What is a transformation that flips a graph or figure across a line?

To flip f(x) across the x-axis simply multiply the right side of f(x)
by -1. That is, form g(x) = -f(x) 

To flip f(x) across the y-axis simply substitute -x for x in the right
side of f(x). That is, form g(x) = f(-x)

To flip f(x) across the horizontal line y = k, replace x by 2k-x,

that is, form g(x) = f(2k-x)

To flip f(x) across the vertical line x = k, subtract the right side
of f(x) from 2k, that is, form g(x) = 2k - f(x)

To flip f(x) across the line y = x, replace f(x) on the left by y,
interchange x and y, then solve for y.  Then replace y by g(x).

Flips across other lines are a lot more complicated but can be done.
Post again if you need to do that.

Edwin