In mathematics, extrapolation is a type of estimation, beyond the original observation range, of the value of a variable on the basis of its relationship with another variable. It is similar to interpolation, which produces estimates between known observations, but extrapolation is subject to greater uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean extension of a method, assuming similar methods will be applicable. Extrapolation may also apply to human experience to project, extend, or expand known experience into an area not known or previously experienced. By doing so, one makes an assumption of the unknown (for example, a driver may extrapolate road conditions beyond what is currently visible and these extrapolations may be correct or incorrect). The extrapolation method can be applied in the interior reconstruction problem.

A sound choice of which extrapolation method to apply relies on a priori knowledge of the process that created the existing data points. Some experts have proposed the use of causal forces in the evaluation of extrapolation methods. Crucial questions are, for example, if the data can be assumed to be continuous, smooth, possibly periodic, etc.

Linear extrapolation means creating a tangent line at the end of the known data and extending it beyond that limit. Linear extrapolation will only provide good results when used to extend the graph of an approximately linear function or not too far beyond the known data.

If the two data points nearest the point ${\displaystyle x_{*}}$ to be extrapolated are ${\displaystyle (x_{k-1},y_{k-1})}$ and ${\displaystyle (x_{k},y_{k})}$, linear extrapolation gives the function:

(which is identical to linear interpolation if ${\displaystyle x_{k-1}<x_{*}<x_{k}}$). It is possible to include more than two points, and averaging the slope of the linear interpolant, by regression-like techniques, on the data points chosen to be included. This is similar to linear prediction.

A polynomial curve can be created through the entire known data or just near the end (two points for linear extrapolation, three points for quadratic extrapolation, etc.). The resulting curve can then be extended beyond the end of the known data. Polynomial extrapolation is typically done by means of Lagrange interpolation or using Newton's method of finite differences to create a Newton series that fits the data. The resulting polynomial may be used to extrapolate the data.

High-order polynomial extrapolation must be used with due care. For the example data set and problem in the figure above, anything above order 1 (linear extrapolation) will possibly yield unusable values; an error estimate of the extrapolated value will grow with the degree of the polynomial extrapolation. This is related to Runge's phenomenon.

A conic section can be created using five points near the end of the known data. If the conic section created is an ellipse or circle, when extrapolated it will loop back and rejoin itself. An extrapolated parabola or hyperbola will not rejoin itself, but may curve back relative to the X-axis. This type of extrapolation could be done with a conic sections template (on paper) or with a computer.