Answer:
[tex]x^2 = -16y[/tex]
Step-by-step explanation:
Given
[tex]Vertex = (0,0)\\Focus = (0,-4)[/tex]
Required
Equation of the parabola (in standard form)
The standard form of a parabola is [tex](x - h)^2 = 4p (y - k),[/tex]
Such that
Vertex = (h,k)
Focus = (h, k + p)
For the vertex
This implies that (h,k) = (0,0)
h = 0 and k = 0
For the focus
This implies that (h, k + p) = (0, -4)
[tex]h = 0\\k + p = -4[/tex]
Recall that [tex]k = 0;[/tex]
Hence, [tex]0 + p = -4[/tex]
[tex]p = -4[/tex]
Substitute [tex]p = -4[/tex], [tex]h = 0\ and\ k = 0[/tex] in the given formula
[tex](x - h)^2 = 4p (y - k),[/tex] becomes
[tex](x - 0)^2 = 4 * -4 (y - 0),[/tex]
[tex](x)^2 = 4 * -4 (y),[/tex]
[tex]x^2 = -16 (y),[/tex]
[tex]x^2 = -16y[/tex]
Hence,, the standard form is [tex]x^2 = -16y[/tex]
