Sequences of transformations | Year 7

01 / LEARN

The idea, step by step

The second step acts on the first image

In a sequence, transform the original to make the first image, then apply the next transformation to that image. Naming intermediate points P′ and P″ prevents you from accidentally starting both steps at P.

Write the instructions in order before calculating. The instruction “reflect, then translate” is different from “translate, then reflect”. The same two ingredients need not give the same final image.

Some sequences simplify

Two translations can be combined by adding their horizontal movements and their vertical movements. Two reflections in the same axis return every point to its starting position. Two quarter-turns in the same direction about the same centre make a half-turn.

These simplifications depend on the exact instructions. Rotations about different centres cannot be combined merely by adding angles. Always check the centre, mirror line and direction.

Keep an intermediate record
Stage	Coordinates of P
Start	(2,1)
Reflect in y-axis	(−2,1)
Move 3 right and 2 down	(1,−1)

A common trap

Applying both instructions to the original coordinates.

Instead: The second instruction starts at the first image. Keep an intermediate coordinate pair.

02 / WORKED EXAMPLES

See the thinking

Example 1

Reflect, then translate

Reflect $P (2, 1)$ in the y-axis, then move the image 3 right and 2 down.

The reflection gives $P^{'} (- 2, 1)$ .
Translate that image: $(- 2 + 3, 1 - 2) = (1, - 1)$ .
Do not apply the translation to the original point.

P^{''} (1, - 1)

Example 2

Reverse the order

Start again at $P (2, 1)$ . Move 3 right and 2 down first, then reflect in the y-axis.

The translation gives $(5, - 1)$ .
Reflecting in the y-axis changes the sign of x, giving $(- 5, - 1)$ .
This differs from $(1, - 1)$ , so these two orders give different results.

(- 5, - 1)

; changing the order changes this result.

Check your understanding

Move $(1, - 2)$ 2 left and 3 up, then move it 5 right and 1 down. What is the final point?

Show answer

$(4, 0)$

The combined move is 3 right and 2 up: $(1 + 3, - 2 + 2) = (4, 0)$ .

03 / WORKSHEET

Your turn

6 questions · 14 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Reflect $(3, 2)$ in the x-axis, then move 1 left and 4 up.
2 marks
Move $(- 2, 1)$ 3 right and 2 down, then reflect in the y-axis.
2 marks
Reflect $(4, - 1)$ in the x-axis and then in the y-axis. Give the final point and one equivalent rotation.
3 marks
Combine a move of 5 left and 2 up with a move of 3 right and 4 down.
2 marks
A point is reflected in the y-axis twice. Explain the final result.
2 marks
Start at $(1, 2)$ . Compare “move 2 up, then reflect in the x-axis” with “reflect in the x-axis, then move 2 up”.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

Reflect $(3, 2)$ in the x-axis, then move 1 left and 4 up.
2 marks
Reflect to $(3, - 2)$ , then translate to $(2, 2)$ .
$(2, 2)$
Move $(- 2, 1)$ 3 right and 2 down, then reflect in the y-axis.
2 marks
The first image is $(1, - 1)$ ; the reflection gives $(- 1, - 1)$ .
$(- 1, - 1)$
Reflect $(4, - 1)$ in the x-axis and then in the y-axis. Give the final point and one equivalent rotation.
3 marks
After x-axis reflection: $(4, 1)$ .
After y-axis reflection: $(- 4, 1)$ .
$(- 4, 1)$ ; a 180° turn about the origin.
Combine a move of 5 left and 2 up with a move of 3 right and 4 down.
2 marks
Horizontal change: $- 5 + 3 = - 2$ .
Vertical change: $2 - 4 = - 2$ .
2 units left and 2 units down.
A point is reflected in the y-axis twice. Explain the final result.
2 marks
The first reflection reverses x; the second reverses that sign again.
The y-coordinate never changes.
It returns to its original position.
Start at $(1, 2)$ . Compare “move 2 up, then reflect in the x-axis” with “reflect in the x-axis, then move 2 up”.
3 marks
First order: $(1, 2) \to (1, 4) \to (1, - 4)$ .
Reverse order: $(1, 2) \to (1, - 2) \to (1, 0)$ .
The final points differ.
First order: $(1, - 4)$ . Reverse order: $(1, 0)$ .

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Curriculum reference & lesson scope

Australian Curriculum v9 / South Australian Curriculum

Translations; reflections in coordinate axes; rotations about a point; recording transformations with coordinates; sequences of transformations; explaining what stays unchanged.

Official curriculum source ↗

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