# interpolate and extrapolate

To interpolate is to estimate the value of a point that lies *between* two known values of a function; this is
often done by approximating a line or a smooth curve between the values, which is the literal
meaning of the word. *Inter* is the Latin prefix for "between," *polire* translates as "to adorn or polish," so together they mean "to smooth between." **Extrapolate** was created as an extension of interpolate
to suggest the smoothing of a line *outside* the known points. This
operation is often done in statistics when patterns are studied over time
to predict future events.

Interpolation and extrapolation are techniques used in mathematical analysis to estimate undetermined values of a dependent variable, a number of values
of which, corresponding to determined values of an independent variable,
are known. This is done by finding a function *f (x)* of the independent
variable *x* such that, for any value, *x _{a}*, the
known corresponding value

*y*of the dependent variable

_{a}*y*closely approximates to the value

*f (x*. In interpolation it is then assumed that, within the range covered by the known values of the variables,

_{a})*y = f (x)*for all intermediate values of

*x*and

*y*. In extrapolation, one assumes this relationship to hold outside the range of known values – a rather less justifiable assumption. The simplest (and most commonly used) technique is that of drawing the best straight line or curve through a set of points on a graph, and assuming that it represents a genuine relationship between the two variables in question. Others include the use of partial difference equations.