A quadrilateral with points A(1,4), B(5,4), C(4,3), and D(2,2) is reflected about the y-axis to form quadrilateral A'B'C'D'. Which of these sequences of transformations will also carry the quadrilateral ABCD onto A'B'C'D'? A. Rotating the quadrilateral 90 degrees clockwise about the origin and then reflecting it across the x-axis B. Rotating the quadrilateral 180 degrees about the origin and then reflecting it across the x-axis C. Rotating the quadrilateral 90 degrees counterclockwise about the origin and then reflecting it across the line y=x D. Rotating the quadrilateral 180 degrees about the origin and then reflecting it across the line y=x

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We are asked to find a transformation from the list that is equivalent, in the sense that it maps the given quadrilateral to the same image, as a reflection over the y-axis. We are given quadrilateral ABCD with A(1,4), B(5,4), C(4,3), and D(2,2).

Note that a reflection over the y-axis maps each point (x,y) in the plane to (-x,y). (This can be written as `R_y:(x,y)->(-x,y)` .) Thus we know the image of the transformation: A'(-1,4), B'(-5,4), C'(-4,3), and D'(-2,2).

B is the correct answer.

A composition of transformations is a sequence of transformations performed from right to left. Compositions are, in general, not commutative, so order matters.

Consider the alternatives:

A....

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