10.2 Reflection

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: reflecting shapes in mirror lines, including y = x and y = −x, and describing reflections. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

• Reflect a shape in a vertical or horizontal mirror line
• Reflect a shape in the lines y = x and y = −x
• Find the image of a point after a reflection
• Describe a reflection by giving the equation of the mirror line

Key words

• Reflection: A flip in a mirror line. The image is the mirror image of the object.
• Mirror line: The line of reflection. Each point and its image are the same distance from it, on opposite sides.
• Perpendicular: At right angles. A point moves to its image at right angles to the mirror line.
• Invariant point: A point that does not move. Points on the mirror line stay where they are.

Reflecting in a mirror line

Take each vertex in turn. Count how far it is from the mirror line, measured at right angles to the line. Put the image vertex the same distance away on the other side. The image is congruent to the object but flipped: it faces the opposite way.

• the y-axis (x = 0); What happens to (x, y) x changes sign: (−x, y); Example: (3, 2) goes to (−3, 2)
• the x-axis (y = 0); What happens to (x, y) y changes sign: (x, −y); Example: (3, 2) goes to (3, −2)

Mirror lines x = a and y = b

For a vertical line such as x = 2, count squares across. For a horizontal line such as y = −1, count squares up or down.

Worked example: reflecting points

Reflect (5, 1) in the line x = 2. It is 3 squares to the right of x = 2, so the image is 3 squares to the left: (−1, 1).
Reflect (3, 3) in the line y = −1. It is 4 squares above y = −1, so the image is 4 squares below: (3, −5).

Answer: (−1, 1) and (3, −5)

Reflecting in y = x and y = −x

The line y = x goes diagonally through (0, 0), (1, 1), (2, 2) … Reflecting in it swaps the coordinates. Reflecting in y = −x swaps them and changes both signs.

Key fact: In y = x: (x, y) → (y, x). In y = −x: (x, y) → (−y, −x).

Worked example: diagonal mirror lines

Reflect (1, 3) in y = x: swap to get (3, 1).
Reflect (1, 3) in y = −x: swap and change both signs to get (−3, −1).

Answer: (3, 1) and (−3, −1)

Describing a reflection

To describe a reflection fully, say reflection and give the equation of the mirror line. The mirror line is exactly halfway between each point and its image.

Worked example: finding the mirror line

P has a vertex at (1, 5) and the matching vertex of Q is at (7, 5).
Halfway between x = 1 and x = 7 is x = (1 + 7) ÷ 2 = 4.

Answer: reflection in the line x = 4

Common mistakes

• Reflecting in the wrong line: x = 2 is a vertical line, y = 2 is a horizontal line.
• Measuring from the edge of the grid instead of from the mirror line.
• For y = x, sliding the shape across instead of swapping the coordinates of each vertex.
• Describing a reflection without the equation of the mirror line, e.g. just “it is reflected”.

Revision questions

Try these, then check your answers below each one.

Question 1. The point (3, −5) is reflected in the x-axis. Write down the coordinates of its image.

Answer: (3,5)

Working: (x, y) → (x, -y), so (3, -5) → (3, 5).

Question 2. The point (−2, 6) is reflected in the y-axis. Write down the coordinates of its image.

Answer: (2,6)

Working: (x, y) → (-x, y), so (-2, 6) → (2, 6).

Question 3. The point (4, 1) is reflected in the line y = x. Write down the coordinates of its image.

Answer: (1,4)

Working: (x, y) → (y, x), so (4, 1) → (1, 4).

Related lessons

• Previous lesson: 10.1 Translation
• Next lesson: 10.3 Rotation
• In this chapter: 10.4 Enlargement
• In this chapter: 10.5 Describing Enlargements
• In this chapter: 10.6 Combining Transformations

Continue in the full course

This lesson is section 10.2 (EGCMF1002) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.