Reflections in coordinate axes | Year 7

01 / LEARN

Understand the idea.

The axis is a mirror line

A reflection uses a mirror line. Each point and its image lie the same perpendicular distance from that line, on opposite sides. The line joining them meets the mirror line at a right angle.

The coordinate axes make these distances easy to see. The $x$ -axis swaps above and below while keeping horizontal position. The $y$ -axis swaps left and right while keeping height. A point on the mirror line stays fixed.

Coordinate rules and checks

For an $x$ -axis reflection, keep $x$ and reverse the sign of $y$ . For a $y$ -axis reflection, reverse the sign of $x$ and keep $y$ . Think about which position crosses the mirror before using a rule.

Reflect every labelled vertex and join the matching image vertices. The shape retains its lengths, angles, perimeter and area, but the clockwise order of its vertices reverses. Zero stays zero, so points on the chosen axis do not move.

\begin{aligned} x-axis: & (x, y) \mapsto (x, - y) \\ y-axis: & (x, y) \mapsto (- x, y) \end{aligned}

Equal-scale coordinate grid with original triangle A(-4,1), B(-2,1), C(-3,3) in sage and dashed reflected triangle A prime (4,1), B prime (2,1), C prime (3,3) in gold. Corresponding vertices are equally far from the y-axis.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Reflect a point in the x-axis

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

Worked solution

1
Identify the unchanged coordinate.
$x^{'} = 4$
Reflection in the horizontal axis keeps the horizontal position.
2
Reverse the vertical displacement.
$y^{'} = - (- 2) = 2$
Two units below the mirror becomes two units above it.
3
Record the matching image point.
$P^{'} (4, 2)$
The original and image have equal perpendicular distances from the x-axis.

Answer

$P^{'} (4, 2)$

EXAMPLE 02

Reflect a triangle in the y-axis

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

Worked solution

1
Write the y-axis rule.
$(x, y) \mapsto (- x, y)$
The mirror swaps left and right without changing height.
2
Reflect A.
$A^{'} = (- (- 4), 1) = (4, 1)$
A lies four units left of the mirror, so its image lies four units right.
3
Reflect B.
$B^{'} = (- (- 2), 1) = (2, 1)$
B retains its height of one unit.
4
Reflect C.
$C^{'} = (- (- 3), 3) = (3, 3)$
C retains its height of three units.

Answer

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

03 / YOUR TURN

Try it yourself.

Check your understanding

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

Show answer & reasoning

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

The point $(- 3, 0)$ lies on the $x$ -axis. Reversing its zero vertical coordinate gives $- 0 = 0$ , so it stays at $(- 3, 0)$ .

Keep practising

6 worksheet questions · 13 marks

Open worksheet 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 ↗

In this lesson

Understand the idea.

The axis is a mirror line

Coordinate rules and checks

See the method in action.

Reflect a point in the x-axis

Identify the unchanged coordinate.

Reverse the vertical displacement.

Record the matching image point.

Reflect a triangle in the y-axis

Write the y-axis rule.

Reflect A.

Reflect B.

Reflect C.

Try it yourself.

Check your understanding

Keep practising