Question

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

Answer 1

Answer:

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

Step-by-step 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:

$f(x)=-5(x-4)^2-1=-5(x^2-2(x)(4)+4^2)-1\\\\=-5(x^2-8x+16)-1=-5x^2+(-5)(-8x)+(-5)(16)-1\\\\=-5x^2+40x-80-1=-5x^2+40x-81$