale factor is therefore $-0.5$.*

*The transformation that maps ABCD to A''B''C''D'' is an **enlargement** with scale factor *$\underline{-0.5}$*, centre **O**.*