Year 9 · Unit 6 Transformations · Lesson 5 of 6

Does the order of two transformations matter?

One transformation can follow another. The order can change the result, each type needs its own details to be described fully, and only the first three keep size and shape.

Geometry and TrigonometryNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. From Lesson 4. Enlarge (3, −2) by scale factor 4 from the origin.
    Answer(12, −8). Both coordinates are multiplied by 4.
  2. From Lesson 3. Turn (1, 4) through 90° clockwise about the origin.
    Answer(4, −1). (x, y) goes to (y, −x).
  3. From Lesson 1. Translate (−2, 3) by the vector (5, −7).
    Answer(3, −4). −2 + 5 = 3 and 3 + (−7) = −4.
Words used in this lesson
combined transformation
two or more transformations one after another. For example: reflect, then translate.
order
which move comes first. For example: the order matters.
congruent
exactly the same size and shape. For example: a shape and its translation.
similar
the same shape, possibly a different size. For example: a shape and its enlargement.
describe fully
give every detail needed to repeat the move. For example: reflection in the line y = x.

The tile printer's order of work

In a workshop in Lisbon, a printer stamps one triangular motif, A (−2, 1), B (1, 1) and C (−2, 3), on a tile grid. For the next tile she mirrors the motif in the vertical centre line (the y-axis) and then slides it 2−2, that is, 2 right and 2 down. Her apprentice does the slide first and the mirror second.

Predict first: will they stamp in the same place? Work out where corner A ends up each way for this slide, then use the figure with a different slide.

3
(5, −1)A″ in this order
(−1, −1)A″ in the other order
differentthe two orders give

Reflect in the y-axis, then translate by (3, −2): A″ is (5, −1). The other order gives (−1, −1), a different place.

Show me the steps
  1. Reflect first, then translate by (3, −2): A (−2, 1) goes to (2, 1) and then to (5, −1).
  2. Translate first, then reflect: A goes to (1, −1) and then to (−1, −1). That is not where the other order put it.
  3. Translation (0, −2): both orders give A″ (2, −1), because the translation has no sideways move for the mirror to disturb.
  4. Translation (−3, −2): reflect then translate gives (−1, −1) and translate then reflect gives (5, −1). The values have swapped from the first state.

To combine transformations, do the first, then apply the second to the image. The order can change the result: reflecting and then translating is not the same as translating and then reflecting.

To describe a transformation fully: a translation needs its vector; a reflection its mirror line; a rotation its centre, angle and direction; an enlargement its centre and scale factor.

Translations, reflections and rotations keep lengths and angles (congruence transformations). An enlargement keeps angles and multiplies lengths (similarity transformation).

Naming what you see

Compare the pre-image with each image: what moved, and what did not change?

translationtype of move
3 and 2long and short side of the image
congruentthe two triangles are

Translation, vector (3, −4): the sides are still 3 and 2 and the right angle is 90°, so the image is congruent.

Show me the steps
  1. Translation, vector (3, −4): sides 3 and 2 stay 3 and 2, so the image is congruent.
  2. Reflection in the line y = x: the sides are still 3 and 2 (the long side now runs up and down), so it is congruent.
  3. Rotation 90° clockwise about the origin: the sides are still 3 and 2, so it is congruent.
  4. Enlargement, scale factor 2, centre the origin: the sides become 6 and 4. The right angle is unchanged, but the sizes are not, so it is similar and not congruent.

Always apply the second move to the image of the first. A common slip is to start the second move from the original point. Write the point after the first move, and use it. Also, a description such as “a reflection” or “a rotation” is not complete: name the mirror line, or the centre, angle and direction.

Worked examples

Reflect P (3, 2) in the x-axis, then translate by −41.

  1. Reflect in the x-axis: (3, 2) → (3, −2).
  2. Translate the image: (3 + (−4), −2 + 1) = (−1, −1).

The same two moves in the other order. Translate first: (3, 2) → (−1, 3). Then reflect in the x-axis: (−1, −3). That is not (−1, −1), so the order mattered.

The triangle (1, 1), (3, 1), (1, 2) goes to (−1, 1), (−3, 1), (−1, 2). Describe it fully.

  1. Every y-coordinate is the same and every x-coordinate has changed sign.
  2. Lengths are unchanged, so it is a congruence transformation: a reflection in the y-axis.

The points (2, 1) and (3, 4) go to (−1, 2) and (−4, 3). Describe it fully. Test the quarter-turn rule (x, y) → (−y, x): (2, 1) goes to (−1, 2) and (3, 4) goes to (−4, 3). Both match, so it is a rotation of 90° anticlockwise about the origin. One point alone is not enough: a translation by −31 also takes (2, 1) to (−1, 2), but not (3, 4) to (−4, 3).

Your go. The first step is given. Turn U (4, −1) through 180° about the origin, then reflect the image in the y-axis.

  1. The half turn: (4, −1) → (−4, 1).
  2. Now reflect (−4, 1) in the y-axis, and write the final image.
Check the answer

The y-axis changes the sign of x: (−4, 1) becomes (4, 1).

Check before you go on

1. The point (2, 3) is reflected in the x-axis, and the image is then translated by (1, −2). Where does it end up?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. The point (1, 5) is translated by (4, −3), and the image is then reflected in the y-axis. The final image is (x, y). What is x?

3. The point (3, 1) is translated by (−5, 2), and the image is then turned 180° about the origin. What is the y-coordinate of the final image?

Level 2 · Building

4. A transformation sends (5, 2) to (2, −5) and (1, 3) to (3, −1). Which one is it?

5. The point (2, 1) is enlarged by scale factor 3 from the origin, and the image is then translated by (−4, 5). What is the x-coordinate of the final image?

6. Triangle PQR with P (1, 1), Q (2, 1) and R (1, 3) goes to P′ (−1, 1), Q′ (−2, 1) and R′ (−1, 3). Which is a complete description of the transformation?

Level 3 · Stretch

7. A triangle of area 5 cm² is enlarged by scale factor 2, then reflected in the x-axis, then translated. What is the area of the final image, in cm²?

8. The point (a, 3) is reflected in the line y = x, and the image is then translated by (2, −1). The final image is (5, 0). What is a?

Something to think about

Two translations always give the same result whichever you do first. Why is this not true for a reflection followed by a translation in the first figure? When is it still true?

One way to think about it

A translation adds the same numbers to every point, and adding is the same in either order. A reflection changes a coordinate's sign, so adding before or after the sign change gives different results: −x + a and −(x + a) are different. It is still true when a = 0, as the third step shows.

Exit ticket

9. The point (−3, 2) is turned 90° clockwise about the origin, and the image is then reflected in the line y = x. Where does it end up?