# Describe Extrapolation and Interpolation. How are the terms related to a graph ?

First of all, both intrapolation and extrapolation are processes of approximation or estimation -- that is, these are both processes that give us ways to know to a certain extent, or approximate, certain values that we have not directly measured.

Intrapolation means to find a value that is intermediate, or between, at least two known values. For instance, let's take a simple example. If you know have the following values:

x   |  f(x)

0   |  0.223

1   |  0.494

2   |  0.76

3   |  1.05

and you want to know the value of f(x) when x = 2.5, then you have to do intrapolation. There is no 2.5 in your data set, but 2.5 is a value beterrn 2 and 3. Since it's an intermediate value, you can simply use the behavior of the graph at x = 2 and x = 3 to 'guess' or approximate the value at x = 2.5.

Extrapolation on the other hand, would be essentially the same principle but it would be approximation for values beyond what you have. Beyond here may means values below your starting point, or higher than your highest value. Using the same example, this could mean predicting f(x) for x = -4 or x = 5 -- both values are beyond the x values that you have from your data set (0 to 3).

In summary, both intrapolation and extrapolation are methods of approximation of certain values using a given data set or function. Intrapolation means using intermediate values (those between certain points in your set), while extrapolation means using values beyond your set (or domain if we're talking about functions).

Graphically, intrapolation would be approximating the x and y coordinates of an arbitrary point in your graph. On the other hand, extrapolation would mean extending your graph beyond (to the left or right) its domain, trying to approximate / predict how it would behave in those areas. (A sample image to illustrate this is attached in the references)