## What you need to know

There are 4 types of transformation: **translation, rotation, reflection, **and **enlargement**. All the info you need on enlargement can be found here (similar shapes revision), so in this topic we’ll focus on the first three. To do this, we will go through how to translate, rotate, and reflect shape A, as seen on the right.

**Translation **(sometimes called ‘shifting’) is the process of moving something around. In the case of our 2D axes, that means moving a shape some distance in the x direction and some distance in the y direction.

Translations are often described using vectors, where the top value represents the movement in x (positive means right, negative means left), and the bottom value represents the movement in y (positive means up, negative means down). For example, the vector

** Example: **Translate shape A by the vector \begin{pmatrix}-4\\1\end{pmatrix}.

means “3 spaces

**left**, and 2 spaces

**up**”. Let’s see an example.

The vector in the question has a -4 on top and a 1 on the bottom, which means we need to translate this shape 4 spaces to the left, and 1 space up.

One way to do this is by moving the corners one-by-one. If you shift each corner 4 spaces left and 1 space up, all that remains is to join up your new set of corners, and you get the translated shape. The resulting shape is shown on the right (red).

The next type of transformation is **rotation**. To rotate a shape, you need a point that you can rotate about, the direction you’re rotating (clockwise/anti-clockwise), and how much you’re rotating (90, 180, or 270 degrees). Let’s see an example.

**Example: **Rotate shape A anti-clockwise 90\degree about (1, 1).

You are allowed to use **tracing paper** when answering these questions, and it is helpful to do so.

Firstly, mark the point of rotation onto the axes (red). Now, anti-clockwise means we are going to rotate in the opposite direction to the hands of a clock. Furthermore, a 90\degree rotation means a **quarter turn**, i.e. by a quarter of a complete 360\degree rotation. To do this on tracing paper, trace over shape A, and place your pencil on the point of rotation. Then, keep your pencil fixed, twist the paper one quarter turn anti-clockwise. The place where your traced shape ends up is the result of the rotation. The resulting shape is shown above (orange).

You may feel comfortable without tracing paper, which is great, but if you aren’t, don’t worry – you can always ask for it in an exam.

The final type of transformation is **reflection**. To reflect a shape, all you need is a mirror line. You aren’t allowed a mirror in an exam, but you are again allowed tracing paper, and that can help.

**Example: **Reflect shape A in the line y=0.

Firstly, recognise that the line y=0 is the x axis, and mark this on the axes (red).

Then, to do this transformation with tracing paper, we need to trace both the shape A and the mirror line y=0 onto the tracing paper. Then, flip the paper over, and perfectly line up the mirror line on your tracing paper with the mirror line on the page, such that the shape you drew on the tracing paper is on the other side of the line to the one on the page.

That new shape (green) is precisely the result of doing the reflection. Draw it on the page, and you should get the picture on the right.

A way to check your answer – as well as a method for doing this without tracing paper – is to check that each corner of the shape is the same distance from the mirror line as the corresponding corner on the other shape. For example, the bottom left corner of A is one above the mirror line, and indeed one corner of the reflected shape is one below the mirror line, and so on.

You may also be asked to look at a transformed shape and describe what was done to it. In answering these questions, it’s important that you give all the answers in a clear and proper way, without missing out any information – just like how they are given to you in a question. In general, the proper way of describing each transformation would be:

**Translation by the vector**\begin{pmatrix}x\text{ direction shift}\\y\text{ direction shift}\end{pmatrix},**Rotation by**[90, 180, or 270 degrees] [clockwise or anti-clockwise]**about the point**[coordinates of rotation point],**Reflection in the line**[equation of mirror line].

If you are on the **higher course **there are a few extra things to consider.

Firstly, you may be asked to do multiple transformations to one shape. This doesn’t require new techniques, all it needs is for you to: do one transformation, draw the new shape, then apply the next transformation to that new shape.

Secondly, you’re expected to understand the concept of **invariance**. If something is **invariant**, then that means it doesn’t change. In terms of transformations, an **invariant point** is any point on the shape that **hasn’t moved after the transformation** has been done.

For example, when we rotated shape A earlier, the bottom right corner of it didn’t move. As a result, the bottom right corner is an **invariant point**.

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### Example Questions

1) Reflect shape B in the line y=1. Mark the resulting shape with a C.

Firstly, we must draw the line y=1 onto the graph. Then, you can either choose to use tracing paper or, if you’re confident without it, just go right into the reflection.

If you’re using tracing paper, you should firstly trace over the shape and the mirror line. Then, flip over the tracing paper, and line up perfectly the mirror line on the page with the one on the tracing paper such that the trace of the shape is on the opposite side of the line to the original shape.

Then, the trace of the shape is the result of the reflection. Draw that shape onto the original axes, mark it with a C and you should get the resulting picture below.

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2) Describe fully the transformation that takes shape D onto shape E.

Firstly, the two shapes look the same and have the same orientation, so it wouldn’t make much sense for them to have been rotated or reflected.

Indeed, E is just the result of shifting D upwards and to the right. We must pick a corner and see how far it has moved. Looking at the bottom right corners of each shape, we can see that it has been shifted 6 spaces to the right and 3 spaces up, so the full description of the transformation is:

Translation by the vector \begin{pmatrix}6\\3\end{pmatrix}

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3) (HIGHER ONLY) a) In this order, performing the following two transformations to shape F.

- Rotation 180\degree clockwise* about (1, 2),
- Reflection in the line y=x

Mark the resulting shape with a G.

b) State the number of invariant points.

*when it’s a rotation, it doesn’t actually matter where it’s clockwise or anti-clockwise, the result is the same.

Firstly, mark the point of rotation on the axes (here, it is a red dot). Then, rotate the shape 180\degree. If you’re using tracing paper, trace the shape onto the tracing paper and place your pencil onto the rotation point. Then, twist the paper one half-turn, and where the traced shape has moved is the result of your rotation. The result of this first transformation is shown below.

Now, we need to apply the second transformation to the result of the first one (here, the dashed grey shape).

So, we will start by drawing on the mirror line y=x (orange). Then, if you’re using tracing paper, trace both the mirror line and the shape onto the tracing paper. After that, flip the tracing paper over, and line up the mirror line on the tracing paper perfectly with the one on the paper. The location of the traced shape is the result of the reflection.

You can check it is correct by seeing if the corners of each of the shapes are the same distance from the reflection line. If you’re confident, then mark the shape G. The result is shown below.

b) None of the points on F remain in the same place after being transformed onto G, so the number of invariant points is zero.

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