5.3 Combined Transformation
A combined transformation involves performing more than one transformation on an object to produce an image. The order of notation is written from right to left, where the combined transformation \(AB\) means transformation \(B\) is carried out first, followed by transformation \(A\).
Learning Objectives
- Determine the image and object of a combined transformation.
- Make and verify the conjecture about the commutative law in combined transformations.
- Describe combined transformations and determine an equivalent single transformation where possible.
- Solve problems involving combined transformations including the relationship between area of object, scale factor, and area of image \(\left(\text{Area of image} = k^2 \times \text{Area of object}\right)\).
Flow of Combined Transformation \(AB\) vs Finding the Object
Forward mapping follows \(B \to A\), while reverse mapping to find the original object follows \(A^{-1} \to B^{-1}\).
Forward: Find the Image
Backward: Find the Object
Brief Explanation
Order of Combined Transformation
The combination of transformation \(A\) and transformation \(B\) written as \(AB\) means: transformation \(B\) followed by transformation \(A\). Similarly, \(P^2\) means transformation \(P\) is performed twice in succession (\(P\) followed by \(P\)).
Commutative Law
Combined transformation \(AB\) satisfies the commutative law if the image under \(AB\) and the image under \(BA\) are the same. Most combined transformations do not satisfy the commutative law (\(AB \neq BA\)).
| Transformation | Inverse / Reverse Operation |
|---|---|
| Translation \(\begin{pmatrix} a \\ b \end{pmatrix}\) | Translation \(\begin{pmatrix} -a \\ -b \end{pmatrix}\) |
| Rotation of \(\theta^\circ\) clockwise | Rotation of \(\theta^\circ\) anticlockwise about the same centre |
| Enlargement with scale factor \(k\) | Enlargement with scale factor \(\frac{1}{k}\) about the same centre |
| Reflection across a line | Reflection across the same mirror line |
Try Answering First
Read the question, answer it in your mind, then press “Check Answer”.
What does the combined transformation \(MN\) mean in terms of execution order?
If transformation \(P\) is an enlargement with scale factor \(2\) at centre \((8, 0)\), what transformation is used to find the object directly from its image?
An object with an area of \(5\text{ cm}^2\) undergoes an enlargement with scale factor \(k = 3\). What is the area of its image?
Common Mistakes
- Performing transformations from left to right for \(AB\) (doing \(A\) first instead of \(B\)).
- Forgetting to reverse the order of operations when finding the object from an image under combined transformation \(AB\) (must undo \(A\) first, then undo \(B\)).
- Changing the mirror line when undoing a reflection (a reflection simply reflects back across the exact same line).
- Forgetting that scale factor is squared when calculating the area of the image (\(\text{Area} = k^2 \times \text{Area of object}\), not \(k \times \text{Area}\)).
Misconceptions
Combined transformations are always commutative (\(AB = BA\)).
Most combinations yield completely different images when the order is switched.
An enlargement with a negative scale factor cannot be equivalent to a combined transformation.
A \(180^\circ\) rotation followed by an enlargement can be equivalent to a single enlargement with a negative scale factor.
Summary
- Order of Execution: Combined transformation \(AB\) means transformation \(B\) followed by transformation \(A\).
- Finding the Object: Reverse the sequence and use opposite transformations: undo \(A\) first, then undo \(B\).
- Commutative Law: Valid only if image of \(AB\) equals image of \(BA\); generally false for most transformation pairs.
- Describing Combined Transformations: State transformation \(B\) in full, then state transformation \(A\) in full.
- Area Calculation: \(\text{Area of image} = k^2 \times \text{Area of object}\).
Short Activity
Complete all parts first, then press the check answer button.
A. Objective Quiz
1 Which statement correctly describes the execution of the combined transformation \(XY\)?
2 Under combined transformation \(UV\), if \(U\) is a translation and \(V\) is a reflection, which transformation is applied first to find the object when the final image is given?
B. Fill in the Blanks
3 Combined transformation \(AB\) satisfies the commutative law if the images produced by \(AB\) and \(BA\) are .
Answer for question 34 To determine the object from an image under an enlargement with scale factor \(k = 4\), perform an enlargement with scale factor at the same centre.
Answer for question 4C. Drag and Drop: Match Each Transformation with Its Reverse Operation
Drag each transformation to the correct reverse operation. On a phone, tap the answer choice first, then tap the matching box.
