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:

1. The angle by which the shape has been rotated.
2. The direction of rotation (clockwise or anticlockwise).
3. 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:

1. The centre of enlargement.
2. The scale factor.

Enlarging a shape

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

procedure

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.