What is the image point of (-4,6)(−4,6) after the transformation r_{y=x}\circ R_{270^{\circ}}r y=x ​ ∘R 270 ∘ ​ ?

Answer 1

The 1st transformation that occurs is reflecting the point over the line y = x. So, you can graph this to help you visualize what's happening, which I recommend.

Think of it like this: if we reflected this point (-4, 6) over the x-axis, we would end up at the coordinates (-4,-6) - only the y-value changes & it becomes the opposite of 6. If we reflected it over the y-axis, we get the point (4,6) - only the x-value changes & it becomes the opposite of -4. Applying both of these changes is the same as reflecting it over the line y=x. So, if our original point is (-4,6) then after the reflection, it is (4,-6).

Now, the 2nd transformation is to rotate our new image/point 270 degrees about the origin. Each time we rotate a point 90 degrees here is what happens:

(X,Y) rotated 90 degrees becomes (-Y,X)

So, we start with (4,-6) 270 degrees is 90 degrees 3 times over.

(4,-6) rotated 90 degrees is [-(-6), 4] or (6,4)

(6,4) rotated 90 degrees is (-4,6)

(-4,6) rotated 90 degrees is (-6,-4)

After both transformations, we land at point (-6,-4)