While looking for the equation to fits the following points:
| x | y |
|---|---|
| 0 | 0.2 |
| 50 | 1 |
| 100 | 5 |
I found (following https://www.youtube.com/watch?v=nLar40Dd_Us&t=263s):
0.064 * x**2 - 0.16 * x + 0.2
Seems to do it.
But then I noticed I actually want the equation with all positive terms, so it will be increasing all the way.
How do I find the equation that fits the points, with only positive terms?
If you know you want to have a quadratic, call it $ax^2 + bx + c$. Then, the 3 points give you 3 simultaneous equations:
$$0.2 = c$$ $$1 = 50^2 a + 50b + c$$ $$5 = 100^2 a + 100b + c$$
Clearly, the first equation reduces this to just 2 simultaneous equations:
$$2500a + 50b + 0.2= 1$$ $$10000a + 100b + 0.2 = 5$$
which has the unique solution you gave in your question. If you want it to be a quadratic, this is the unique quadratic going through those 3 points.
You could find a polynomial of higher degree, for example a cubic, of which there are a whole family passing through your points.