Question

Analyzing Student Work about a Sequence of Transformations
An image is 2 units to the left of the pre-image.

Santana says the image is formed from the pre-image by a reflection across the y-axis followed by a translation 2 units to the right, in that order. Is Santana’s thinking correct?

Answer 1

No — Santana is not correct.

Let a pre-image point be (x,y). Reflecting across the y-axis gives (-x,y); translating 2 units to the right gives (-x+2,y). For the image to be 2 units left of the pre-image we would need (-x+2,y) = (x-2,y) for all x, which implies 2x = 4 and so x = 2. That holds only for points on the line x = 2, not in general.

Counterexample: take pre-image (0,0). Santana’s sequence gives (0,0) → (0,0) → (2,0), which is 2 units right, not 2 units left. The transformation "image 2 units left of the pre-image" is simply a translation 2 units left, not a reflection across the y-axis followed by a right translation.