To find the quadratic function in the form $f(x) = a(x - h)^2 + k$ using the given graph, let's identify the key components from the graph:

Step 1: Identify the Vertex

From the graph, the vertex of the parabola is at the point $(-2, 6)$. Thus, $h = -2$ and $k = 6$.

Step 2: Use Another Point on the Graph

To find the value of $a$, use another point on the parabola. The point $(0, -6)$ is on the graph.

Step 3: Substitute into the Vertex Form

Substitute $h$, $k$, and the point $(0, -6)$ into the vertex form to solve for $a$.

$f(x) = a(x - (-2))^2 + 6 = a(x + 2)^2 + 6$

Substitute $x = 0$ and $f(x) = -6$:

$-6 = a(0 + 2)^2 + 6$

Simplify and solve for $a$:

$-6 = a \cdot 4 + 6$

$-6 - 6 = 4a$

$-12 = 4a$

$a = -3$

Final Function

The quadratic function is:

$f(x) = -3(x + 2)^2 + 6$

Would you like more details or have any questions?

Related Questions:

1. How can you find the equation of a parabola given three points?
2. What is the significance of the vertex in a parabola?
3. How does the value of $a$ affect the shape of the parabola?
4. How do you convert the vertex form of a quadratic function to the standard form?
5. What methods can be used to find the roots of a quadratic function?

Tip: The vertex form of a quadratic function is especially useful for identifying transformations like shifts, stretches, and reflections of the parabola.