# Origin of the notation $(x-h)$ and $y-k$ in shifting

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$. More realistically, the choice is just arbitrary. – JavaMan Dec 7 '11 at 17:42 Well, "h" for horizontal, and "k" for, erm... – David Mitra Dec 7 '11 at 17:45 ## 2 Answers The choice of letters is arbitrary, and different authors use different ones. - It is absolutely arbitrary. Usually people use$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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