Recording transformations with coordinates | Year 7

01 / LEARN

The idea, step by step

Match each original point with its image

A coordinate table records the original and image beside one another. Labels matter: A corresponds to A′, not to whichever image point looks closest. An arrow shows a mapping, not an equation between the two coordinate pairs.

For a translation, subtract each original coordinate from its image coordinate. Every vertex must have the same horizontal change and the same vertical change. For reflections and rotations, check the correct mirror line or centre instead of searching for one translation distance.

Choose a rule, then test it

A single pair of points may fit more than one transformation. More matching points help you test which description fits the whole shape. State all the information: a reflection needs its mirror line, and a rotation needs its centre, angle and direction.

Use a sketch as a check. A correct coordinate rule must preserve the shape for translations, reflections and rotations. If one point does not fit, reconsider the proposed rule or check for a recording error.

A translation recorded for three vertices
Original	Image	Change
A(−4,−2)	A′(−1,0)	3 right, 2 up
B(−2,−2)	B′(1,0)	3 right, 2 up
C(−4,1)	C′(−1,3)	3 right, 2 up

A common trap

Deciding from one vertex and not checking the rest.

Instead: Test every corresponding pair. Different transformations can share one original–image pair.

02 / WORKED EXAMPLES

See the thinking

Example 1

Recognise a translation from a table

A maps from $(- 4, - 2)$ to $(- 1, 0)$ , B from $(- 2, - 2)$ to $(1, 0)$ , and C from $(- 4, 1)$ to $(- 1, 3)$ . Describe the transformation.

For A, the change is 3 right and 2 up.
B has the same change: $1 - (- 2) = 3$ and $0 - (- 2) = 2$ .
C also has change $(3, 2)$ .

A translation 3 units right and 2 units up.

Example 2

Recognise a reflection

The table maps $P (2, 1)$ to $P^{'} (- 2, 1)$ and $Q (4, 3)$ to $Q^{'} (- 4, 3)$ . Which coordinate-axis reflection fits?

For both points, x changes sign and y stays the same.
The points keep their heights and swap left and right.
This is the y-axis rule, $(x, y) \mapsto (- x, y)$ .

A reflection in the y-axis.

Check your understanding

A table maps $(1, 2)$ to $(4, 0)$ and $(- 2, 1)$ to $(1, - 1)$ . What is the translation?

Show answer

3 units right and 2 units down.

Both matching pairs have change $(3, - 2)$ .

03 / WORKSHEET

Your turn

6 questions · 12 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Complete the images of $A (- 1, 2)$ and $B (2, - 2)$ under $(x, y) \mapsto (x + 2, y - 1)$ .
2 marks
$P (- 3, 1)$ maps to $(1, - 2)$ and $Q (- 1, 4)$ maps to $(3, 1)$ . Find the common translation.
2 marks
A translation maps a point to $(2, 5)$ by moving 1 left and 3 up. Find the original.
2 marks
Which coordinate-axis reflection maps $(3, - 2)$ to $(- 3, - 2)$ and $(- 1, 4)$ to $(1, 4)$ ?
2 marks
Use $(x, y) \mapsto (- y, x)$ to find the image of $(4, 1)$ . Name the turn.
2 marks
One proposed translation maps $A (0, 1)$ to $(2, 4)$ and $B (2, 1)$ to $(4, 5)$ . Explain why the record contains a mistake.
2 marks

Read the worked solutions

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

Complete the images of $A (- 1, 2)$ and $B (2, - 2)$ under $(x, y) \mapsto (x + 2, y - 1)$ .
2 marks
Add 2 to x and subtract 1 from y for each point.
$A^{'} (1, 1), B^{'} (4, - 3)$
$P (- 3, 1)$ maps to $(1, - 2)$ and $Q (- 1, 4)$ maps to $(3, 1)$ . Find the common translation.
2 marks
Subtract original coordinates from image coordinates for both pairs.
4 units right and 3 units down.
A translation maps a point to $(2, 5)$ by moving 1 left and 3 up. Find the original.
2 marks
Undo 1 left by adding 1 to x; undo 3 up by subtracting 3 from y.
$(3, 2)$
Which coordinate-axis reflection maps $(3, - 2)$ to $(- 3, - 2)$ and $(- 1, 4)$ to $(1, 4)$ ?
2 marks
Only the x-coordinates change sign for both pairs.
Reflection in the y-axis.
Use $(x, y) \mapsto (- y, x)$ to find the image of $(4, 1)$ . Name the turn.
2 marks
Substitute x=4 and y=1 into the mapping.
$(- 1, 4)$ ; 90° anticlockwise about the origin.
One proposed translation maps $A (0, 1)$ to $(2, 4)$ and $B (2, 1)$ to $(4, 5)$ . Explain why the record contains a mistake.
2 marks
A single translation must give the same coordinate change to both vertices.
A changes by $(2, 3)$ , but B changes by $(2, 4)$ .

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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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