Sure, I can help you with that! To find the equation of a parabola given its vertex and a point it passes through, we can use the vertex form of the equation. The vertex form is given by:
y = a(x - h)^2 + k
where (h, k) represents the vertex of the parabola.
In this case, the vertex is (-5, 7), so we have h = -5 and k = 7. We also know that the parabola passes through the point (-3, -1).
To find the value of a, we can substitute the coordinates of the point (-3, -1) into the equation and solve for a.
-1 = a(-3 - (-5))^2 + 7
Simplifying the equation:
-1 = a(2)^2 + 7
-1 = 4a + 7
4a = -1 - 7
4a = -8
a = -2
Now that we have the value of a, we can substitute it back into the vertex form equation:
y = -2(x - (-5))^2 + 7
Simplifying further:
y = -2(x + 5)^2 + 7
So, the equation of the parabola with a vertex at (-5, 7) and passing through (-3, -1) is y = -2(x + 5)^2 + 7.