Transformations Extended
Diagonal mirrors: reflecting in y = x swaps the coordinates: (a, b) → (b, a). Reflecting in y = −x swaps AND flips signs: (a, b) → (−b, −a).
Negative sf: the image appears on the OPPOSITE side of the centre, upside down. Distances still scale by |sf|. With centre at the origin, sf −2 sends (3, 1) to (−6, −2). An enlargement sf −1 lands exactly where a 180° rotation about the same centre would.
Combinations: do transformation 1, then transformation 2, then ask what single move makes the same journey. Classic results: two reflections in parallel lines = a translation; two reflections in crossing lines = a rotation about their crossing point.
Reflect (4, −1) in the line y = x.
Swap: image (−1, 4).
Reflect (2, 5) in the line y = −x.
Swap and negate: image (−5, −2).
Enlarge (3, 1) with centre (0, 0) and sf −2.
Multiply by −2: image (−6, −2) — other side of the centre, inverted.
Enlarge (5, 4) with centre (1, 2) and sf −1.
Centre → point: 4 across, 2 up. sf −1 → 4 back, 2 down from the centre: (1 − 4, 2 − 2) = (−3, 0). Same as rotating 180° about (1, 2).
Shape S is reflected in the x-axis, then the image is reflected in the y-axis. What single transformation is this?
Try a point: (a, b) → (a, −b) → (−a, −b). Both signs flipped = rotation 180° about the origin.
Q1. Reflect (−3, 7) in the line y = x.
👀 Show answer
Q2. Reflect (6, 2) in the line y = −x.
👀 Show answer
Q3. Enlarge (2, 3) with centre (0, 0) and sf −3.
👀 Show answer
(−6, −9) ✓ A1
Q4. Enlarge (7, 3) with centre (3, 1) and sf −\(\dfrac{1}{2}\).
👀 Show answer
× (−½): 2 back, 1 down ✓ M1
(3 − 2, 1 − 1) = (1, 0) ✓ A1
Q5. Triangle T is rotated 90° anticlockwise about the origin, then the image is rotated a further 90° anticlockwise about the origin. Describe the single transformation that maps T to the final image.
👀 Show answer
rotation, 180°, about the origin ✓ A2
Q6. A shape is reflected in the y-axis, then the image is reflected in the x-axis. Describe the single equivalent transformation.
👀 Show answer
rotation, 180°, about the origin ✓ A2
- y = x and y = −x rules mixed up: y = x just swaps; y = −x swaps AND negates both.
- Negative sf drawn on the same side of the centre — it must cross over.
- Answering a combination question with two transformations: the word SINGLE means one.
- Assuming order never matters: for many combinations it does — always track a test point through BOTH steps in the order given.
Try this: Set k to −2. Describe fully the single transformation that maps the blue triangle onto the orange one — centre, type and scale factor.
Papers 2 and 4 love “describe the single transformation” after a combination — 3 marks. Negative-sf drawing questions are a top discriminator. Next: vectors get serious in TV-E2.