Four Transformation Techniques

Transformations change a shape’s position, orientation or size. This guide from Quantum Leap Star Limited explains translation, rotation, reflection and enlargement, using the examples and practice from our Lesson 40 notes.

GCSE Mathematics: learn to draw images accurately on coordinate grids, describe transformations completely, use positive, fractional and negative scale factors, and identify invariant points.

Essential language

• Object: the starting shape.
• Image: the transformed shape.
• Corresponding vertices: matching corners, such as A and A′.
• Congruent: the same shape and size.
• Invariant: unchanged by a transformation.

1. Translation

A translation slides a shape. Every vertex moves the same horizontal and vertical distance. Give the movement as a column vector: the top number is horizontal and the bottom number is vertical. Right and up are positive; left and down are negative.

Worked example: 4 right and 3 up

Translate A(−3, −1), B(−1, −1) and C(−3, 1). The vector has 4 above 3. The coordinate rule is (x, y) → (x + 4, y + 3).

• A′ = (−3 + 4, −1 + 3) = (1, 2)
• B′ = (−1 + 4, −1 + 3) = (3, 2)
• C′ = (−3 + 4, 1 + 3) = (1, 4)

Check: shape, size and orientation stay the same. A nonzero translation has no invariant points.

Try: translate P(2, −3) by 5 left and 2 up.

2. Rotation

A rotation turns a shape about a fixed point. State the centre, angle and direction. For 180°, clockwise and anticlockwise give the same result.

Worked example: 90° anticlockwise about the origin

Rotate A(1, 1), B(4, 1) and C(1, 3) about O(0, 0). Use (x, y) → (−y, x).

• A′ = (−1, 1)
• B′ = (−1, 4)
• C′ = (−3, 1)

Other rules about the origin:

• 90° clockwise: (x, y) → (y, −x)
• 180°: (x, y) → (−x, −y)

Check: each vertex stays the same distance from the centre. The centre is invariant. For a different centre, subtract its coordinates, rotate the displacement, then add the centre back.

Try: rotate P(2, −3) through 90° clockwise about the origin.

3. Reflection

A reflection makes a mirror image. Give the equation of the mirror line. The line is the perpendicular bisector of the segment joining each point to its image.

Worked example: reflection in x = 1

Reflect A(−3, 1), B(−1, 1) and C(−3, 4). In x = a, the rule is (x, y) → (2a − x, y).

• A′ = (2 − (−3), 1) = (5, 1)
• B′ = (2 − (−1), 1) = (3, 1)
• C′ = (2 − (−3), 4) = (5, 4)

• In y = b: (x, y) → (x, 2b − y)
• In y = x: swap coordinates.
• In y = −x: (x, y) → (−y, −x)

Check: matching points have equal perpendicular distances from the mirror line. Every point on the mirror line is invariant.

Try: reflect P(2, −3) in y = x.

4. Enlargement

An enlargement scales a shape. Give the centre and scale factor. Nonzero scale factors preserve angles; lengths multiply by the absolute value of the scale factor.

Worked example: scale factor 2 about the origin

Enlarge A(1, 1), B(3, 1) and C(1, 2). Double each displacement from O(0, 0).

• A′ = (2, 2)
• B′ = (6, 2)
• C′ = (2, 4)

The base changes from 2 units to 4 units, so the scale factor is 4 ÷ 2 = 2.

For centre (a, b) and scale factor k:

x′ = a + k(x − a)
y′ = b + k(y − b)

Fractional scale factors

When 0 < k < 1, the image is smaller and lies between the centre and the object. With k = 1, every point is unchanged.

Try: enlarge P(4, 2) by factor ½ about C(2, 0).

Negative scale factors

A negative scale factor places each image point on the opposite side of the centre.

Example: enlarge A(1, 1), B(2, 1) and C(1, 2) by factor −2 about the origin. Double the distances and reverse their directions.

• A′ = (−2, −2)
• B′ = (−4, −2)
• C′ = (−2, −4)

Lengths double; they do not become negative. To find |k|, divide an image side length by its corresponding object side length. Opposite sides of the centre indicate a negative sign.

To find the centre, extend lines through two pairs of corresponding vertices. Choose pairs giving different lines. Their intersection is the centre.

A scale factor of −1 gives the same result as a 180° rotation about the centre. Corresponding sides are parallel for both positive and negative factors.

Try: enlarge P(−2, 3) by factor −1 about the origin.

Combining transformations: order matters

Start at P(1, 2). Reflect in the y-axis to get (−1, 2), then translate 3 right to get (2, 2).

Reverse the order: translate to (4, 2), then reflect to get (−4, 2). The results differ. Work in the stated order and label intermediate images.

Exam checklist

• Translation: column vector.
• Rotation: angle, direction when needed, and centre.
• Reflection: mirror-line equation.
• Enlargement: signed scale factor and centre.

If asked for a single transformation, give one transformation. Translations, rotations and reflections preserve lengths and angles.

Check your understanding

Try these six questions before opening the answers.

1. Translate (3, −2) by 4 left and 5 up.
2. Rotate (2, 5) through 180° about (1, 1).
3. Reflect (−2, 4) in x = 3.
4. Enlarge (3, 2) by factor −2 about (1, 1).
5. State the invariant points of reflection in y = −1.
6. A corresponding side increases from 3 to 9 units. Object and image are on opposite sides of the centre. Find k.

Show answers and explanations

1. (3 − 4, −2 + 5) = (−1, 3).
2. The displacement from (1, 1) is (1, 4). Reverse it to (−1, −4), then add the centre: (0, −3).
3. (2 × 3 − (−2), 4) = (8, 4).
4. (1 − 2(3 − 1), 1 − 2(2 − 1)) = (−3, −1).
5. All points on y = −1.
6. |k| = 9 ÷ 3 = 3. Opposite sides give k = −3.

Answers to the “Try” prompts

• Translation: (2 − 5, −3 + 2) = (−3, −1).
• Clockwise rotation: (x, y) → (y, −x), giving (−3, −2).
• Reflection in y = x: swap coordinates to get (−3, 2).
• Fractional enlargement: (2 + ½(4 − 2), 0 + ½(2 − 0)) = (3, 1).
• Factor −1 about the origin: (2, −3).

Based on Quantum Leap Star Limited’s revised Lesson 40 notes. Diagrams constructed independently from the example coordinates.