Is this visual explanation of horizontal shifting at least helpful ? ( …if not beautiful…)

With the image below I try to explain in which way substituting (x-a) ( with a> 0) for x in the expression defining a function results in a shift to the right, although " intuition" tells us it should result in a shift to the left. To do this I use the auxiliary idea of "shifting" as "copying" or "imitating".

I would be interested in knowing whether this explanation could be efficient in the classroom at the high school level.

Suppose we have $$y=f(x)=x^2$$ and want to graph $$y=g(x)=f(x-1).$$

The function $$g$$ maps each $$x$$ to the image of $$(x-1)$$ under the function $$f$$.

In other words, each $$x$$ value has an "$$x-1$$ " ( his own "$$x-1$$" ) , and copies his (x-1) 's image under the "old" mapping $$f$$ .

Since each "imitator" ( each $$x$$ value) is to the RIGHT of it's "model", that is it's " $$x-1$$" , the change from the function $$f(x)=x^2$$ to the function $$g(x)=f(x-1)$$ results , for each "old" point of the graph of f, in a translation of $$1$$ unit to the RIGHT. Remark. - I first used the idea of " stealing" , for which I substitute the idea of " copying" or "imitating". I've just seen a post of Hyperpallium using the idea of " sampling" , even better.

Link to Hyperpallium's post : *Seeing* why horizontal shifts are reversed?

• This link is a good post explaining this concept, because at the moment your explanation is overcomplicating the situation quite alot – Hushus46 Mar 25 at 21:46
• I'm voting to close this question as off-topic because it is about teaching mathematics (Math Educators SE) and not based on an understand, or lack thereof, of various mathematical fields. – Chase Ryan Taylor Mar 25 at 21:50
• @ Hushus46 . Thanks for the link. – Ray LittleRock Mar 25 at 21:53

Yes it moves the whole plane by 1 in x dimension. I would like to call it swiping instead of stealing. Nothing is taken, it's just shuffled around a bit.

Imagine swiping a whole image on a mobile phone screen 1 step right but none in up-down direction.

It is a good insight. First step towards geometric transformations, rotations, projections.

Your suggestion is nothing deeper than this: Ignore all the confusing suggestions about "stealing." Certainly not helpful. Certainly not beautiful.

• This drawing is self explanatory. Thanks! – Ray LittleRock Mar 26 at 15:41
• Hmmm.... not even an upvote? – David G. Stork Mar 26 at 17:31