Question

..vertex: (4, -1)
f(3) = -6

Answer 1

`\huge\boxed{f(x)=-5(x-4)^2-1=-5x^2+40x-81}`

Explanation:

The vertex form of an equation of a parabola:

`f(x)=a(x-h)^2+k`

`(h;\ k)` - vertex

We have

vertex
`(4;\ -1)\to h=4;\ k=-1`

`f(3)=-6`

Therefore we have:

`f(x)=a(x-4)^2+(-1)=a(x-4)^2-1`

Substitute
`x=3` and
`f(3)=-6`

`a(3-4)^2-1=-6\qquad|\text{add 1 to both sides}\\\\a(-1)^2=-6+1\\\\a(1)=-5\\\\a=-5`

Finally:

`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`