A direct application of the factor theorem is Lagrange interpolation .
Let a natural number and look at measuring points , , , with the first coordinates different from each other. Then there exists exactly one polynomial function of degree less than or equal to such that the graph of the polynomial function passes through the measuring points (so for ). The interpolation formula can be written as
In an earlier section we already saw how such a function can be constructed for , the straight line two two points: first you make for and a first degree that satisfies and for ). The formula for this is: The requested function is then the sum .
Given two points and , what is the function rule for which the graph goes through these two points?
Lagrange interpolation immediately gives a function definition: Check that and .
So the function definition is The vertical intercept is .
The graph of this function is drawn together with the two given points in the figure below.
Below is an example of a quadratic polynomial function passing through three given points.
Given three points , and , what is the function definition in case the graph of goes through these three points?
The quadratic function that has value in and that has in and value is .
The quadratic function that has value in and that has in and value is .
The quadratic function that has value in and that has in and value is .
The sum of these three quadratic functions is the requested function definition:
In the diagram below you can create points by clicking their positions in the coordinate plane. The graph of the polynomial function passing through these points is then drawn, provided the horizontal coordinates of the specified points are different from each other and no more than six measurement points are specified.