Transformations

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Cambridge O Level Mathematics (Syllabus D) 4024

Transformations

The big idea: Each of the four transformations is pinned down by a short list of facts. Give exactly those facts and the description is complete; leave one out and the marks go with it.

Summary — What This Chapter Covers

Reflection in a straight line: the mirror line, given as an equation such as x = 2, y = −1, y = x or y = −x.
Rotation about a centre through a multiple of 90°: centre, angle and direction.
Enlargement from a centre by a scale factor, which may be positive, fractional or negative.
Translation by a column vector xy: x across, y up.
Questions may combine transformations. Use a ruler for every straight edge, and ask for tracing paper if it helps.

1. Reflection in a Straight Line

Syllabus 7.1 · Paper 1 & 2

The mirror line

A reflection flips a shape over a mirror line. Each image point is the same perpendicular distance from the mirror as the object point, on the other side. The image is the same size and shape (congruent) but reversed.

Mirror line	The point (x, y) maps to
x = a (a vertical line)	(2a − x, y)
y = b (a horizontal line)	(x, 2b − y)
the x-axis (y = 0)	(x, −y)
the y-axis (x = 0)	(−x, y)
y = x	(y, x)
y = −x	(−y, −x)

To find a mirror line, join a point to its image. The mirror line is the perpendicular bisector of that join, so it passes through the midpoint.

Worked Example

Triangle A has vertices (1, 1), (4, 1) and (4, 3).
(a) Reflect triangle A in the line x = −1.
(b) Reflect triangle A in the line y = x.

Practice Question 1

(a) The point P(2, −1) is reflected onto P′(2, 5). Find the equation of the mirror line.
(b) Write down the image of (3, 1) after a reflection in the line y = −x.

2. Rotation through Multiples of 90°

Syllabus 7.1 · Paper 1 & 2

Centre, angle, direction

A rotation turns a shape about a fixed point, the centre of rotation. In this syllabus the angle is always 90°, 180° or 270°. Say clockwise or anticlockwise for 90° (180° needs no direction). Tracing paper is the easiest tool: trace the shape, hold the centre with your pencil point and turn.

Rotations about the origin 90° anticlockwise: (x, y) → (−y, x)
90° clockwise: (x, y) → (y, −x)
180°: (x, y) → (−x, −y)

About any other centre (a, b) subtract the centre → apply the rule → add the centre back

Worked Example

Triangle T has vertices (1, 1), (3, 1) and (3, 2). Rotate T through 90° clockwise about the origin.

Practice Question 2

(a) Rotate the point (5, 2) through 90° anticlockwise about (1, 1).
(b) Rotate the point (4, 3) through 180° about (2, 1).

Finding the centre

Draw the perpendicular bisector of the line joining a point to its image, then do the same for a second point. The centre is where the two bisectors cross. With tracing paper you can test a guess: if the traced shape lands on the image when you turn it, the guess is right.

3. Enlargement: Positive, Fractional and Negative Scale Factors

Syllabus 7.1 · Paper 1 & 2

Centre and scale factor

centre scale factor

Scale factor	What happens
k > 1
0 < k < 1 (a fraction)	smaller

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Transformations