Final Answer:
The plotted points after reflecting ABC over the x-axis, translating by 2 units to the right and 1 unit down, and rotating 180 degrees about the origin are A'(2, 1), B'(4, 1), and C'(0, 3).
Explanation:
Reflecting a point over the x-axis involves changing the sign of the y-coordinate while keeping the x-coordinate unchanged. Thus, the new coordinates for A (2, 1) become A' (2, -1), for B (4, 1) become B' (4, -1), and for C (0, -3) become C' (0, 3).
Next, to translate the points by 2 units to the right and 1 unit down, you add 2 to each x-coordinate and subtract 1 from each y-coordinate. This yields the new coordinates: A'(2, -1) + (2, -1) = A'(4, -2), B'(4, -1) + (2, -1) = B'(6, -2), and C'(0, 3) + (2, -1) = C'(2, 2).
Finally, rotating points 180 degrees about the origin entails taking the negative of both x and y coordinates. Applying this operation gives A'(4, -2) as A'( -4, 2), B'(6, -2) as B'(-6, 2), and C'(2, 2) as C'(-2, -2).
Plotting these final coordinates A'( -4, 2), B'(-6, 2), and C'(-2, -2) on the grid results in the image of the reflected, translated, and rotated triangle A'B'C'. This sequence of transformations—reflection, translation, and rotation—produces the final positions of the points on the grid, depicting the geometric changes applied to the original triangle ABC.