I want to make an equation from ordered pairs

Question

for a project I'm working on I need to make a equation that passes through 15 different points on a graph. To keep it simple if someone can show me how to do it with 4 ordered pairs I can probably figure out the rest. Thank you!!!

Answer 1

Suppose the points are $(x_i,y_i)$ for $i=1,\dots,15$. The simplest approach is to define $p_i(x)=(x-x_1)(x-x_2)\dots(x-x_{i-1})(x-x_{i+1})\dots(x-x_{15})$. Then $p(x_j)=0$ for $j\ne i$ and $p(x_i)\ne0$.

So now you just take $$P(x)=\sum_{i=1}^{15}\frac{y_i}{p_i(x_i)}p_i(x)$$