Question

Which sequence of transformations will change figure PQRS to figure P′Q′R′S′ on the given coordinate grid?

Figure PQRS has P at the ordered pair (1, 2), Q at (2, 1), R at (3, 2), and S at (3, 3).
Figure P′Q′R′S′ has P′ at the ordered pair (1, -2), Q′ at (2, -1), R′ at (3, -2), and S′ at (3, -3).
Which sequence of transformations accomplishes this transformation?

A) Counterclockwise rotation about the origin by 180 degrees followed by reflection over the x-axis
B) Counterclockwise rotation about the origin by 180 degrees followed by reflection over the y-axis
C) Counterclockwise rotation about the origin by 270 degrees followed by reflection over the x-axis
D) Counterclockwise rotation about the origin by 270 degrees followed by reflection over the y-axis

Answer 1

The correct sequence of transformations to change figure PQRS to figure P′Q′R′S′ on the given coordinate grid is option D) Counterclockwise rotation about the origin by 270 degrees followed by reflection over the y-axis.

Step-by-step explanation:

First, let's analyze the transformations needed to change the original figure PQRS to P′Q′R′S′. There's a change in the y-coordinates of the points from positive to negative values while the x-coordinates remain the same. This indicates a reflection over the x-axis. Additionally, the points need to be rotated 180 degrees counterclockwise about the origin.

Option D involves a counterclockwise rotation of 270 degrees (which is equivalent to 180 degrees counterclockwise plus another 90 degrees counterclockwise) followed by reflection over the y-axis. This sequence of transformations precisely matches the required changes in the coordinates from PQRS to P′Q′R′S′.

A 270-degree counterclockwise rotation flips the figure over the x-axis and then shifts it one more quadrant counterclockwise. After this rotation, a reflection over the y-axis will match the negative y-coordinates of the transformed figure. Therefore, D) Counterclockwise rotation about the origin by 270 degrees followed by reflection over the y-axis is the correct sequence of transformations to convert PQRS to P′Q′R′S′ on the coordinate grid.