The problem states that the graph of the quadratic function $y = a(x+3)^2 - 2$ passes through the point $(8, 361)$. We need to find the value of $a$ and the coordinates of the vertex of the graph.

1. Problem Description

The problem states that the graph of the quadratic function $y = a(x+3)^2 - 2$ passes through the point $(8, 361)$. We need to find the value of $a$ and the coordinates of the vertex of the graph.

2. Solution Steps

First, we need to find the value of $a$. Since the graph passes through the point $(8, 361)$, we can substitute $x=8$ and $y=361$ into the equation:

$361 = a(8+3)^2 - 2$

$361 = a(11)^2 - 2$

$361 = 121a - 2$

Add 2 to both sides:

$363 = 121a$

Divide both sides by 121:

$a = \frac{363}{121}$

$a = 3$

Now we have the equation $y = 3(x+3)^2 - 2$.

The vertex form of a quadratic equation is $y = a(x-h)^2 + k$, where the vertex is at the point $(h, k)$.

In our case, the equation is $y = 3(x+3)^2 - 2$, which can be rewritten as $y = 3(x-(-3))^2 + (-2)$.

Therefore, the vertex is at the point $(-3, -2)$.

3. Final Answer

$a = 3$

Vertex is at $(-3, -2)$.