Answer:
[tex]y = - \frac{1}{4} {(x + 2)}^{2} + 5[/tex]
Step-by-step explanation:
The vertex of this parabola is the midpoint of the focus (-2,4) and where the directrix intersects the axis of symmetry of the parabola (-2,6)
This parabola must open downwards due to the position of the directrix and has equation of the form:
[tex] {(x - h)}^{2} = - 4p(y - k)[/tex]
where (h,k) is the vertex.
This implies that:
[tex]h = - 2[/tex]
and
[tex]k = \frac{4 + 6}{2} = 5[/tex]
The value of p is the distance from the vertex to the focus:
[tex]p = |6 - 5| = 1[/tex]
We substitute all the values into the formula to get:
[tex](x - - 2)^{2} = - 4(1){(y - 5)}[/tex]
[tex] {(x + 2)}^{2} = - 4(y - 5)[/tex]
Or
[tex]y = - \frac{1}{4} {(x - 5)}^{2} + 5[/tex]
