Answer:
The required quadratic function is [tex]y=-\frac{1}{12}(x-3)^2-1[/tex].
Step-by-step explanation:
The standard form of the parabola is
[tex](x-h)^2=4p(y-k)[/tex]
Where the focus is (h, k + p) and the directrix is y = k - p.
The directrix of y = 2 and a focus of (3, −4).
[tex](h,k+p)=(3,-4)[/tex]
On comparing both sides we get
[tex]h=3[/tex]
[tex]k+p=-4[/tex] ...... (1)
[tex]y=k-p[/tex]
[tex]k-p=2[/tex] ...... (2)
Add equation (1) and (2).
[tex]2k=-2[/tex]
[tex]k=-1[/tex]
Substitute k=-1 in equation (1).
[tex](-1)+p=-4[/tex]
[tex]p=-3[/tex]
Therefore the equation of parabola is
[tex](x-3)^2=4(-3)(y-(-1))[/tex]
[tex](x-3)^2=-12(y+1)[/tex]
It can be rewritten as
[tex]y=-\frac{1}{12}(x-3)^2-1[/tex]
Therefore the required quadratic function is [tex]y=-\frac{1}{12}(x-3)^2-1[/tex].
