# Lagrange's formula

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## Lagrange's formula

[lə′grān·jəz ‚fȯr·myə·lə]
(mathematics)
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Freeman (1960) notes that "Lagrange's formula is usually laborious to apply in practice" and recommends instead using other finite difference interpolation formulae.
By contrast, Lagrange's formula can be explained in minutes using only high school mathematics, and requires only the coordinates of points.
At the point ([x.sub.1], [y.sub.1]), Lagrange's formula states:
If you appreciate that pattern, it is straightforward to extend Lagrange's formula to find a cubic polynomial passing through four given points, or any higher order scenario you require.
Using the first three points, Lagrange's Formula gives
Lagrange's Formula does not require that the three x-values be evenly spaced, so students can be asked to pick any three of the five points and verify that they can reach the same answer.
A nice algebraic challenge for students is to derive Lagrange's formula for the linear case from this result.

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