How does interpolation estimate values between known data points?

In numerical analysis, interpolation is the process of estimating the value of a function $f(x)$ at an unknown point $x$ by using a set of known data points $(x_i,y_i)$ where $x_0<x<x_n$.1

The core of this method involves determining a simpler, more tractable function—such as a polynomial—whose graph passes through all the given data points. By constructing this interpolating function $p(x)$ such that $p(x_i)=y_i$, one can estimate the value of $f(x)$ at any point between the known data.12

While polynomials are the most common choice, interpolation can also be performed using other functions, including:
* Rational functions: Polynomials divided by polynomials.2
* Trigonometric polynomials: Useful for periodic data.2
* Spline functions: Created by connecting several polynomial functions at their endpoints, which are often preferred over simple polynomials when the number of data points is large.2

Additionally, linear interpolation estimates values by assuming the function behaves as a straight line between two known points.3