A few years ago we saw how we could approximate a function ( and ( by linear and cubic spline interpolation which connect them with straight lines and cubic polynomials respectively, the latter of which yield smooth curves at the cost of somewhat arbitrary choices about their exact shapes.

An alternative approach is to construct a single function that passes through all of the points and, given that values at distinct , it's tempting to use them.

*f*between pairs of points*x*,

_{i}*f*(

*x*))

_{i}*x*,

_{i+1}*f*(

*x*))

_{i+1}An alternative approach is to construct a single function that passes through all of the points and, given that

*n*^{th}order polynomials are uniquely defined by*n*+1

*x*

_{i}