Answer:
[tex](x+5)^{2}=-4(y+3)[/tex]
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.
[tex]-4 = -3+p[/tex]
[tex]p=-4+3[/tex]
[tex]p=-1[/tex]
So the standard form of the parabola is written as.
[tex](x-h)^{2}=4p(y-k)[/tex]
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.
[tex](x-(-5))^{2}=4(-1)(y-(-3))[/tex]
[tex](x+5)^{2}=-4(y+3)[/tex]
Therefore, equation for the parabola with focus at (-5,-4) and vertex at (-5,-3)
[tex](x+5)^{2}=-4(y+3)[/tex]
