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. `R_x @ R_(O,-90^@)` A clockwise rotation of 90 degrees about the origin (note the negative sign) followed by a reflection over the x-axis. This composition maps points from the first quadrant to the first quadrant (e.g., A(1,4) gets sent to (4,-1) by the rotation and then to (4,1) by the reflection). This is equivalent to reflecting over the line y=x.

C. `R_(y=x)@R(O,90^@)` A counterclockwise rotation of 90 degrees about the origin followed by a reflection over the line y=x. This composition sends points from the first quadrant to the fourth quadrant (e.g., A(1,4) gets mapped to (-4,1) by the rotation and then to (1,-4) by the reflection). This is equivalent to a reflection over the x-axis.

D. `R_(y=x)@R_(O,180^@)` A rotation about the origin of 180 degrees (or a half-turn about the origin) followed by a reflection over the line y=x. This composition maps points from the first quadrant to the third quadrant (e.g., A(1,4) gets mapped to (-1,-4) by the rotation and then to (-4,-1) by the reflection).

B is the only composition given that maps first quadrant points to the second quadrant. `R_x @ R_(O,180^@)` is a half-turn followed by a reflection over the x-axis (e.g., A(1,4) gets sent to (-1,-4) by the rotation and then to (-1,4) by the reflection). You should check the other points.

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