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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.

Determine the quadratic function of the form f(x) = a(x − h)^2 + k whose graph is given on the right.

Learn how to determine a quadratic function in vertex form f(x) = a(x - h)^2 + k from a given parabola graph. This problem includes identifying the vertex, using an additional point to solve for 'a', and formulating the quadratic equation. Suitable for students in Grades 10-12.

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