The idea, step by step
Move a point without changing its shape
An ordered pair records horizontal position first, then vertical position. A translation adds the same horizontal change to every x-coordinate and the same vertical change to every y-coordinate. Moving left or down uses a negative change. The original point and its image should have exactly the stated separation.
Draw one point first and count its movement. Apply the same rule to the other vertices of a shape; changing the rule halfway through would stretch or turn it instead of translating it.
Reflect or rotate about a stated reference
Reflection in the y-axis changes the sign of x and leaves y alone. Reflection in the x-axis changes y and leaves x alone. A point on the mirror line stays in place. The word axis matters: reflecting in the y-axis does not mean changing y.
For a 90° anticlockwise turn about the origin, the image of (x,y) is (−y,x). For a clockwise quarter-turn, it is (y,−x). Check one familiar point such as (1,0) before using a rule: anticlockwise sends it to (0,1). The centre must be specified.
A common trap
A reflection in the y-axis changes the sign of y.
Instead: The y-axis is the vertical mirror. Horizontal distances reverse, so x changes sign and y stays the same.
See the thinking
A translation
Translate A(1,2) three units left and one unit down.
- Subtract 3 from the x-coordinate: 1−3=−2.
- Subtract 1 from the y-coordinate: 2−1=1.
- The image has moved left and down, matching the instruction.
A reflection followed by a turn
Reflect P(−3,2) in the y-axis, then rotate the image 90° anticlockwise about the origin.
- The reflection changes −3 to 3, giving (3,2).
- For the quarter-turn, (x,y) becomes (−y,x).
- Apply that to (3,2), not to the original point.
Check your understanding
Reflect (4,−1) in the x-axis.
Show answer
(4,1)
The horizontal coordinate stays 4; the vertical coordinate changes from −1 to 1.
Your turn
6 questions · 12 marks · Work at your own pace.
Read the questions online
- Translate (−2,3) four units right and two units down.2 marks
- Reflect (3,−2) in the y-axis.2 marks
- Reflect (−1,4) in the x-axis.2 marks
- Rotate (2,1) 90° anticlockwise about the origin.2 marks
- A(−3,1) moves to A′(1,−2). State the translation.2 marks
- B(0,−3) is reflected in the y-axis. Explain why it does not move.2 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- Translate (−2,3) four units right and two units down.2 marks
−2+4=2 and 3−2=1.
(2,1) - Reflect (3,−2) in the y-axis.2 marks
Change x to its opposite; leave y unchanged.
(−3,−2) - Reflect (−1,4) in the x-axis.2 marks
Leave x unchanged; change the sign of y.
(−1,−4) - Rotate (2,1) 90° anticlockwise about the origin.2 marks
Use (x,y)→(−y,x).
(−1,2) - A(−3,1) moves to A′(1,−2). State the translation.2 marks
1−(−3)=4; −2−1=−3.
4 units right and 3 units down. - B(0,−3) is reflected in the y-axis. Explain why it does not move.2 marks
Reflection changes x to −x; −0=0.
Its image is (0,−3), because it lies on the mirror line.
Curriculum reference & lesson scope
Australian Curriculum v9 and SA Curriculum Mathematics
Coordinate transformations review; identifying positions using three coordinates; interpreting 3D axes; describing routes and relative positions; comparing diagrams, views and digital representations. This is spatial location, not senior vector algebra.
Official curriculum source ↗