Combined transformations: worked examples and method

Order matters

In reflect then translate, you reflect first. The second transformation acts on the image of the first one.

Track a point

Pick a point and apply the first transformation. Write the result, then apply the second transformation to that result.

Coordinate rules

About the origin: x-axis (x, -y); y-axis (-x, y); y = x swaps to (y, x); 90 degrees anticlockwise (-y, x); 180 degrees (-x, -y). A translation adds the vector.

Parallel mirrors

Two reflections in parallel lines make a translation. It is twice the distance between the lines, at right angles to them.

Crossing mirrors

Two reflections in lines that cross make a rotation about the crossing point. The angle turned is twice the angle between the lines.

Describe fully

A rotation needs a centre, an angle and a direction. A reflection needs the mirror line. A translation needs a vector. An enlargement needs a centre and a scale factor.

Tracking a point through two steps

Coordinate rules make this fast. About the origin, reflection in the x-axis sends (x, y) to (x, -y), reflection in the y-axis sends it to (-x, y), and reflection in the line y = x swaps the coordinates to (y, x). A rotation of 90^° anticlockwise sends (x, y) to (-y, x), and a rotation of 180^° sends it to (-x, -y). A translation by the vector beginpmatrix p \ q endpmatrix adds p to x and q to y. An enlargement of scale factor k about the origin multiplies both coordinates by k.

To find where a point ends up, apply the rules one after the other. A point (4, 1) reflected in the y-axis becomes (-4, 1). A translation by beginpmatrix 2 \ 3 endpmatrix then moves it to (-2, 4). Do this for two or three corners of the shape, or two or three points of your choice. If the transformation is a single one, every point you test will fit the same pattern, and the pattern tells you what it is.

Method: 1. Pick one or two simple points, such as (2, 1) or (3, 0). 2. Apply the first rule, write the result, then apply the second rule to that result. 3. Compare the start and finish points to spot the pattern.

Patterns that name the single transformation

A few combinations have a quick answer. Two translations are one translation whose vector is the sum of the two vectors. Two enlargements about the same centre are one enlargement whose scale factor is the product of the scale factors. Two rotations about the same centre are one rotation through the total angle. Reflections give the most interesting pattern. Two reflections in parallel mirror lines are a translation, at right angles to the lines, by twice the distance between them. Two reflections in mirror lines that cross are a rotation about the crossing point, through twice the angle between the lines.

For example, reflect in the line x = 1 and then in the line x = 4. The lines are 3 units apart, so the result is a translation of 6 units, and it moves the shape from the first mirror line towards the second, which is to the right. To be sure, track the point (0, 2). Reflecting in x = 1 gives (2, 2) and reflecting in x = 4 gives (6, 2). The point has moved 6 to the right, so the single transformation is the translation beginpmatrix 6 \ 0 endpmatrix.

Method: 1. Look at the two transformations and decide whether a pattern applies. 2. Use the pattern to name the single transformation and its size. 3. Check it by tracking one point through both steps and through your single transformation.

Describing the single transformation fully

A full description names the type and gives every detail it needs. A reflection needs the equation of the mirror line, such as y = x or x = 2. A rotation needs the centre, the angle and the direction, clockwise or anticlockwise, although a half turn does not need a direction. A translation needs the vector. An enlargement needs the centre of enlargement and the scale factor. If any detail is missing, the answer is incomplete and loses marks, even when the type is right.

When the pattern is not obvious, find the fixed point. A point that does not move under the combination is the centre of a rotation or an enlargement, or it lies on the mirror line of a reflection. For a half turn followed by a translation, the combination is a half turn about a new centre. The half turn sends (x, y) to (-x, -y) and the translation by beginpmatrix 4 \ -2 endpmatrix then gives (4 - x, -2 - y). A half turn about the point (p, q) gives (2p - x, 2q - y). Matching the two gives p = 2 and q = -1, so the single transformation is a half turn about (2, -1).

Method: 1. Track two points through both transformations to see what has happened. 2. Name the type of the single transformation and give every detail it needs. 3. Check your description by applying it to one of the points you tracked.

Tip: One transformation means one name. If you write the word “and” in your answer you have probably described two.