You start with a graph of

y = g(x)  

As an example I'll use y = |x|

Sometimes you do something to x and sometimes you do something to the whole
right side.  When you do something to the whole right side, it does something to
the graph LIKE you think it should be, but when you do something to x only, it
does just the opposite of what you think it should do.

The first thing is doing something to the whole right side, which is multiplying
the whole right side by -1, getting

y = -g(x) 

The example is y = -|x|

This reflects the graph across the x-axis so that things above the x-axis go
below the x-axis and things below the x-axis go above the x-axis.  That's like
you would think it should be, because it's done to the whole right side, not
just to x.

The next thing is to replace x by 2x.  This is just doing something to x, giving

y = -g(2x)  

The example is y = -|2x|



Normally you would think that multiplying x by 2 would make the curve twice as
wide, but it's doing something to just x, so just the opposite is true!  It
makes the curve narrower by one half.

Finally you are going to do something to the whole right side, which is to add 2
the whole right side. And it does just what you'd think it would.  It shifts the
graph up 2 units like this, like you'd expect. 

y = -g(2x)+2

The example is y = -|2x|+2
  

Do something to the whole right side of the equation, it does just what you'd
think it does.

Do something to just x, it does the EXACT OPPOSITE of what you'd think it should
do.
 
If you were to replace x by x-3 (Subtract 3 from x) it would moves the curve
right 3 units. You would think it would move it left, but when you do something
to x only, it does the opposite from what you'd expect.  

If you replace x by x+3 (Add 3 to x) it moves the curve left 3 units.  You would
think it would move it right, but when you do something to x, it does the
opposite from what you'd expect. 

Edwin