Explanation:
first of all, it is important to remember that f(x) stands for y, it calculates the y coordinate for a given x coordinate.
so, now, by comparing the two points
(1, 1)
(2, -7)
what do we need to do to f(x) just in the "outside" (without knowing the inner details of f(x)) to create (2, -7) ?
the first thing we see : the negative y (-7).
so, the first transformation is : -f(x)
that would move the point to (1, -1)
because the same point is now suddenly at x=2 (instead of 1), what do we need to do to the argument of 2 to bring it back to the original 1 (so that the original function can be performed on the original x=1) ? we need to subtract 1.
so, after the second transformation we have : -f(x-1)
that would move the point now to (2, -1)
so, the only thing missing is to bring y from -1 to -7. we need to subtract -6 from the original result.
so, after the third transformation we have : -f(x-1) - 6
and that finally moves the original point to (2, -7)