Answer:
[tex]\huge\boxed{f(x)=-5(x-4)^2-1=-5x^2+40x-81}[/tex]
Step-by-step explanation:
The vertex form of an equation of a parabola:
[tex]f(x)=a(x-h)^2+k[/tex]
[tex](h;\ k)[/tex] - vertex
We have
vertex [tex](4;\ -1)\to h=4;\ k=-1[/tex]
[tex]f(3)=-6[/tex]
Therefore we have:
[tex]f(x)=a(x-4)^2+(-1)=a(x-4)^2-1[/tex]
Substitute [tex]x=3[/tex] and [tex]f(3)=-6[/tex]
[tex]a(3-4)^2-1=-6\qquad|\text{add 1 to both sides}\\\\a(-1)^2=-6+1\\\\a(1)=-5\\\\a=-5[/tex]
Finally:
[tex]f(x)=-5(x-4)^2-1=-5(x^2-2(x)(4)+4^2)-1\\\\=-5(x^2-8x+16)-1=-5x^2+(-5)(-8x)+(-5)(16)-1\\\\=-5x^2+40x-80-1=-5x^2+40x-81[/tex]
