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.

What does the graph of $g(x) = f(x+1)$ look like?  Well, $g(0)$ is $f(1)$, $g(1)$ is $f(2)$, and so on. Put another way, the point $(1,f(1))$ on the graph of $f(x)$ has become the point $(0,g(0))$ on the graph of $g(x)$, and so on. At this point, drawing an actual graph and showing how the points on the graph of $f(x)$ move
one unit to the left to become points on the graph of $g(x)$ helps the student
understand the concept. Whether the student absorbs the concept well enough
to utilize it correctly later is quite another matter.
