![horizontal shift horizontal shift](https://image3.slideserve.com/6336768/example-horizontal-shifts-l.jpg)
That doesn't mean that there's nothing to teach here. It's almost closer in character to a scientific fact than a mathematical fact. Ultimately, I don't think this has a simple conceptual explanation - it's just a phenomenon that you observe when you start drawing graphs. After you draw the graphs, you can observe that one is obtained by translating the other. $$\cdots \quad (-3\ ) \quad \color$ by plotting the corresponding points. Again, I emphasize that the purpose is to strengthen the student's intuition a rigorous algebraic approach is not what I'm looking for.Īs described in the link, to plot a graph, simply imagine the $x$-axis covered in coconuts, one for every $x$ value, like this: I was hoping someone else in the community had an enlightening way to explain these phenomena. My explanation seems to help some students and mystify others. Thus, the root that used to be at $x=5$ is now at $x=4$, which is a shift to the left. 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. For example, suppose $f(x)$ has a root at $x = 5$. I generally explain this by saying $x$ is getting a "head start". Similarly, $f(2x)$ causes the graph to shrink horizontally, not expand. 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.įor 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.