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Sphinxa [80]
2 years ago
10

WILL MARK BRAINLIEST PLEASE HELP

Mathematics
1 answer:
Marrrta [24]2 years ago
7 0

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







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