Transformations Revision Notes

Reflection produces a mirror image of a shape across a specified line called the line of reflection. When reflecting across a vertical line (e.g., x = k), each point of the shape moves horizontally to the opposite side of the line at the same perpendicular distance. When reflecting across a horizontal line (e.g., y = k), each point moves vertically to the opposite side at the same distance.

To reflect a point across a vertical line x = k, the x-coordinate changes to 2k - x, while the y-coordinate remains the same. For example, reflecting point (3, 5) across x = 1 gives (2*1 - 3, 5) = (-1, 5).

To reflect a point across a horizontal line y = k, the y-coordinate changes to 2k - y, while the x-coordinate remains the same. For example, reflecting point (4, 2) across y = 3 gives (4, 2*3 - 2) = (4, 4).

• Reflection flips the shape over the line of reflection.
• Distances perpendicular to the line remain equal on both sides.
• Coordinates transform by subtracting from twice the line's coordinate.

Rotation turns a shape around a fixed point called the centre of rotation through a specified angle, here multiples of 90° (90°, 180°, 270°, or 360°). The direction is usually anticlockwise unless stated otherwise.

When rotating about the origin (0,0), the coordinates of each point transform as follows:

90° anticlockwise: (x, y) → (-y, x) 180° anticlockwise: (x, y) → (-x, -y) 270° anticlockwise: (x, y) → (y, -x) 360° anticlockwise: (x, y) → (x, y) (full turn) For rotation about other points such as vertices or midpoints, first translate the shape so the centre of rotation is at the origin, perform the rotation, then translate back. This ensures accurate positioning of the rotated shape without distortion or resizing. Example: Rotate point (3, 2) 90° anticlockwise about the origin: new point is (-2, 3).

• Rotation preserves shape and size but changes orientation.
• Coordinates transform using specific rules depending on the angle.
• Rotation about points other than the origin requires translation steps.

Enlargement changes the size of a shape by scaling distances from a fixed centre point, called the centre of enlargement. The scale factor (k) determines how much the shape grows or shrinks.

If k > 1, the shape enlarges (gets bigger). If 0 < k < 1, the shape reduces (gets smaller). Negative scale factors are not considered here.

To find the image of a point (x, y) under enlargement from centre (x_c, y_c) with scale factor k, use: (x', y') = (x_c + k(x - x_c), y_c + k(y - y_c)) Example: Enlarge point (4, 3) from centre (2, 1) with scale factor 1.5: New x = 2 + 1.5(4 - 2) = 2 + 3 = 5 New y = 1 + 1.5(3 - 1) = 1 + 3 = 4 Image point is (5, 4). Enlargement preserves shape but changes size and distances from the centre proportionally.

• Enlargement scales distances from the centre by the scale factor.
• Positive scale factors enlarge or reduce the shape without flipping.
• Coordinates transform by moving along vectors from the centre scaled by k.

Translation moves every point of a shape the same distance in the same direction, defined by a vector. A vector is written as 9a b9, where a is the horizontal movement and b is the vertical movement.

To translate a point (x, y) by vector (a, b), add the vector components: (x', y') = (x + a, y + b) Example: Translate point (3, 5) by vector (2, -1): New point is (3 + 2, 5 - 1) = (5, 4). Translation does not change the shape or orientation, only its position.

Vectors can be represented graphically by arrows showing direction and length corresponding to a and b.

• Translation shifts shapes without rotation or resizing.
• Coordinates change by adding vector components.
• Vectors describe direction and distance of translation.