TITLE: Is there some intuition for Lagrange interpolation formula?
QUESTION [8 upvotes]: How do I prove the Lagrange interpolation formula is true as stated in this link?
I ask this because the article isn't self contained on intuition of each step in the proof, please don't use things too advanced because I am just a high school student. Thanks in advance.

REPLY [3 votes]: I first saw Lagrange's interpolation formula in tenth grade and calculus was not part of the tenth grade curriculum. However, the second form is fairly straight forward:
$$
p(x)=\sum_{i=1}^n\;y_i\prod_{\genfrac{}{}{0}{1}{j\not=i}{j=1}}^n\frac{x-x_j}{x_i-x_j}\tag{1}
$$
For each $i$, consider
$$
q_i(x)=\prod_{\genfrac{}{}{0}{1}{j\not=i}{j=1}}^n\frac{x-x_j}{x_i-x_j}\tag{2}
$$
Plug in $x=x_j$ where $j\not=i$. Since $\dfrac{x-x_j}{x_i-x_j}$ is a factor in $(2)$, $q_i(x_j)=0$.
Plug in $x=x_i$. Each term in $(2)$ is then $\dfrac{x_i-x_j}{x_i-x_j}=1$; thus, $q_i(x_i)=1$.
Therefore, $\displaystyle p(x_i)=\sum_{j=1}^n\;y_jq_j(x_i)=y_i$.