The graph represents a parabola, and the equation given is $y = ax^2 + bx + c$, with the range of the parabola being $y \geq -4$. The vertex is at the point $(6, -4)$, and one of the points on the parabola is $(12, 8)$.

We are tasked with finding the value of $a$.

Step-by-step approach:

1. Vertex form of the parabola: Since the vertex is given at $(6, -4)$, we can start by writing the equation in vertex form: $y = a(x - h)^2 + k$ where $(h, k) = (6, -4)$. This gives us: $y = a(x - 6)^2 - 4$

2. Using the point $(12, 8)$: Now, substitute $x = 12$ and $y = 8$ into the equation to find $a$: $8 = a(12 - 6)^2 - 4$ Simplify the equation: $8 = a(6)^2 - 4$ $8 = 36a - 4$ Solve for $a$: $8 + 4 = 36a$ $12 = 36a$ $a = \frac{12}{36} = \frac{1}{3}$

Thus, the value of $a$ is $\frac{1}{3}$.

Answer: The correct option is A) $\frac{1}{3}$.

Would you like a more detailed explanation or help with another question?

Here are some related questions:

1. How do you find the vertex of a parabola from its equation in standard form?
2. What is the significance of the value of $a$ in a parabola's equation?
3. How do you convert from vertex form to standard form for a parabola equation?
4. What does the range of a parabola tell you about its graph?
5. How do you determine the direction a parabola opens based on its equation?

Tip: Always double-check the vertex and points on the graph to ensure accuracy in solving for the constants in the equation.