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Does anyone know the origin of the notation $(x-h)$ and $(y-k)$ when shifting functions in algebra? Why $h$ and $k$?

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Guess: $h$ is sometimes used as a difference, as in the definition of the derivative. Presumably, $k$ was chose because $i$ and $j$ were bad choices, so it was the "next" good candidate after $h4. – Thomas Andrews Dec 7 '11 at 17:39
Random guess: Perhaps it is because the vertex of a quadratic function is related to curvature (a vertex is a place where the instantaneous rate of change of curvature is zero), and Gaussian curvature, for example uses $k_1$ and $k_2$. Rather than $k_1$ and $k_2$, perhaps authors felt they should use $h$ and $k$. More realistically, the choice is just arbitrary. – JavaMan Dec 7 '11 at 17:42
Well, "h" for horizontal, and "k" for, erm... – David Mitra Dec 7 '11 at 17:45

The choice of letters is arbitrary, and different authors use different ones.

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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.

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