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, xa, the known corresponding value ya of the dependent variable y closely approximates to the value f (xa). In interpolation it is then assumed that, within the range covered by the known values of the variables, 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.