Sequences of transformations | Year 7

01 / LEARN

Understand the idea.

The second step acts on the first image

In a sequence of transformations, the second instruction acts on the first image. Record the intermediate point as $P^{'}$ and the final point as $P^{''}$ . This makes the order visible and prevents you from starting over halfway through.

Read words such as “then” carefully. A reflection followed by a translation can finish somewhere different from the same translation followed by the reflection. Calculate each stage before deciding whether the orders give the same result.

Some sequences simplify

Two translations combine by adding horizontal changes and vertical changes separately. Two reflections in the same mirror line undo each other. Two quarter-turns in the same direction about the same centre combine into a half-turn.

These shortcuts depend on the details. Adding rotation angles is valid here only when the centre is unchanged. Keep the intermediate record when the transformations differ, then compare the final image with any shortcut you propose.

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)

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Reflect, then translate

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

Worked solution

1
Apply the reflection first.
$P (2, 1) \mapsto P^{'} (- 2, 1)$
Reflection in the y-axis reverses the horizontal coordinate.
2
Translate the reflected image.
$P^{''} = (- 2 + 3, 1 - 2) = (1, - 1)$
The translation starts at the reflected coordinates, not the original point.
3
Check the final displacement from the intermediate point.
$(1 - (- 2), - 1 - 1) = (3, - 2)$
The second stage really moved three units right and two units down.

Answer

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

EXAMPLE 02

Reverse the order

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

Worked solution

1
Translate the original point first.
$P^{'} = (2 + 3, 1 - 2) = (5, - 1)$
This order starts with the slide, so the intermediate coordinates differ.
2
Reflect this new image.
$P^{''} = (- 5, - 1)$
The y-axis reflection reverses five to negative five and keeps the height.
3
Compare the two finishing points.
$(- 5, - 1) \neq (1, - 1)$
For these instructions, reversing the order changes the horizontal result.

Answer

$(- 5, - 1)$ ; changing the order changes this result.

03 / YOUR TURN

Try it yourself.

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 & reasoning

$(4, 0)$

The first slide gives $(1 - 2, - 2 + 3) = (- 1, 1)$ . The second gives $(- 1 + 5, 1 - 1) = (4, 0)$ ; equivalently, the combined slide is three right and two up.

Keep practising

6 worksheet questions · 14 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

The second step acts on the first image

Some sequences simplify

See the method in action.

Reflect, then translate

Apply the reflection first.

Translate the reflected image.

Check the final displacement from the intermediate point.

Reverse the order

Translate the original point first.

Reflect this new image.

Compare the two finishing points.

Try it yourself.

Check your understanding

Keep practising