Determine $a, b$ & $c$ in a (possible) Quadratic equation

Question

Given $f(x)=x$:

1 - $f(5)=2$


2 - $f(3)=3$

How to determine the coefficients $a, b ,c$ in the polynomial equation like: $$f(x) = ax^2 + bx + c$$

Answer 1

$2=a5^2+5b+c=25a+5b+c$

$3=a3^2+3b+c=9a+3b+c$

Subtracting the two equations, we have

$-1=16a+2b$

Without another point, there are infinitely many solutions. Quadratic equations are defined by $3$ points, as lines are defined by $2$.

Answer 2

Any two points define a straight line so allow $a=0$ the $y=x-8$ is a solution. This also implies an infinite solutions due to the free variable a. For example

$-0.153661 x^2+0.229292 x+5.69508=y$ and $y = 2.84754 + 0.229292 x - 0.307323 x^2$.

Both of the above are generated by adding a third point then calculating. Its a good exercise to see what third points can be added to the parabola with those constraints and why.