Moving the graph to the right -- What do you think?

[0kelvin]

I'm studying calculus alone with textbooks. The part about moving the graphs to the right or to the left struck me because they just have a list of rules, properties and make you relate the graph with the corresponding equation. I know what is the rate of change and I thought I could do better than the textbook.

I vectorized this to explain why: f(x - n) moves the parabola to the right.

Not satisfied I though. f(x - 2) does remind me of the concept of a composite function. Can I draw something to explain this and relate it to the rate of change?

 

[anuttarasammyak]

With a graph described with an equation $f(x,y)=0$ given, another graph
$$f(x-a,y-b)=0$$
is a translation of that graph with vector (a,b) on x-y plane. For an example say (0,0) is on the original graph, it is translated to (a,b) on the new one.

 

[PeroK]

I'm studying calculus alone with textbooks. The part about moving the graphs to the right or to the left struck me because they just have a list of rules, properties and make you relate the graph with the corresponding equation. I know what is the rate of change and I thought I could do better than the textbook.

Take a function that is zero everywhere except the origin:$$f(x)=\begin{cases} 1 & x = 0 \\ 0 & x \ne 0 \end{cases}$$Now define $g(x) = f(x -2)$. Note that $g(2) = f(0) = 1$, hence:$$g(x)=\begin{cases} 1 & x = 2 \\ 0 & x \ne 2 \end{cases}$$And we see that $g(x)$ is $f(x)$ moved to the right.

 

[robphy]

Here's a related example from physics: a traveling wave disturbance on a string.

Suppose a disturbance has a profile F(x) along a string.
[In physicist's notation...]
F(x-vt) describes that disturbance translating (traveling without distortion) to the right with constant velocity v.

At t=0, consider the disturbance at the string location x=1: F(1).
After a time t, F(1)=F(x-vt) where 1=x-vt.
Since t increases, x must increase to keep x-vt=1. (Indeed, x=vt+1.)
...and similarly for other locations.
Thus, the disturbance moves to the right.

See https://www.desmos.com/calculator/bjt6dleg5h
from
https://www.physicsforums.com/threa...mean-in-the-wave-equation.836348/post-5254546