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Reflections

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How to reflect points and shapes in a mirror line on a grid, including the axes and diagonal lines, and how to find, name and describe a mirror line.

What a reflection is

A reflection flips a shape over a mirror line. The image is exactly the same size and shape as the original, but it faces the other way, like your reflection in a mirror.

In the exam you are usually asked to reflect a shape on a grid, for 1 or 2 marks. The mirror line may be drawn for you, or given as an equation such as x=3 or y=-1. You may also be asked to write down the equation of a mirror line, or to describe fully the single transformation that maps one shape onto another, for 2 marks.

Every point and its image are the same distance from the mirror line, on opposite sides. Measure the distance at right angles to the mirror. Here the mirror is the y-axis. Corner L is 3 squares to the right of it, so its image L′ is 3 squares to the left, at the same height.

The image of a point is written with a small dash, called a prime: the image of L is L′, read "L prime". A point that lies on the mirror line does not move.

Naming the mirror line

  • The x-axis is the line y=0.
  • The y-axis is the line x=0.
  • A line such as x=2 is vertical: every point on it has an x-coordinate of 2.
  • A line such as y=-1 is horizontal: every point on it has a y-coordinate of -1.

So x= a number is a vertical line, going up and down, and y= a number is a horizontal line, going across. Here x=2 goes up through 2 on the x-axis, and y=-1 goes across through -1 on the y-axis.

For a horizontal mirror, count squares up or down, and the x-coordinate stays the same. For a vertical mirror, count squares left or right, and the y-coordinate stays the same.

Worked example 1: Reflecting a point in the x-axis

Reflect the point P(4,3) in the x-axis. Write down the coordinates of its image, P′.

  1. The x-axis is horizontal, so count squares up or down. P is 3 squares above the x-axis.

  2. Your turn. Write down the coordinates of P′.

    Write your answer as (x, y)

    P′ is 3 squares below the x-axis, straight underneath P: (4,-3)

  3. Answer: P′(4,-3)

Worked example 2: Reflecting a point in a horizontal line

Reflect the point B(-3,5) in the line y=2.

  1. y=2 is a horizontal line through 2 on the y-axis. So count squares up or down, and keep the x-coordinate, -3.
  2. B is at height 5, so it is 5-2=3 squares above the mirror.

  3. Your turn. What is the y-coordinate of B′?

    B′ is 3 squares below the mirror: 2-3=-1

  4. Answer: B′(-3,-1)

Reflecting a whole shape

To reflect a shape, reflect each corner, or vertex, one at a time. Then join the new corners with a ruler, in the same order as the original.

Corners far from the mirror move a long way; corners close to it move only a little. Check your image: it is the same size and shape, the same distance from the mirror on the other side, and it faces the other way.

Tracing paper helps, and you can ask for it in the exam. Trace the shape and the mirror line, turn the paper over, line the mirror line up again, and see where the shape lands.

Worked example 3: Reflecting a triangle in a vertical line

Triangle DEF has vertices D(5,1), E(7,1) and F(5,3). Reflect the triangle in the line x=4.

  1. x=4 is a vertical line, so count squares left or right. Reflect one corner at a time.
  2. D is 1 square right of the mirror, so D′ is 1 square left of it: D′(3,1)
  3. E is 3 squares right of the mirror, so E′ is 3 squares left of it: E′(1,1)

  4. Your turn. Write down the coordinates of F′.

    Write your answer as (x, y)

    F is 1 square right of the mirror, so F′ is 1 square left of it: F′(3,3)

  5. Answer: the image has vertices D′(3,1), E′(1,1) and F′(3,3)

Diagonal mirror lines

Every point on the line y=x has two equal coordinates, such as (1,1), (2,2) and (-3,-3). Every point on the line y=-x has coordinates with opposite signs, such as (1,-1), (2,-2) and (-3,3).

For a diagonal mirror, the distance is still measured at right angles to the line, which means going diagonally across the squares. Here P(3,-2) reflects in y=x to P′(-2,3).

  • Reflecting in y=x swaps the coordinates: (3,-2) becomes (-2,3).
  • Reflecting in y=-x swaps the coordinates and changes both signs: (3,-2) becomes (2,-3).

After using a shortcut, check on the grid that the point and its image are the same distance from the mirror, on opposite sides.

Worked example 4: Reflecting a point in y = −x

Reflect the point R(2,3) in the line y=-x.

  1. For the mirror y=-x, swap the coordinates and change both signs.
  2. Swap the coordinates: (2,3) becomes (3,2)

  3. Your turn. Now change both signs. Write down the coordinates of R′.

    Write your answer as (x, y)

    Changing both signs of (3,2) gives R′(-3,-2)

  4. Check on the grid: R and R′ are the same distance from the mirror, on opposite sides, and the line joining them crosses the mirror at right angles.
  5. Answer: R′(-3,-2)

Finding and describing a mirror line

To find a mirror line, pick a corner of the shape and the matching corner of its image. The mirror line is exactly halfway between them.

If the matching corners have the same y-coordinate, the mirror is vertical: x= the number halfway between their x-coordinates. If they have the same x-coordinate, the mirror is horizontal: y= the number halfway between their y-coordinates.

To find the number halfway between two numbers, count the gap and go half of it from either end. For example, -1 and 5 are 6 apart, so halfway is 3 along from -1, which is 2.

To describe fully a reflection, give two things: the word reflection and the equation of the mirror line, for example "reflection in the line x=2". Use the word reflection, not flip, and give only one transformation.

Worked example 5: Finding the equation of a mirror line

Triangle Q is a reflection of triangle P. Write down the equation of the mirror line.

  1. The corner (1,1) of P matches the corner (-5,1) of Q. They have the same y-coordinate, so the mirror line is vertical.

  2. Your turn. What number is halfway between 1 and -5?

    1 and -5 are 6 apart, and half of 6 is 3: 1-3=-2

  3. Check with another pair: (2,1) and (-6,1) are each 4 squares from x=-2, on opposite sides.
  4. Answer: x=-2

Worked example 6: Describing a reflection fully

Describe fully the single transformation that maps triangle P onto triangle Q.

  1. Q is the same size and shape as P, but it faces the other way. To fit a tracing of P onto Q, you have to turn the tracing paper over. So the transformation is a reflection, not a rotation.
  2. Match the corners. The right angle of P is at (1,4) and the right angle of Q is at (4,1). Then (2,4) goes to (4,2), and (1,2) goes to (2,1).
  3. Each time, the two coordinates have swapped round.

  4. Your turn. Write down the equation of the mirror line.

    Type powers with ^ and fractions with /, for example x^2 or (x+1)/3

    Swapping the coordinates is a reflection in the line y=x

  5. Check on the grid: each corner of P and its matching corner of Q are the same distance from y=x, on opposite sides.
  6. Answer: reflection in the line y=x

Common mistakes

  • Using the wrong mirror line. Read the equation carefully: x=3 is a vertical line and y=3 is a horizontal line.
  • Writing the mirror line's equation wrongly. A horizontal line through 1 on the y-axis is y=1, not x=1 and not just 1.
  • Putting the image in the wrong place. Count the squares to the mirror for every corner, so each image corner is the same distance away on the other side.
  • Mixing up a reflection and a rotation. A reflection flips the shape over a line, so the image faces the other way; a rotation only turns it.
  • Leaving out the mirror line when describing. The word reflection on its own gets only one of the two marks; give the equation of the line too.
  • Giving two transformations or the wrong word. Write reflection, not flip or mirroring, and describe only one transformation.

Practice questions

More on this topic: Reflections worksheet with full solutions