Congruency, Enlargement and Combined Transformations · 5.3.1
Images under combined transformations
Students learn to apply two or more transformations, translation, reflection, rotation, or enlargement, one after another in the stated order to find the final image of an object, and to work backwards from a final image to identify the original object.
The official learning standard (5.3.1)
“Determine the image and object of a combined transformation.”
What it means
Students learn to apply two or more transformations, translation, reflection, rotation, or enlargement, one after another in the stated order to find the final image of an object, and to work backwards from a final image to identify the original object.
How it is examined
Paper 2 questions mainly ask students to draw or state the coordinates of an image after applying two given transformations in sequence, such as a translation followed by a reflection. Occasionally, students must find an intermediate or original object from a given final image.
Worked example
Triangle A has vertices A(1,1), B(3,1) and C(1,2). Transformation T is a translation by the vector (2,3), and transformation R is a reflection in the y-axis.
Find the coordinates of the final image after triangle A undergoes T followed by R.
- Apply T (translation by vector (2,3)) to each vertex: A(1,1)→(1+2, 1+3)=(3,4); B(3,1)→(3+2, 1+3)=(5,4); C(1,2)→(1+2, 2+3)=(3,5). This gives the intermediate image A₁B₁C₁.
- Apply R (reflection in the y-axis, where (x,y)→(−x,y)) to A₁B₁C₁: A₁(3,4)→(−3,4); B₁(5,4)→(−5,4); C₁(3,5)→(−3,5).
- Combine the results: the final image A₂B₂C₂ has coordinates A₂(−3,4), B₂(−5,4), C₂(−3,5).
Source:DSKP KSSM Mathematics Form 4 and 5 (Versi English)
Frequently asked questions
Does the order of transformations matter?
Yes, combined transformations are generally not commutative, so performing T then R usually gives a different image from performing R then T. Always follow the exact order stated in the question.
How do I find the object if only the final image and the transformations are known?
Work backwards by applying the inverse of each transformation in reverse order. For example, if the object underwent T then R, apply the inverse of R first, then the inverse of T, to the final image.
What transformations can be combined?
Any of translation, reflection, rotation, and enlargement can be combined in sequence. SPM problems commonly combine two of these, requiring students to find images vertex by vertex using the rules for each transformation in turn.