Question

Darien drew a quadrilateral on a coordinate grid. He rotated the figure 180 degrees and then translated it left 4 units. What are the coordinates of the image of point P?

Answer 1

If we take a point (x,y) and rotate it 180 degrees about the origin, the two coordinates change sign. That is, the point (x,y) becomes (or moves to) (-x,-y).

If you then take that point which is now (-x,-y) and translate it (slide it over) 4 units left you are taking the x-coordinate and moving it left 4 which is taking the x-coordinate and subtracting 4. The point (-x,-4) when you subtract 4 from the -x becomes (-x-4,-y)

So the two transformations (rotation of 180 degrees followed by a translation 4 units left) take a point (x,y) and move it to its image (-x-4,-y)

Now we are asked for the image of point P. The coordinates of P can be found from the diagram. If you start at the origin (where the x and y axis cross or the center of the plane you are given) to travel to P you move 5 units right and 2 units up. That is, the coordinates of P are (5,2)

So for P (x,y) = (5,2)

The image of P is (-x-4,-y) which is (-5-4, -2). That is, the image of P is (-9,-2)