Recording transformations with coordinates | Year 7

01 / LEARN

Understand the idea.

Match each original point with its image

A coordinate table pairs each original vertex with its matching image. Follow the labels: $A$ maps to $A^{'}$ even if another image point is closer. A mapping arrow records movement rather than claiming the coordinates are equal.

To test a translation, subtract original coordinates from image coordinates. Each pair must give the same horizontal and vertical changes. Check negative coordinates carefully: subtracting a negative can increase the difference.

Choose a rule, then test it

To recognise an axis reflection, look for one coordinate staying fixed while the other changes sign. To test a rotation, use the stated centre and turning rule. Test every available pair, since one pair can fit several transformations.

A complete description includes the movement for a translation, the mirror line for a reflection, or the centre, angle and direction for a rotation. If one vertex fails your proposed rule, check the data before describing the whole shape.

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

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

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.

Worked solution

1
Find the change for A.
$\begin{aligned} Δ x & = - 1 - (- 4) = 3 \\ Δ y & = 0 - (- 2) = 2 \end{aligned}$
Subtract the starting coordinates from the finishing coordinates.
2
Test the same change at B.
$(1 - (- 2), 0 - (- 2)) = (3, 2)$
A translation must move B by the same amounts as A.
3
Test C and describe the movement.
$(- 1 - (- 4), 3 - 1) = (3, 2)$
Every recorded vertex moves three units right and two units up.

Answer

A translation 3 units right and 2 units up.

EXAMPLE 02

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?

Worked solution

1
Compare the coordinates of P.
$(2, 1) \mapsto (- 2, 1)$
The horizontal coordinate reverses sign and the height stays one.
2
Check Q against the same pattern.
$(4, 3) \mapsto (- 4, 3)$
The second pair also reverses only the horizontal coordinate.
3
Identify the mirror rule.
$(x, y) \mapsto (- x, y)$
Keeping height and reversing left and right is reflection in the y-axis.

Answer

A reflection in the y-axis.

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

3 units right and 2 units down.

The changes are $(4 - 1, 0 - 2) = (3, - 2)$ and $(1 - (- 2), - 1 - 1) = (3, - 2)$ . Both points move three units right and two units down.

Keep practising

6 worksheet questions · 12 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

Match each original point with its image

Choose a rule, then test it

See the method in action.

Recognise a translation from a table

Find the change for A.

Test the same change at B.

Test C and describe the movement.

Recognise a reflection

Compare the coordinates of P.

Check Q against the same pattern.

Identify the mirror rule.

Try it yourself.

Check your understanding

Keep practising