Write the equation for a parabola with the focus at (–1, 4) and the equation of the directrix x = 5.

A standard form of the equation of a rotated parabola is
(y - k)² = 4p(x - h)
where
(h, k) is the location of the vertex.
(h+p, k) is the location of the focus.
The directrix is the line x = h - p

Because the focus is at (-1, 4), therefore
h + p = -1         (1)
k = 4               (2)

Because the directrix is x = 5, therefore
h - p = 5          (3)

Add equations(1) and (3) to obtain
h + p + (h - p) = -1 + 5
2h = 4
h = 2
From (1), obtain
p = -1 - h = -1 - 2 = -3

The equation of the parabola is
(y - 4)² = -12(x - 2)

   (y - 4)² = -12(x - 2)