Answer
$a=-3, b=6,c=1$
Work Step by Step
When the vertex of a graph is at $(h, k)$, then the general form for the quadratic function can be expressed as: $f(x) = a(x-h)^2+k~~~(1)$
We are given that the vertex is at $(h,k)=(1,4)$
Therefore, $f(x)= a(x-1)^2+k \implies f(x) =a(x-1)^2 +2~~~ (2)$
Plug the point $(-1, -8)$ into equation (2) to obtain:
$-8 = a(-1-1)^2+4 \implies a=-3$
Thus, the equation of the function is:
$f(x)=-3(x-1)^2 +4=-3x^2+6x+1$
On comparing $f(x)=-3x^2+6x+1$ with $f(x) = ax^2+bx+c$, we get:
$a=-3, b=6,c=1$
