In more general terms, what we have done here is to approximate a given function $f(x)$ with a polynomial of a certain degree. One can show that given $n+1$ distinct points $x_0,\dots, x_n\in[a,b]$ and $n+1$ values $y_0,\dots,y_n$ there exists a unique polynomial $P_n(x)$ with the property $$\begin{equation*} P_n(x_j) = y_j\hspace{0.5cm} j=0,\dots,n \end{equation*}$$