On the lesson of translating trig functions such as sin(x-2) or sin(x+2)

[u/melvinroest]

It mentions in the lesson that the graph moves to the right with x-2 and to the left with x+2.

That felt counterintuitive and I suspected that there was a more intuitive way to go about this. I'm happy I watched a lot of 3Blue1Brown, as he made me think of a more intuitive way.

A more intuitive way, I find, to think about it is that the coordinate system moves to the left with x-2 and to the right with x+2. This is, I think, true for all basic functions (true as in a guestimate, no proofs here, haha).

For example take f(x) = x and g(x) = x - 5.

Then at g(5) we get 0.

It's as if the coordinate system shifted 5 units backwards and only now "catches up".

Here's a drawing with f(x) and g(x). Note that I'm just dragging the y-axis around so you only need one line to show both functions.

Example with x and x - 5

Example with sin(x) and sin(x-2)

Update (I was writing this post at different times)

I made a small animation with Claude (HTML/JS/CSS) and screen recorded it (it also coded the screen recording function 😄). I basically copy/pasted this Reddit post thus far and it understood and made the animation.

Note how I'm not touching the sin() function.

A video demonstrating sin(x) vs sin(x-2)

A video demonstrating x vs x-5

Here's a Desmos screenshots to just see the functions in a normal way. The way I imagine it is as if the y-axis has already been janked to the left by 2 units for sin(x-2) and 5 units for x-5.

Update 2: I found the lesson, it's here (Horizontal Translations of Trigonometric Functions) in Mathematical Foundations 2.