To determine the quadratic function of the given graph, let's analyze the information visible in the graph:

1. Vertex: The graph has a maximum point (vertex) at $(-3, 3)$.
2. Direction: The parabola opens downward, indicating a negative coefficient for $x^2$.
3. Y-intercept: Based on the graph, the y-intercept occurs approximately at $(0, -6)$.

Given these observations, we can assume the function is in the vertex form of a quadratic function: $f(x) = a(x - h)^2 + k$ where $(h, k)$ is the vertex.

Step-by-Step Solution:

1. Substitute the vertex $(-3, 3)$: $f(x) = a(x + 3)^2 + 3$

2. Determine the value of $a$ using the y-intercept $(0, -6)$: $-6 = a(0 + 3)^2 + 3$ $-6 = 9a + 3$ $-9 = 9a$ $a = -1$

3. Construct the function: $f(x) = -1(x + 3)^2 + 3$

4. Expand the function (optional) to standard form: $f(x) = - (x^2 + 6x + 9) + 3$ $f(x) = -x^2 - 6x - 9 + 3$ $f(x) = -x^2 - 6x - 6$

Final Answer:

The quadratic function is: $f(x) = -x^2 - 6x - 6$

Would you like more details on any step?

Here are some additional questions related to this topic:

1. What if the vertex was different? How would that change the function?
2. How do we determine the direction of a parabola from the function?
3. What are other forms of representing a quadratic function besides vertex form?
4. How can you find the axis of symmetry from a quadratic equation?
5. How does changing the coefficient $a$ affect the parabola's width?

Tip: The vertex form is particularly useful for identifying the maximum or minimum point of a parabola directly from the equation.