A counterclockwise quarter-turn Q about the origin is applied to the point (x, y). What are the coordinates of the image point? Answer in the form Q(x, y) = (ax + by, cx + dy).

Answer 1

If by a quarter-turn you mean a counterclockwise roation of 90°, then that would simply be

(x,y) ----> (-y,x)

You can illustrate that by drawing a right-angled triangle in the first quadrant, with sides +x and +y
Now do a 90° rotation to put the triangle in the second quadrant.
Notice that the base is now y, but of course it is going to the left, so -y
and the height is the original x of the first triangle.