Final answer:
The equation of the parabola with a vertex at (-6, -4) and passing through (-8, -8) is y = -(x + 6)^2 - 4.
Step-by-step explanation:
To write the equation of the parabola with a vertex at (-6, -4) and that passes through the point (-8, -8), we can use the vertex form of a parabola's equation:
y = a(x - h)^2 + k
where (h, k) is the vertex of the parabola. In this case, h = -6 and k = -4, so our equation starts to take the form:
y = a(x + 6)^2 - 4.
To find the value of 'a', we use the point that the parabola passes through (-8, -8). Plugging these values into the equation gives:
-8 = a(-8 + 6)^2 - 4
Solving for 'a' yields:
-8 = a(4)^2 -4
-8 = 4a - 4
-4 = 4a
a = -1
Now we have all the information needed to write the final equation of the parabola:
y = -(x + 6)^2 - 4
This is the equation of the parabola with our given conditions.