Estimate parameters of a quadratic function

Question

Suppose that we have two data points which tell us about the output of some function $f(x)$:

1. $(0, 50)$
2. $(10, 150)$

We know that the function is quadratic (so it's something like $ax^2 + bx + c$). The question is: how do we estimate the parameters $a, b$ and $c$ in such a way that the resulting function matches our data best?

Does this have to be regression, or is there some easier way?

Answer 1

Assume $f(x) = a x^2 + b x + c$. The points give the equations $$ 50 = f(0) = c \\ 150 = f(10) = 100 a + 10 b + c = 100 a+ 10 b + 50 \iff \\ 10a + b = 10 $$ So your $c$ is determined, but $a$ and $b$ are not, you have to choose either $a$ or $b$.