How to extrapolate a reasonable price given data points? I work as a math tutor, and the kids get cards for getting work done. They then are able to exchange those cards for gift cards. The exchange policies being (10,5),(19,10),(27,15),(35,20),(42,25),(55,35),(75,50) |(cards,gift card balance). Recently a student had 100 cards, which was exchanged for a 100$ gift card, but I felt that it should've actually been more than that given the exchange policies. So to get a more proper exchange rate, I thought it would be a good idea to use newtons method to extrapolate a better exchange rate for 100 cards, but I couldn't figure out or find out how to do this with data. So I'd like to know how one would do this or if there is a better way.

Using a naive polynomial approximation, I found that $f(x) = \frac{x}2 + \frac{x^2}{400} $ gives a reasonable approximation of your data points. Using this, I found that the pair (100,75) is a reasonable extrapolation for 100 cards.

![Naive Polynomial Approximation](