Congruency, Enlargement and Combined Transformations · 5.3.2
Commutative law in transformations
This standard asks students to test whether performing two transformations in a different order (T then U, versus U then T) produces the same image. By trying examples with translations, reflections, rotations and enlargements, students form and check a conjecture about when combined transformations commute.
The official learning standard (5.3.2)
“Make and verify the conjecture about commutative law in combined transformation.”
What it means
This standard asks students to test whether performing two transformations in a different order (T then U, versus U then T) produces the same image. By trying examples with translations, reflections, rotations and enlargements, students form and check a conjecture about when combined transformations commute.
How it is examined
Paper 2 typically gives an object and two named transformations, asking students to find the image under each order and state whether they commute. Paper 1 may test the idea with a short conceptual or true/false style item on a specific pair of transformations.
Worked example
Transformation P is a reflection in the x-axis. Transformation Q is a translation by the vector (3, −1).
Point A has coordinates (2, 4). Find the image of A under P followed by Q, and the image of A under Q followed by P.
Hence determine whether P and Q commute.
- P then Q: reflect A(2, 4) in the x-axis to get (2, −4); translate (2, −4) by (3, −1) to get (5, −5).
- Q then P: translate A(2, 4) by (3, −1) to get (5, 3); reflect (5, 3) in the x-axis to get (5, −3).
- Compare the two images: (5, −5) is not equal to (5, −3).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Does a combined transformation always commute?
No. In general the order matters, performing T then U can give a different image from U then T.
Only special pairs, such as two translations, two enlargements with the same centre, or two rotations about the same centre, always commute.
How do I check whether two transformations commute?
Apply both transformations to the same object in one order, then in the reverse order, and compare the two resulting images. If the coordinates (or the position and orientation of the shape) match, the pair commutes; if not, order matters.
Which types of combined transformations do commute?
Two translations always commute because vector addition is commutative. Two enlargements sharing the same centre commute, since their scale factors simply multiply.
Two rotations about the same centre also commute, as their angles simply add together.