KBAT Problems
KBAT Problem: A Combined Transformation
A KBAT problem where a shape is moved by two transformations in turn, order matters, and you must describe the single transformation that does the same job.
Apply them in order
A shape is reflected, then rotated. Carry out the first transformation to get an intermediate image, then the second on that image.
Doing them in the stated order, not swapping, is where care earns marks.
Describe the single equivalent
The KBAT part asks for the one transformation with the same overall effect. Compare the final image to the original and describe it fully, type, and every detail such as centre, angle or axis.
A description missing a detail loses the mark.
Understand the problem
A shape is transformed by P then Q, where P is a reflection in the y-axis and Q is a reflection in the line x = 3. Take the point A(1, 2).
Find its image after P then Q, and describe the single transformation with the same effect. What it really asks: apply the two in the stated order, then recognise the combined result as one named transformation with all its details.
Plan and solve
- Apply P first. Reflecting A(1, 2) in the y-axis sends (x, y) to (−x, y), giving A′(−1, 2).
- Apply Q to A′. Reflecting in x = 3 sends (x, y) to (6 − x, y), so A′(−1, 2) goes to A″(6 − (−1), 2) = (7, 2).
- So overall A(1, 2) maps to A″(7, 2): the point has moved 6 units in the positive x-direction with y unchanged.
- The single equivalent transformation is a translation by the vector (6, 0).
Check and a variant
Check with a second point so you are describing the whole transformation, not one lucky point. Take B(2, −1): P gives (−2, −1), then Q gives (6 − (−2), −1) = (8, −1), again a move of 6 in x, confirming the translation (6, 0).
A variant the examiner could add: do Q then P instead. That sends (x, y) to (x − 6, y), a translation by (−6, 0) in the opposite direction, proof that the order of transformations changes the result.
Frequently asked questions
Why do two reflections combine into a translation?
When the two mirror lines are parallel, reflecting twice slides every point across in the same direction, which is exactly a translation. The distance moved is twice the gap between the lines: here the lines are 3 apart, so the shift is 6.
If the mirror lines intersect instead, two reflections combine into a rotation.
Does the order of the two transformations really matter?
Usually yes. As the variant shows, P then Q gives a shift of +6 while Q then P gives −6, opposite directions.
Only in special cases do two transformations give the same result either way. Always apply them strictly in the order stated, and never swap them to make the arithmetic feel easier.
How complete must my description of the single transformation be?
Fully complete, or you lose the mark. Name the type, then give every defining detail: a translation needs its vector, a rotation needs centre, angle and direction, a reflection needs its axis, and an enlargement needs centre and scale factor.
A description missing even one detail is treated as incomplete, however correct the rest is.