Answer
$a=-3$
$b=6$
$c=1$
Work Step by Step
If the vertex of a graph is at $(m,n)$ then the vertex form for the quadratic function is $f(x)=a(x-m)^2+n$.
The vertex is at $(m,n)=(1,4)$. so the function is:
$f(x)=a(x-m)^2+n
\\f(x)=a(x-1)^2+4$
$(-1,-8)$ is on the graph, hence plugging in the values into the function above gives:
$-8=a\cdot(-1-1)^2+4\\
-8=a(-2)^2+4
-8=a(4)+4\\
-8=4a+4\\
-12=4a\\
-3=a.$
Thus, the function is:
$f(x)=-3\cdot (x-1)^2+4\\
f(x)=-3x^2+6x+1$
Therefore, $a=-3$, $b=6$, and $c=1$.
