G8: Combined Transformations
Describe the changes and invariance achieved by combinations of transformations
📋 Key Concepts
Definition: Combined transformations occur when two or more transformations are applied one after another to a shape.
Invariance: A point, line, or shape is invariant if it remains unchanged after a transformation.
📝 Common Combinations
Two reflections: A reflection followed by another reflection is equivalent to a rotation or translation.
Two rotations: Two rotations about the same point are equivalent to a single rotation with angle = sum of angles.
Two translations: Two translations are equivalent to a single translation (vector addition).
Example 1
Reflect in the line x = 0, then reflect in the line y = 0.
Solution:
This is equivalent to a 180° rotation about the origin.
Point (a, b) → (-a, b) → (-a, -b)
Which is the same as a 180° rotation: (a, b) → (-a, -b)
📝 Describing Combined Transformations
Example 2
Triangle A is reflected in the y-axis to get B. B is then reflected in the x-axis to get C. Describe the single transformation from A to C.
Solution:
Reflection in y-axis: (x, y) → (-x, y)
Reflection in x-axis: (-x, y) → (-x, -y)
Combined: (x, y) → (-x, -y)
This is a 180° rotation about the origin (0, 0).
📝 Invariant Points
Definition: A point is invariant if it maps to itself under a transformation.
Example 3
Find the invariant points for a reflection in the line y = x.
Solution:
Points on the line y = x are invariant.
If x = y, then (x, x) → (x, x) after reflection.
All points on the mirror line are invariant.
Example 4
Find invariant points for a rotation of 180° about the origin.
Solution:
Only the centre of rotation is invariant: (0, 0).
(0, 0) → (0, 0) ✓
📝 Invariant Lines
Definition: A line is invariant if it maps to itself (though individual points may move along it).
Example 5
Find the invariant line for a reflection in the line x = 3.
Solution:
The line x = 3 itself is invariant (every point stays on the line).
Any line perpendicular to the mirror line is also invariant as a set.
📝 Combination Rules
| First Transformation | Second Transformation | Equivalent Single Transformation |
|---|---|---|
| Reflection in line 1 | Reflection in parallel line 2 | Translation |
| Reflection in line 1 | Reflection in line 2 (at angle θ) | Rotation (angle 2θ) |
| Rotation by α | Rotation by β (same centre) | Rotation by (α + β) |
| Translation by vector a | Translation by vector b | Translation by (a + b) |
Example 6
Reflect in y = x, then reflect in y = -x. What single transformation is this?
Solution:
y = x and y = -x are perpendicular (angle = 90°)
Two reflections in lines at angle θ = rotation by 2θ
This equals a 180° rotation about the origin.
❓ Practice Questions
Q1: What single transformation is: reflect in x = 0, then reflect in y = 0?
Q2: Find the invariant point for a rotation of 90° about (2, 3).
Q3: Two translations:
| 3 |
| -2 |
| -1 |
| 5 |
Q4: What points are invariant under reflection in the x-axis?
Q5: A rotation of 120° followed by rotation of 240° about the same point. What is the single transformation?
✅ Answers
- 180° rotation about the origin
- The centre of rotation: (2, 3)
2 3 - All points on the x-axis (where y = 0)
- Rotation of 360° (or identity - no change)
🎯 Exam Tips
- Work through transformations one step at a time
- Draw diagrams showing each stage
- Check your answer by applying the transformations to coordinates
- Remember: mirror line points are always invariant for reflections
- Only the centre is invariant for rotations (except 360°)
- Two parallel mirrors = translation, two angled mirrors = rotation
🧠 Problem-Solving Strategies
Problem-Solving
Two reflections in parallel lines distance d apart equals a translation of 2d perpendicular to the lines. Two reflections in intersecting lines equals a rotation of twice the angle between the lines, about the intersection point. Apply transformations step by step, tracking each vertex. The order of combined transformations matters — transformation A then B is not always the same as B then A.
Multi-Step Problem
Reflect point P(3,4) in the line x = 2, then reflect the image in the line x = 6. Describe the single transformation that maps P directly to the final image.
Solution: First reflection in x = 2: (3,4) maps to (1,4) since 3 is 1 unit right of x = 2, so image is 1 unit left. Second reflection in x = 6: (1,4) maps to (11,4) since 1 is 5 units left of x = 6, so image is 5 units right. Single transformation: translation by vector (8, 0). Distance between parallel lines = 6 - 2 = 4, so translation = 2 x 4 = 8 in the x-direction. Verified.
⚠️ Common Errors
Watch Out!
1. Wrong: Thinking the order of combined transformations does not matter Correct: Order usually matters. A rotation then a translation generally gives a different result from the translation then the rotation. Always apply in the stated order.
2. Wrong: Combining two rotations and assuming the result is always a rotation Correct: Two rotations about different centres can combine to give a translation (if angles sum to zero) or a different rotation. Only rotations about the SAME centre combine by adding angles.
3. Wrong: When combining a reflection and a rotation, treating it as just a reflection Correct: A reflection followed by a rotation (or vice versa) may result in a glide reflection or another transformation entirely. Track the effect on each point carefully.
✍️ 6-Mark Exam Question
Extended Answer
6 marks: Triangle T has vertices A(0,0), B(2,0), C(0,2). (a) Reflect T in the line y = x to get T1. State the coordinates of T1. (b) Reflect T1 in the x-axis to get T2. State the coordinates of T2. (c) Find the single transformation that maps T directly to T2. Prove your answer is correct.
(a) Reflection in y = x swaps coordinates. T1: A(0,0), B(0,2), C(2,0).
(b) Reflection in x-axis changes sign of y. T2: A(0,0), B(0,-2), C(2,0).
(c) The two mirror lines y = x and y = 0 (x-axis) meet at the origin at 45 degrees. The single transformation is a rotation of 2 x 45 = 90 degrees clockwise about the origin. Check: (0,0) stays at (0,0). (2,0) rotates 90 degrees clockwise to (0,-2). (0,2) rotates 90 degrees clockwise to (2,0). All match T2. Verified.
Mark scheme: M1 reflection in y=x, A1 coordinates of T1, M1 reflection in x-axis, A1 coordinates of T2, M1 identifying rotation, A1 90 degrees clockwise about origin, A1 verification
📊 AO3: Reason & Interpret
Reasoning and Interpretation
A student reflects a shape in the line x = 0 (y-axis), then reflects the image in the line y = 0 (x-axis).
(a) Describe the single transformation that is equivalent to these two reflections.
(b) Would the result be the same if the reflections were done in the opposite order?
(c) A third reflection is then applied in the line y = x. The student says "Three reflections must equal one reflection." Is this always true?
Answers: (a) The mirror lines are perpendicular and meet at the origin. The single transformation is a rotation of 180 degrees about the origin (equivalent to point reflection through the origin). (b) Yes — two reflections in perpendicular lines always combine to a 180 degree rotation, regardless of order. (c) No — three reflections can combine to give a reflection or a glide reflection, depending on the lines. For example, reflecting in x = 0, then y = 0, then y = x is equivalent to a single reflection in a different line, not always the case. The student's claim is not always true.
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