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