Explaining Horizontal Shifting and Scaling I always find myself wanting for a clear explanation (to a college algebra student) for the fact that horizontal transformations of graphs work in the opposite way that one might expect.
For example, $f(x+1)$ is a horizontal shift to the left (a shift toward the negative side of the $x$-axis), whereas a cursory glance would cause one to suspect that adding a positive amount should shift in the positive direction. Similarly, $f(2x)$ causes the graph to shrink horizontally, not expand.
I generally explain this by saying $x$ is getting a "head start". For example, suppose $f(x)$ has a root at $x = 5$. The graph of $f(x+1)$ is getting a unit for free, and so we only need $x = 4$ to get the same output before as before (i.e. a root). Thus, the root that used to be at $x=5$ is now at $x=4$, which is a shift to the left.
My explanation seems to help some students and mystify others. I was hoping someone else in the community had an enlightening way to explain these phenomena. Again, I emphasize that the purpose is to strengthen the student's intuition; a rigorous algebraic approach is not what I'm looking for.

It may help to try to comprehend the abridged version of Kaz's answer: https://math.stackexchange.com/a/133348/53259. 
Because we want to work with $(x,y)$, the key is to remain focused on the $xy$-axis in black. It truly helps to use new variables for the new coordinate axes. So define $\color{green}{\left\{ \begin{array}{rcl}
s = x + 1 \\
t = y + 3
\end{array}\right.} \iff \left\{ \begin{array}{rcl}
x = s - 1 \\
y = t - 3
\end{array}\right. $.
Thus, this shows the source of the bewilderment. We should remain thinking of the black and not the green.        

