Transformations

A reflection flips a shape over a mirror line. Each point ends up the same distance from the line, on the other side.

To describe a reflection fully, give the equation of the mirror line:

• x = a is a vertical line, and y = b is a horizontal line
• the y-axis is x = 0, and the x-axis is y = 0
• y = x and y = −x are the diagonal lines through the origin.

Example: describe fully the single transformation that maps triangle A onto triangle B.

1. Each corner has moved to the same distance on the other side of the y-axis
2. A reflection in the line x = 0 (the y-axis)

A rotation turns a shape about a fixed point. To describe it fully, give:

1. the angle (such as 90° or 180°)
2. the direction (clockwise or anticlockwise; 180° doesn't need one)
3. the centre of rotation, as a coordinate.

About the origin, 90° clockwise takes (x, y) to (y, −x), and 180° takes (x, y) to (−x, −y).

Example: rotate the point (3, 1) by 90° clockwise about the origin.

1. (x, y) goes to (y, −x)
2. (1, −3)

A translation slides a shape without turning it. It's described by a column vector, written here as (3 over −2): the top number is how far right (negative for left), and the bottom number is how far up (negative for down).

1. Move each point across by the top number.
2. Move it up or down by the bottom number.

Example: translate the point (2, 5) by the vector (3 over −2).

1. Across: 2 + 3 = 5
2. Up: 5 + (−2) = 3
3. (5, 3)

Example: a shape moves 4 left and 1 up. Write the column vector.

1. Left is negative: −4. Up is positive: 1
2. (−4 over 1)

An enlargement changes a shape's size. To describe it fully, give the scale factor and the centre of enlargement.

1. Measure (or count) from the centre to each corner.
2. Multiply that distance by the scale factor, in the same direction.
3. Plot the new corners and join them.

A scale factor between 0 and 1 makes the shape smaller.

Example: enlarge triangle A, with corners (1, 1), (2, 1) and (1, 3), by scale factor 2 with centre (0, 0).

1. From (0, 0), each corner is twice as far away
2. (2, 2), (4, 2) and (2, 6): triangle B

Example: enlarge the point (4, 2) by scale factor ½, centre (0, 0).

1. Half the distance from the centre: (2, 1)

A negative scale factor puts the image on the other side of the centre, upside down.

1. Find the movement from the centre to the point.
2. Multiply it by the scale factor, including the minus sign.
3. Add the result to the centre.

Example: enlarge the point (2, 1) by scale factor −2, centre (0, 0).

1. From the centre: 2 across, 1 up
2. × −2: 4 left, 2 down
3. (−4, −2)

Example: enlarge the point (2, 1) by scale factor −2, centre (1, 1).

1. From the centre: 1 across, 0 up
2. × −2: 2 left, 0 up
3. (1 − 2, 1 + 0) = (−1, 1)

Doing one transformation after another can give the same result as a single transformation.

Example: reflect a shape in the x-axis, then in the y-axis. Describe the single transformation that has the same effect.

1. Reflecting in the x-axis takes (x, y) to (x, −y)
2. Reflecting in the y-axis then takes it to (−x, −y)
3. A rotation of 180° about the origin

An invariant point is a point that stays in the same place after a transformation. For a reflection, every point on the mirror line is invariant; for a rotation, the centre is.

Example: which points are invariant under a reflection in y = x?

1. Points on the mirror line stay where they are
2. Every point on y = x, such as (2, 2)