Inscrib-tion Inspiration

Given an arbitrary equilateral triangle inscribed in the parabola represented by $y = x^2$ with all three vertices touching, if the locus of its centroid can be represented by the equation $f(x) = ax^2 + b$ , where $\gcd(a,b) = 1$ and $a > 0$ , find $f(4)$ .

Inspiration.