Reflections in coordinate axes | Year 7

01 / LEARN

The idea, step by step

The axis is a mirror line

A reflection places each image point on the opposite side of a mirror line at the same perpendicular distance. The segment joining a point to its image meets the mirror line at a right angle.

Reflecting in the x-axis keeps the horizontal position but swaps above and below. Reflecting in the y-axis keeps the vertical position but swaps left and right. A point already on the mirror line does not move.

Coordinate rules and checks

For an x-axis reflection, keep x and change the sign of y. For a y-axis reflection, change the sign of x and keep y. Zero does not change when its sign is reversed.

Check a shape by reflecting each labelled vertex. A reflection reverses the clockwise or anticlockwise order of the vertices, but keeps side lengths, angles, perimeter and area.

x-axis: (x, y) \mapsto (x, - y), y-axis: (x, y) \mapsto (- x, y)

A common trap

Changing both signs for an x-axis reflection.

Instead: The x-axis reflection changes only the y-coordinate. Changing both signs gives a half-turn about the origin instead.

02 / WORKED EXAMPLES

See the thinking

Example 1

Reflect a point in the x-axis

Reflect $P (4, - 2)$ in the x-axis.

The x-coordinate remains 4.
The point is 2 units below the x-axis, so its image is 2 units above it.
Change y from $- 2$ to 2.

P^{'} (4, 2)

Example 2

Reflect a triangle in the y-axis

Reflect $A (- 4, 1)$ , $B (- 2, 1)$ and $C (- 3, 3)$ in the y-axis.

Change the sign of every x-coordinate.
Keep the y-coordinates 1, 1 and 3.
The image lies the same distance to the right of the y-axis as the original lies to the left.

A^{'} (4, 1), B^{'} (2, 1), C^{'} (3, 3)

Check your understanding

Reflect $(- 3, 0)$ in the x-axis. Does it move?

Show answer

$(- 3, 0)$ ; it stays fixed.

The point lies on the mirror line because its y-coordinate is zero.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Reflect $(5, - 3)$ in the x-axis.
2 marks
Reflect $(- 2, 4)$ in the y-axis.
2 marks
Reflect $(0, - 5)$ in the y-axis. Explain why the result is unchanged.
2 marks
A point is reflected in the x-axis to $(2, - 4)$ . Find the original point.
2 marks
Reflect $A (1, 1)$ , $B (4, 1)$ and $C (1, 3)$ in the y-axis. Give the image coordinates. Draw and label the image on the supplied grid.
3 marks
Jai says reflecting $(2, 3)$ in the y-axis gives $(2, - 3)$ . Explain the mistake and give the correct image.
2 marks

Read the worked solutions

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

Reflect $(5, - 3)$ in the x-axis.
2 marks
Keep x as 5; reverse the sign of y.
$(5, 3)$
Reflect $(- 2, 4)$ in the y-axis.
2 marks
Reverse the sign of x; keep y as 4.
$(2, 4)$
Reflect $(0, - 5)$ in the y-axis. Explain why the result is unchanged.
2 marks
The point lies on the y-axis, so its perpendicular distance to the mirror line is zero.
$(0, - 5)$
A point is reflected in the x-axis to $(2, - 4)$ . Find the original point.
2 marks
Reflection undoes itself; reverse the sign of the image y-coordinate.
$(2, 4)$
Reflect $A (1, 1)$ , $B (4, 1)$ and $C (1, 3)$ in the y-axis. Give the image coordinates. Draw and label the image on the supplied grid.
3 marks
Reverse each x-coordinate and keep each y-coordinate.
$A^{'} (- 1, 1), B^{'} (- 4, 1), C^{'} (- 1, 3)$
Jai says reflecting $(2, 3)$ in the y-axis gives $(2, - 3)$ . Explain the mistake and give the correct image.
2 marks
Jai changed y, which reflects in the x-axis.
For the y-axis, x changes sign and y stays the same.
$(- 2, 3)$

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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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