Question

WILL MARK BRAINLIEST PLEASE HELP

Answer 1

Answer:

$f(x)=-\frac{1}{8}(x-4)^2+3$

Step-by-step explanation:

The directrix given to us has equation,

$y=5$ and the focus is $(4,1)$.  

This means that the axis of symmetry of the parabola is parallel to the y-axis and has equation $x=4$, because it must go through the focus.

This axis of symmetry of the parabola will meet the directrix at $(4,5)$ .

The vertex of this parabola is the midpoint of the point of intersection of the axis of symmetry and the focus.

Thus,

$V(h,k)=(\frac{4+4}{2},\frac{5+1}{2})$

$V(h,k)=(4,3)$.

The equation is given by $(x-h)^2=4p(y-k)$.

$(x-4)^2=4p(y-3)$.

$|p|$ is the distance between the vertex and the focus, which is 2.

This implies that,

$p=-2$ or $p=2$

But the position of the directrix and the vertex implies that  the parabola opens downwards.

$\therefore p=-2$.

The equation of the parabola now becomes;

$(x-4)^2=4(-2)(y-3)$.

$(x-4)^2=-8(y-3)$

We solve for y to obtain,

$y=-\frac{1}{8}(x-4)^2+3$

or

$f(x)=-\frac{1}{8}(x-4)^2+3$