A vertical transformation is doing something to y, the vertical axis.
A horizontal transformation is doing something to x, the horizontal axis.

When we do a vertical transformation, we do something to y, and y equals the
whole right side of the equation. So we do something to the WHOLE FUNCTION,
which is the WHOLE RIGHT SIDE of the equation for y.
 
Vertical transformations are done "the way they seem". 
Something is done to the WHOLE function, not to x only.

To shift a graph upward, we ADD to the WHOLE function, which is the WHOLE RIGHT
SIDE.
To shift a graph downward, we SUBTRACT from the WHOLE function, the WHOLE RIGHT
SIDE.
To stretch a graph vertically, we MULTIPLY the WHOLE function, the WHOLE RIGHT
SIDE, by a number > 1.
To compress a graph vertically, we MULTIPLY the WHOLE function, the WHOLE RIGHT
SIDE, by a number < 1
To reflect a graph in (or across) the x- axis, this is a vertical 
transformation
(of flipping the graph vertically, we multiply the WHOLE function, the WHOLE
RIGHT SIDE by -1.

Those seem logical.  But horizontal transformations are "reversed from the way
they seem."  That's because we are compensating.
Something is done to x only.  That is, we replace x by something.   

To shift a graph to the RIGHT, we SUBTRACT from x only. 
To shift a graph LEFT, we ADD to x only.
[Those two seem backward, don't they? But they're not!]
To stretch a graph horizontally, we MULTIPLY x only by a number < 1. 
To compress a graph horizontally, we MULTIPLY x only by a number > 1.
[Those two also seem backward, don't they? But they're not!]
To reflect a graph in (or across) the y- axis, this is a vertical 
transformation (of flipping the graph horizontally, we multiply x only by -1.
[This is the only one that doesn't "seem" reversed].

This is a horizontal compression, so it's "the reverse of what it seems".
So we multiply x only by a number greater than 1.  Hey, but 4/9 is less than 1.
Its reciprocal of a number less than 1 is a number greater than 1, so we
multiply x only by the reciprocal of 4/9, which is 9/4.  So we replace x only by
9/4x

 

This is a vertical stretch, so it is "as it seems", we multiply the entire
function by a number greater than one.  9/8 is greater than 1, so we multiply
the entire function (the whole right side) by 9/8:

To flip the graph about the y-axis is a horizontal change (swapping the right
and left quadrants.) So we multiply x only by -1.

     

     <--answer

[If you would like more instruction to understand the reasoning behind why it's
the opposite of "what it seems" for horizontal transformations, just ask me in
the thank-you note form below, and I'll get back to you by email.  But most
likely, you'll just take my word for it.] J

Edwin