Final thoughts. A geometry-based approach.

Certain formulas become famous because they accurately describe the relationship between the x-coordinate and the y-coordinate on certain shapes AS MEASURED FROM THE ORIGIN.

If you go and move your shape away from the origin to a new place where it doesn't want to be, then you have to describe the relationship in terms of distances to the moved origin.

This approach has the benefit of being non-verbal and giving geometric (concrete?) meaning to the expression x-a.

Final thoughts. A geometry-based approach.

Certain formulas become famous because they accurately describe the relationship between the x-coordinate and the y-coordinate on certain shapes AS MEASURED FROM THE ORIGIN.

But students should probably just plot lots of points and learn the theorem through many repeated direct experiences with the chart data (which do not lie!). Then, after the comfort level is high, for those asking why, there's this kind of lesson. But I don't think it's a good intro to the topic.

In practice, it's just something you get comfortable with or memorize. When confronted with f(x-3), you kinda just plug in 3 for x, compute f(0), plot that and just go with the flow. The graph has already moved to the right, it's happening before my very eyes, which is exactly what I expected.

But IMO this isn't actually "helpful" in the traditional sense. I don't recommend that students use this sort of reasoning in practice. It's a rhetorical thing: math DOES make sense. It is possible to understand everything. You don't have to just memorize it as a counterintuitive trick.

With f(x-3), we subtract 3 from the clock time (move the clocks back, so to speak), so everything happens the same way, but 3 hours LATER. It's a delay. Sorry, everyone, the plane is delayed 3 hours, we can all relax for awhile. Just subtract 3 from the clock time, and follow the old plan.

With f(x+3), we add 3 to the clock time (move the clocks forward, so to speak), so everything happens the same way, but 3 hours EARLIER. Suddenly we need to leave the house 3 hours earlier than we thought! Quick, advance the clock 3 hours and then follow the old plan!

Read this and discussed with a few friends. I appreciate what you're doing, but found the old/new notation confusing.

My approach was to bring time into it.

f(x) is a plan to do certain y-values at certain TIMES.

Here's the video version:
youtu.be/h0anjfhfkjs?...