The equation for a parabola with vertex \((h, k)\) is
\(y=a(x-h)^{2}+k\)
We plug in:
\(y=a(x-4)^{2}+75\)
We find \(a\) by plugging in the point \((0, 27)\):
\(27=a(0-4)^{2}+75\)
\(\Rightarrow 27=a(-4)^{2}+75\)
\(\Rightarrow 27=16a+75\)
\(\Rightarrow 16a=-48\)
\(\Rightarrow a=-3\)
So our equation is
\(y=-3(x-4)^{2}+75\)
To find x-intercepts, we set \(y=0\):
\(-3(x-4)^{2}+75=0\)
\(\Rightarrow -3(x-4)^{2}=-75\)
\(\Rightarrow (x-4)^{2}=25\)
\(\Rightarrow x-4=5, -5\)
\(\Rightarrow x=9, -1\)
So the x-intercepts are 9 and -1.
