If the argument x of a function f(x) is replaced by x-|Dx|,
then the new function f(x-|Dx|) is the old function
f(x) shifted by |Dx| to the right.

Any traveling wave moving at constant speed v can be written in the form of
f(x-vt) or f(x+vt). If you cannot write a function in the form of f(x-vt) or f(x+vt), then the function is not a traveling wave.

Shifting a function along the x-axis

If the argument x of a function f(x) is replaced by x+|Dx|,
then the new function f(x+|Dx|) is the old function
f(x) shifted by |Dx| to the left.

To have the shape of a function f(x) move to the right at
constant speed v, replace the argument of the function x by x-|Dx|
= x-vt.
The form of the function f(x-vt)
moves to the right at constant speed v.
The form of the function f(x+vt)
moves to the left at constant speed v.