Geometric Transformations of Points

In this problem, we need to find the final image of the point negative one comma two after it undergoes two successive transformations: first a translation, and then a rotation.

Let's identify our starting point and the first transformation. The point is at negative one comma two, and we are translating it by the vector one comma negative one.

To perform a translation, we simply add the transformation coordinates to the original point's coordinates. So, for the x coordinate, we have negative one plus one.

And for the y coordinate, we have two plus negative one, or two minus one.

This gives us the intermediate point, P prime, at zero comma one. Let's keep that in mind as we move to the next step.

Now for Step 2. The problem mentions a 'rotation' applied to the translation, but specifies coordinates two comma one. Looking closely at the question, it asks for the image of the point after translation followed by the rotation shown as choice a, b, c, or d. Wait, let's re-read carefully.

The text says 'followed by the translation (2, 1)'. It seems there is a typo in the prompt versus the visual options. Let's follow the printed question text: image of (-1, 2) by translation (1, -1) followed by translation (2, 1).