Suppose x is substituted into the function y=f(−x)−2. The order of operations tells us what happens to x, in our case the order is, −1 multiplies it , f is taken of it , −2 is added to it. In this case there are two transformations to the original graph.
The f(−x) part of the equation means that to get the same y value as the original graph the new x value will have to be the negative of what it was before (so that when you multiply the new x value by −1 you get the old one).
Since the x value is displayed in the horizontal direction, this means we stretch or scale horizontally by a factor of −1.
Notice that x is modified twice before the function f gets to it. If we want the same y value as the original graph we need to feed f with the same input. Let xo be the old x value and xn be the new x value. To get the same y values we want f(xo)=f(−xn+0), that is, we require xo=−xn+0. Rearranging we see xn=−xo+0.
Method one (shift then scale)
The equation xn=−xo+0 tells us to take the old x then multiply this result by −1.
Method two (scale then shift)
We could rewrite xn=−xo+0 as xn=−xo−0. This equation tells us to take the old x value, multiply it by −1, then
The f(x)−2 part of the equation means that the y value of each point on the graph will be 2 units less than they were before. Since the y value is displayed in the vertical direction, this means we shift vertically by −2 units.