Coordinate transformations review | Year 8

01 / LEARN

Understand the idea.

Move a point without changing its shape

An ordered pair $(x, y)$ gives horizontal position first and vertical position second. For a translation, change each coordinate by the stated movement: right and up add; left and down subtract. Begin with the point's actual coordinates, including any negative signs.

When translating a whole shape, use the same coordinate changes for every vertex. Check one point by counting the movement on a grid, then apply that rule consistently. Equal movements preserve the shape's lengths, angles and orientation.

(x, y) \mapsto (x + a, y + b)

An equal-scale unit coordinate grid shows A at (1,2) in sage and A prime at (-2,1) in gold. The grid spans -4 to 4 on both axes; the labels state the given coordinates of each point.

Reflect or rotate about a stated reference

A reflection needs a mirror line. In the $y$ -axis, use $(x, y) \mapsto (- x, y)$ because left and right reverse. In the $x$ -axis, use $(x, y) \mapsto (x, - y)$ because above and below reverse. Points on the mirror line remain fixed.

A quarter-turn also needs a centre and direction. About the origin, $90^{\circ}$ anticlockwise uses $(x, y) \mapsto (- y, x)$ , while clockwise uses $(x, y) \mapsto (y, - x)$ . For successive transformations, apply the second rule to the first image.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

A translation

Translate A(1,2) three units left and one unit down.

Worked solution

1
Write the horizontal change.
$x^{'} = 1 - 3 = - 2$
Three units left decreases the horizontal coordinate by 3.
2
Write the vertical change.
$y^{'} = 2 - 1 = 1$
One unit down decreases the vertical coordinate by 1.
3
Combine the image coordinates in order.
$A^{'} = (- 2, 1)$
The first entry gives horizontal position and the second gives vertical position.

Answer

A′(−2,1)

EXAMPLE 02

A reflection followed by a turn

Reflect P(−3,2) in the y-axis, then rotate the image 90° anticlockwise about the origin.

Worked solution

1
Reflect the point in the y-axis.
$(- 3, 2) \mapsto (3, 2)$
Reflection in the vertical axis reverses x and leaves y unchanged.
2
Write the anticlockwise quarter-turn rule.
$(x, y) \mapsto (- y, x)$
This rule applies to a 90-degree anticlockwise rotation about the origin.
3
Rotate the reflected image.
$(3, 2) \mapsto (- 2, 3)$
The second transformation starts from the intermediate point, giving the final image $(- 2, 3)$ .

Answer

Final image (−2,3).

03 / YOUR TURN

Try it yourself.

Check your understanding

Reflect (4,−1) in the x-axis.

Show answer & reasoning

(4,1)

The $x$ -axis is the horizontal mirror. Keep $x = 4$ and reverse the sign of $y = - 1$ , giving $(4, 1)$ .

Keep practising

6 worksheet questions · 12 marks

Open worksheet Worked solutions

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 $(x, y) \to (- y, x)$ .
$(- 1, 2)$
$A (- 3, 1)$ 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.

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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 ↗

In this lesson

Understand the idea.

Move a point without changing its shape

Reflect or rotate about a stated reference

See the method in action.

A translation

Write the horizontal change.

Write the vertical change.

Combine the image coordinates in order.

A reflection followed by a turn

Reflect the point in the y-axis.

Write the anticlockwise quarter-turn rule.

Rotate the reflected image.

Try it yourself.

Check your understanding

Keep practising