Animations of the Graphical Transformations

All the vertical transformations of y=f(x) (including reflecting about the x-axis, which is -f(x)) involve modifying what is happening outside the function!
All the horizontal transformations of y=f(x) (including reflecting about the y-axis, which is f(-x)) involve modifying what is happening inside the function!
Vertical translation and vertical stretch are the easiest to remember, so I suggest you remember the other ones by relating them to those two. For example, y=f(3x) would be a horizontal (since it modified the inside) compression (since it does the opposite of what the vertical would do) of the graph of y=f(x) by a factor of 3.

Vertical Translation	Vertical Stretch	Vertical Compression
Dash: f(x) Solid: f(x)+a, a from 0 to 5 Note: Adding a>0 outside function moves the function up. Subtracting would move down.	Dash: f(x) Solid: a f(x), a from 1 to 5 Note: Multiplying by a>1 outside function stretches the function vertically.	Dash: f(x) Solid: a f(x), a from 1 to 0 Note: Multiplying by 0<a<1 outside function compresses the function vertically.

Horizontal Translation	Horizontal Stretch	Horizontal Compression
Dash: f(x) Solid: f(x+a), a from 0 to 5 Note: Adding a>0 inside function moves the function to the left. Subtracting would move to the right.	Dash: f(x) Solid: f(ax), a from 1 to 0 Note: Multiplying by 0<a<1 inside function stretches the function horizontally.	Dash: f(x) Solid: f(ax), a from 1 to 5 Note: Multiplying by a>1 inside function compresses the function horizontally.