High School Maths · Year 2 · Unit 12 · Lesson 5 of 5
Combined Transformations: Two Moves, One Answer
A flag A is reflected in the y-axis, and then the image is reflected again in the line x = 2.
The final flag C faces the same way as A. What single transformation takes A straight to C?
Not a reflection: C is the same way round as A. Every point has moved 4 squares to the right, so it is a translation by 40. Two mirrors that are parallel always give a translation of twice the gap between them.
Master the essentials
Order
"P followed by Q" means do P first, then Q. Changing the order can change the answer.
First P, then Q
Parallel mirrors
Two reflections in parallel lines make a translation: twice the gap, at right angles to the mirrors.
x = 2 then x = 6: 80
Perpendicular mirrors
Two reflections in lines at right angles make a half turn about the point where they cross.
x = 2 then y = −1: 180° about (2, −1)
Mirrors at 45°
Reflections in two lines through O at 45° make a quarter turn about O. Track a point to find the direction.
y-axis then y = x: 90° clockwise
Two translations
Add the two column vectors.
3−1 + −54 = −23
Two enlargements
With the same centre, multiply the scale factors.
k = 3 then k = −½: k = −1.5
Invariant point
A point that does not move. For a reflection: every point on the mirror. For a rotation or an enlargement: only the centre.
In y = x, (6, 6) stays put
Use the rule
Follow (x, y) through both moves. Then match the result to a rule you know.
(x, y) → (−x, y) → (−x, −y): a half turn
See it done
1. Triangle T has corners (1, 1), (3, 1) and (3, 2). T is reflected in the y-axis, and the image is then reflected in the x-axis. Describe fully the single transformation that maps T onto the final image.
- Reflect in the y-axis: (x, y) → (−x, y).
- Then reflect in the x-axis: (−x, y) → (−x, −y).
- (x, y) → (−x, −y) is a half turn about the origin. Check: (3, 1) → (−3, −1) ✓.
Rotation of 180° about (0, 0)
2. A shape is rotated through 90° clockwise about the origin, and then reflected in the line y = x. Describe fully the single transformation.
- Rotate 90° clockwise: (x, y) → (y, −x).
- Reflect in y = x (swap): (y, −x) → (−x, y).
- (x, y) → (−x, y) is a reflection in the y-axis. Check: (2, 5) → (5, −2) → (−2, 5) ✓.
Reflection in the line x = 0 (the y-axis)
3. Find the invariant point of an enlargement with scale factor 3, centre the origin, followed by a translation by −46.
- Follow (x, y): (x, y) → (3x, 3y) → (3x − 4, 3y + 6).
- Invariant means the point maps to itself: 3x − 4 = x and 3y + 6 = y.
- 2x = 4, so x = 2. 2y = −6, so y = −3.
- Check: (2, −3) → (6, −9) → (2, −3) ✓. (The two moves together are an enlargement, scale factor 3, centre (2, −3).)
(2, −3)
Common mistake: Doing the moves in the wrong order. Reflect (2, 5) in the x-axis and then rotate 90° anticlockwise about O: (2, 5) → (2, −5) → (5, 2). In the other order: (2, 5) → (−5, 2) → (−5, −2), a different point. "P followed by Q" means P first.
10 questions with instant feedback
Choose the correct answer. Follow one point, or the rule (x, y), through both moves in the right order, then name the single transformation. Questions 1–3 are warm-ups, 4–7 are at exam standard and 8–10 are challenge questions. After you answer, the feedback shows the full solution.
Warm-up (The basic skill: one or two steps.)
1.Which single transformation is the same as a translation by 25 followed by a translation by −7−1?
Correct.
Solution
- Two translations make one translation: add the vectors.
- Tops: 2 + (−7) = −5. Bottoms: 5 + (−1) = 4.
- Translation by −54.
Not quite. This subtracts the second vector. Add the two vectors.
Solution
- Two translations make one translation: add the vectors.
- Tops: 2 + (−7) = −5. Bottoms: 5 + (−1) = 4.
- Translation by −54.
Answer: A. Translation by −54.
Not quite. The bottom is 5 + (−1) = 4. Watch the sign.
Solution
- Two translations make one translation: add the vectors.
- Tops: 2 + (−7) = −5. Bottoms: 5 + (−1) = 4.
- Translation by −54.
Answer: A. Translation by −54.
Not quite. This multiplies the parts. Add the two vectors.
Solution
- Two translations make one translation: add the vectors.
- Tops: 2 + (−7) = −5. Bottoms: 5 + (−1) = 4.
- Translation by −54.
Answer: A. Translation by −54.
2.Which point is invariant under a reflection in the line x = 3?
Not quite. This point is on y = 3, not x = 3. It moves to (1, 3).
Solution
- Points on the mirror line do not move.
- Every point on x = 3 has x-coordinate 3.
- (3, −5) is on the line, so it is invariant.
Answer: C. (3, −5).
Not quite. Its x-coordinate must be 3, not −3. It moves to (9, 3).
Solution
- Points on the mirror line do not move.
- Every point on x = 3 has x-coordinate 3.
- (3, −5) is on the line, so it is invariant.
Answer: C. (3, −5).
Correct.
Solution
- Points on the mirror line do not move.
- Every point on x = 3 has x-coordinate 3.
- (3, −5) is on the line, so it is invariant.
Not quite. The origin is not on x = 3. It moves to (6, 0).
Solution
- Points on the mirror line do not move.
- Every point on x = 3 has x-coordinate 3.
- (3, −5) is on the line, so it is invariant.
Answer: C. (3, −5).
3.A shape is enlarged by scale factor 4, centre the origin. The image is then enlarged by scale factor −½, centre the origin. Which single transformation does the same?
Not quite. This adds the scale factors. Multiply them.
Solution
- Same centre, so multiply the scale factors.
- 4 × (−½) = −2.
- Enlargement, scale factor −2, centre (0, 0).
Answer: B. Enlargement, scale factor −2, centre (0, 0).
Correct.
Solution
- Same centre, so multiply the scale factors.
- 4 × (−½) = −2.
- Enlargement, scale factor −2, centre (0, 0).
Not quite. This divides 4 by ½. Multiply: 4 × (−½).
Solution
- Same centre, so multiply the scale factors.
- 4 × (−½) = −2.
- Enlargement, scale factor −2, centre (0, 0).
Answer: B. Enlargement, scale factor −2, centre (0, 0).
Not quite. This loses the minus sign. 4 × (−½) = −2.
Solution
- Same centre, so multiply the scale factors.
- 4 × (−½) = −2.
- Enlargement, scale factor −2, centre (0, 0).
Answer: B. Enlargement, scale factor −2, centre (0, 0).
Exam standard (Several steps, like real exam questions.)
4.A shape is reflected in the line x = 2, and the image is then reflected in the line x = 7. Describe fully the single transformation.
Not quite. This is the gap between the mirrors. The shape moves twice the gap.
Solution
- Follow (x, y): in x = 2, x → 4 − x.
- Then in x = 7: 4 − x → 14 − (4 − x) = x + 10.
- So (x, y) → (x + 10, y): a translation by 100.
Answer: D. Translation by 100.
Not quite. This is the other order. Reflecting in x = 2 first moves the shape to the right.
Solution
- Follow (x, y): in x = 2, x → 4 − x.
- Then in x = 7: 4 − x → 14 − (4 − x) = x + 10.
- So (x, y) → (x + 10, y): a translation by 100.
Answer: D. Translation by 100.
Not quite. Two reflections never make one reflection: the shape is flipped twice, so it ends up the same way round.
Solution
- Follow (x, y): in x = 2, x → 4 − x.
- Then in x = 7: 4 − x → 14 − (4 − x) = x + 10.
- So (x, y) → (x + 10, y): a translation by 100.
Answer: D. Translation by 100.
Correct.
Solution
- Follow (x, y): in x = 2, x → 4 − x.
- Then in x = 7: 4 − x → 14 − (4 − x) = x + 10.
- So (x, y) → (x + 10, y): a translation by 100.
5.Triangle F is rotated through 90° anticlockwise about the origin, and then reflected in the x-axis. Describe fully the single transformation that does the same.
Correct.
Solution
- Rotate 90° anticlockwise: (x, y) → (−y, x).
- Reflect in the x-axis: (−y, x) → (−y, −x).
- (x, y) → (−y, −x) is a reflection in y = −x. Check: (3, 1) → (−1, 3) → (−1, −3) ✓.
Not quite. This comes from doing the reflection first. Rotate first.
Solution
- Rotate 90° anticlockwise: (x, y) → (−y, x).
- Reflect in the x-axis: (−y, x) → (−y, −x).
- (x, y) → (−y, −x) is a reflection in y = −x. Check: (3, 1) → (−1, 3) → (−1, −3) ✓.
Answer: A. Reflection in the line y = −x.
Not quite. A rotation and a reflection together flip the shape, so the result cannot be a rotation.
Solution
- Rotate 90° anticlockwise: (x, y) → (−y, x).
- Reflect in the x-axis: (−y, x) → (−y, −x).
- (x, y) → (−y, −x) is a reflection in y = −x. Check: (3, 1) → (−1, 3) → (−1, −3) ✓.
Answer: A. Reflection in the line y = −x.
Not quite. A rotation and a reflection together flip the shape, so the result cannot be a rotation.
Solution
- Rotate 90° anticlockwise: (x, y) → (−y, x).
- Reflect in the x-axis: (−y, x) → (−y, −x).
- (x, y) → (−y, −x) is a reflection in y = −x. Check: (3, 1) → (−1, 3) → (−1, −3) ✓.
Answer: A. Reflection in the line y = −x.
6.The table shows three points and their images under a rotation. Find the invariant point.
| Point | Image |
|---|---|
| (0, 0) | (4, −2) |
| (2, 1) | (5, −4) |
| (−1, 3) | (7, −1) |
Not quite. The origin moves to (4, −2), so it is not invariant.
Solution
- From the table the rule is (x, y) → (y + 4, −x − 2). Check: (2, 1) → (5, −4) ✓.
- Invariant: x = y + 4 and y = −x − 2.
- Substitute: x = −x − 2 + 4, so 2x = 2, x = 1 and y = −3. Check: (1, −3) → (−3 + 4, −1 − 2) = (1, −3) ✓.
Answer: C. (1, −3).
Not quite. This is the image of the origin, not a point that stays put.
Solution
- From the table the rule is (x, y) → (y + 4, −x − 2). Check: (2, 1) → (5, −4) ✓.
- Invariant: x = y + 4 and y = −x − 2.
- Substitute: x = −x − 2 + 4, so 2x = 2, x = 1 and y = −3. Check: (1, −3) → (−3 + 4, −1 − 2) = (1, −3) ✓.
Answer: C. (1, −3).
Correct.
Solution
- From the table the rule is (x, y) → (y + 4, −x − 2). Check: (2, 1) → (5, −4) ✓.
- Invariant: x = y + 4 and y = −x − 2.
- Substitute: x = −x − 2 + 4, so 2x = 2, x = 1 and y = −3. Check: (1, −3) → (−3 + 4, −1 − 2) = (1, −3) ✓.
Not quite. This is the midpoint of (0, 0) and its image. That is not a fixed point for a quarter turn.
Solution
- From the table the rule is (x, y) → (y + 4, −x − 2). Check: (2, 1) → (5, −4) ✓.
- Invariant: x = y + 4 and y = −x − 2.
- Substitute: x = −x − 2 + 4, so 2x = 2, x = 1 and y = −3. Check: (1, −3) → (−3 + 4, −1 − 2) = (1, −3) ✓.
Answer: C. (1, −3).
7.Find the invariant point of an enlargement with scale factor −2, centre the origin, followed by a translation by 9−6.
Not quite. Sign slip: x = −2x + 9 gives 3x = 9, so x = +3.
Solution
- Follow (x, y): (x, y) → (−2x, −2y) → (−2x + 9, −2y − 6).
- Invariant: −2x + 9 = x and −2y − 6 = y.
- 3x = 9, so x = 3; 3y = −6, so y = −2.
Answer: D. (3, −2).
Not quite. This is the image of the origin, not a point that stays put.
Solution
- Follow (x, y): (x, y) → (−2x, −2y) → (−2x + 9, −2y − 6).
- Invariant: −2x + 9 = x and −2y − 6 = y.
- 3x = 9, so x = 3; 3y = −6, so y = −2.
Answer: D. (3, −2).
Not quite. This solves −2x + 9 = 0. The image must equal the point itself: −2x + 9 = x.
Solution
- Follow (x, y): (x, y) → (−2x, −2y) → (−2x + 9, −2y − 6).
- Invariant: −2x + 9 = x and −2y − 6 = y.
- 3x = 9, so x = 3; 3y = −6, so y = −2.
Answer: D. (3, −2).
Correct.
Solution
- Follow (x, y): (x, y) → (−2x, −2y) → (−2x + 9, −2y − 6).
- Invariant: −2x + 9 = x and −2y − 6 = y.
- 3x = 9, so x = 3; 3y = −6, so y = −2.
Challenge (Harder problems that mix ideas or need reasoning.)
8.A shape is reflected in the line y = 1, and the image is then reflected in the line x = −2. Describe fully the single transformation.
Not quite. The mirrors cross at x = −2, y = 1, so the centre is (−2, 1). The coordinates are swapped here.
Solution
- The mirrors y = 1 and x = −2 are at right angles.
- They cross at (−2, 1).
- Two perpendicular mirrors make a half turn about the crossing point. Check: (0, 0) → (0, 2) → (−4, 2), and 180° about (−2, 1) also sends (0, 0) to (−4, 2) ✓.
Answer: D. Rotation of 180° about (−2, 1).
Not quite. Parallel mirrors give a translation. These mirrors are at right angles.
Solution
- The mirrors y = 1 and x = −2 are at right angles.
- They cross at (−2, 1).
- Two perpendicular mirrors make a half turn about the crossing point. Check: (0, 0) → (0, 2) → (−4, 2), and 180° about (−2, 1) also sends (0, 0) to (−4, 2) ✓.
Answer: D. Rotation of 180° about (−2, 1).
Not quite. The angle of turn is twice the angle between the mirrors: 2 × 90° = 180°.
Solution
- The mirrors y = 1 and x = −2 are at right angles.
- They cross at (−2, 1).
- Two perpendicular mirrors make a half turn about the crossing point. Check: (0, 0) → (0, 2) → (−4, 2), and 180° about (−2, 1) also sends (0, 0) to (−4, 2) ✓.
Answer: D. Rotation of 180° about (−2, 1).
Correct.
Solution
- The mirrors y = 1 and x = −2 are at right angles.
- They cross at (−2, 1).
- Two perpendicular mirrors make a half turn about the crossing point. Check: (0, 0) → (0, 2) → (−4, 2), and 180° about (−2, 1) also sends (0, 0) to (−4, 2) ✓.
9.A shape is reflected in the line x = 1, and the image is then reflected in x = k. The result is a translation by −80. Find k.
Not quite. This gives a translation of +8. The shape moves left, so the second mirror is to the left of x = 1.
Solution
- Follow x: in x = 1, x → 2 − x. Then in x = k: 2 − x → 2k − (2 − x) = x + 2k − 2.
- So 2k − 2 = −8.
- 2k = −6, so k = −3.
Answer: C. −3.
Not quite. The translation is 2(k − 1) = 2k − 2. This leaves out the −2.
Solution
- Follow x: in x = 1, x → 2 − x. Then in x = k: 2 − x → 2k − (2 − x) = x + 2k − 2.
- So 2k − 2 = −8.
- 2k = −6, so k = −3.
Answer: C. −3.
Correct.
Solution
- Follow x: in x = 1, x → 2 − x. Then in x = k: 2 − x → 2k − (2 − x) = x + 2k − 2.
- So 2k − 2 = −8.
- 2k = −6, so k = −3.
Not quite. This moves the mirror by the whole translation. The gap is half of it.
Solution
- Follow x: in x = 1, x → 2 − x. Then in x = k: 2 − x → 2k − (2 − x) = x + 2k − 2.
- So 2k − 2 = −8.
- 2k = −6, so k = −3.
Answer: C. −3.
10.Which working shows that a reflection in the x-axis followed by a reflection in the line y = x is a rotation of 90° anticlockwise about the origin?
Not quite. Swapping (x, −y) gives (−y, x), not (y, −x). And (y, −x) is a clockwise turn.
Solution
- Reflect in the x-axis: (x, y) → (x, −y).
- Reflect in y = x (swap): (x, −y) → (−y, x).
- (x, y) → (−y, x) is 90° anticlockwise about O ✓.
Answer: B. (x, y) → (x, −y) → (−y, x), which is the rule for 90° anticlockwise about O.
Correct.
Solution
- Reflect in the x-axis: (x, y) → (x, −y).
- Reflect in y = x (swap): (x, −y) → (−y, x).
- (x, y) → (−y, x) is 90° anticlockwise about O ✓.
Not quite. This reflects in y = x first, and (y, −x) is a clockwise turn.
Solution
- Reflect in the x-axis: (x, y) → (x, −y).
- Reflect in y = x (swap): (x, −y) → (−y, x).
- (x, y) → (−y, x) is 90° anticlockwise about O ✓.
Answer: B. (x, y) → (x, −y) → (−y, x), which is the rule for 90° anticlockwise about O.
Not quite. Not true: parallel mirrors make a translation. The turn is twice the angle between the mirrors.
Solution
- Reflect in the x-axis: (x, y) → (x, −y).
- Reflect in y = x (swap): (x, −y) → (−y, x).
- (x, y) → (−y, x) is 90° anticlockwise about O ✓.
Answer: B. (x, y) → (x, −y) → (−y, x), which is the rule for 90° anticlockwise about O.
Two mirrors, one turn
Triangle A has corners (1, 1), (4, 1) and (1, 3).
1. Reflect A in the line x = −1. Write down the corners of the image B.
2. Reflect B in the line y = 2. Write down the corners of the image C.
3. Describe fully the single transformation that maps A onto C, and give its invariant point.
- In x = −1, x → −2 − x and y stays the same.
- B: (−3, 1), (−6, 1) and (−3, 3).
- In y = 2, y → 4 − y and x stays the same.
- C: (−3, 3), (−6, 3) and (−3, 1).
- The mirrors x = −1 and y = 2 are at right angles and cross at (−1, 2).
- So A → C is a rotation of 180° about (−1, 2). Check: (1, 1) → (−3, 3), and (−1, 2) is the midpoint of these ✓.
- The only invariant point is the centre, (−1, 2).
Next: the Lesson 5 exit quiz, then the unit test. The unit test has 18 multiple-choice questions and five structured problems, 44 marks in all. No calculator. Allow 70 minutes.