$$f(x) = -0.0000x^9+0.0000x^8-0.0008x^7+0.0061x^6-0.0090x^5-0.1077x^4+0.4684x^3-0.1703x^2-1.6762x+2.0000$$
Show Scaled Basis Polynomials

## About Lagrange Interpolation

Lagrange interpolation is used to find a unique polynomial of least degree that best fits a set of distinct data points. To find the polynomial that passes through a set of $$k+1$$ data points $(x_0, y_0), (x_1, y_1), \ldots, (x_k, y_k)$ begin by finding the basis polynomials for each point. The scaled basis polynomial $$L_i(x)$$ corresponding to the point $$(x_i,y_i)$$ is given by $L_i(x) = y_i \frac{(x-x_0)}{(x_i-x_0)}\cdot\frac{(x-x_1)}{(x_i-x_1)}\cdots\frac{(x-x_{i-1})}{(x_i-x_{i-1})}\cdot\frac{(x-x_{i+1})}{(x_i-x_{i+1})}\cdots \frac{(x-x_k)}{(x_i-x_k)}$ Each basis polynomial will then have a root at each $$x_j \neq x_i$$, and will pass through the point $$(x_i,y_i)$$. The final interpolating polynomial $$f(x)$$ that passes through all data points is a linear combination of these basis polynomials: $f(x) = L_0(x) + L_1(x) + \cdots + L_k(x)$

## Using the Applet

Drag the data points around the graph, and the Lagrange interpolation polynomial will be updated as a solid black line. Remove a data point by double clicking the point you'd like to remove. Double click the background to add a point (up to 10). You can also choose to hide or display the scaled basis polynomials generated by the data points as dashed lines.