One after another
Combining transformations just means doing them in sequence. Carry out the first transformation to get an image, then treat that image as the starting shape for the second, and so on. Take it one step at a time and label each stage clearly, for example the original, then image 1, then image 2.
- Combined transformation
- The overall effect of carrying out two or more transformations one after another. Very often a single transformation produces exactly the same final image.
Order matters
In general, swapping the order of two transformations changes where the shape ends up. Reflecting then translating is usually not the same as translating then reflecting. So always follow the order the question gives, and never merge the steps in your head.
Finish one before starting the next
Complete the first transformation fully and draw its image before you begin the second. Trying to do both at once is where most mistakes creep in.
A worked example
A shape is reflected in the \(x\)-axis, and then that image is reflected in the \(y\)-axis. Describe the single transformation that has the same overall effect. Test it on the point \((2,3)\).
Naming the single move
To describe the single equivalent transformation, compare the final image with the original and ask which one move links them. Some combinations have well known results:
| Combination | Single equivalent |
|---|---|
| Two translations | one translation (add the vectors) |
| Two reflections in parallel lines | a translation |
| Two reflections in lines that cross | a rotation about the crossing point |
Where this is assessed
Carrying out each step accurately is Criterion A (Knowing and understanding). Describing the single equivalent transformation with every detail, the type plus its vector, angle, centre or mirror line, is Criterion C (Communicating). Working through multi-step design or animation problems reaches Criterion D (Applying maths in context).
Check yourself
1. A shape is translated by the vector (2,0) and then by (0,3). What single translation is equivalent? +
Add the two vectors: \(\begin{pmatrix} 2 \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 3 \end{pmatrix} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}\). The single translation is by \(\mathbf{\begin{pmatrix} 2 \\ 3 \end{pmatrix}}\).
2. A shape is rotated 90° anticlockwise about the origin, then rotated another 90° anticlockwise about the origin. What single transformation is that? +
The two quarter turns add to a half turn, so it is a rotation of \(180^\circ\) about the origin.
3. Does the order of two transformations usually matter? +
Yes. In general, changing the order changes the final image, so you must follow the order the question sets out. A few special pairs happen to give the same result either way, but you should not assume it.
Part of the Standard Maths library. Spotted an error or want a topic added? That feedback makes the notes better.