Does anyone know the origin of the notation $(x-h)$ and $(y-k)$ when shifting functions in algebra? Why $h$ and $k$?
It is absolutely arbitrary. Usually people use $h$ and $k$ for conic section equations and their shifts, so then you'll rarely see $\sin(x-k)$ and see instead $\sin(\phi-\psi)$.
For integrals you might see that $u$ and $v$ are preffered, and for complex numbers you'll see $a$ , $b$ and $\rho$ , $\theta$.
For limits you'll see $t$ and $x$ are mostly used, and for trigonometric limits $\theta$ will pop again.
And what about differential equations? Some authors use $y$ and $x$, some use $f$ and $x$, and for systems some use $u$, $v$, while some use $x_1$ and $x_2$ as functions.
It is arbitrary, but we try to keep some convention to make things easier.