Answer:
![(x+5)^{2}=-4(y+3)](https://tex.z-dn.net/?f=%28x%2B5%29%5E%7B2%7D%3D-4%28y%2B3%29)
Step-by-step explanation:
Given:
Focus point = (-5, -4)
Vertex point = (-5, -3)
We need to find the equation for the parabola.
Solution:
Since the x-coordinates of the vertex and focus are the same,
so this is a regular vertical parabola, where the x part is squared. Since the vertex is above the focus, this is a right-side down parabola and p is negative.
The vertex of this parabola is at (h, k) and the focus is at (h, k + p). So, directrix is y = k - p.
Substitute y = -4 and k = -3.
![-4 = -3+p](https://tex.z-dn.net/?f=-4%20%3D%20-3%2Bp)
![p=-4+3](https://tex.z-dn.net/?f=p%3D-4%2B3)
![p=-1](https://tex.z-dn.net/?f=p%3D-1)
So the standard form of the parabola is written as.
![(x-h)^{2}=4p(y-k)](https://tex.z-dn.net/?f=%28x-h%29%5E%7B2%7D%3D4p%28y-k%29)
Substitute vertex (h, k) = (-5, -3) and p = -1 in the above standard form of the parabola.
So the standard form of the parabola is written as.
![(x-(-5))^{2}=4(-1)(y-(-3))](https://tex.z-dn.net/?f=%28x-%28-5%29%29%5E%7B2%7D%3D4%28-1%29%28y-%28-3%29%29)
![(x+5)^{2}=-4(y+3)](https://tex.z-dn.net/?f=%28x%2B5%29%5E%7B2%7D%3D-4%28y%2B3%29)
Therefore, equation for the parabola with focus at (-5,-4) and vertex at (-5,-3)
![(x+5)^{2}=-4(y+3)](https://tex.z-dn.net/?f=%28x%2B5%29%5E%7B2%7D%3D-4%28y%2B3%29)