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The problem asks you to write the equation of the quadratic function based on the graph provided.

### Analysis:
1. **Vertex**: The graph shows that the vertex of the parabola is at the origin, \((0, 0)\).
2. **Symmetry**: The parabola opens upwards, meaning the coefficient of \(x^2\) is positive.
3. **Points on the Graph**: The graph crosses the x-axis at points \((-3, 0)\), \((0, 0)\), and \((3, 0)\).

Given that the graph crosses the x-axis at \(x = -3\), \(x = 0\), and \(x = 3\), these are the roots of the quadratic equation.

### Equation Formation:
The factored form of a quadratic equation is given by:
\[
y = a(x - r_1)(x - r_2)
\]
where \(r_1\) and \(r_2\) are the roots.

In this case, the roots are \(-3\), \(0\), and \(3\). So, the equation becomes:
\[
y = a(x + 3)(x)(x - 3)
\]

### Simplification:
Since the vertex is at the origin and there is no vertical stretch (as seen from the graph), we can assume \(a = 1\). Therefore, the equation is:
\[
y = (x + 3)(x)(x - 3)
\]

Expanding this:
\[
y = x(x^2 - 9)
\]
\[
y = x^3 - 9x
\]

However, this is not a quadratic equation but a cubic one, which implies that my earlier assumption about the roots needs reconsideration.

Instead, considering only two roots and the symmetry around the origin:
\[
y = x(x - 3)(x + 3)
\]

This simplifies to:
\[
y = x(x^2 - 9) = x^3 - 9x
\]
Given that the question asks for a quadratic equation, the correct interpretation would focus on the roots -3 and 3. This symmetry indicates that the quadratic expression involves just \(x^2 - 9\).

The equation in standard form, therefore, is:
\[
y = x^2 - 9
\]

### Final Answer:
The equation of the quadratic function is:
\[
y = x^2 - 9
\]

Would you like any additional details or have any questions?

### Related Questions:
1. How do you derive the quadratic equation if given only the vertex and a point on the parabola?
2. What does the discriminant of a quadratic equation tell you about its roots?
3. How can you determine the direction in which a parabola opens by looking at its equation?
4. What is the significance of the axis of symmetry in a quadratic graph?
5. How would the equation change if the parabola were shifted up or down?

### Tip:
Always verify the equation you derive by substituting known points from the graph to ensure accuracy.

Write the equation of the quadratic function in factored form based on the graph provided.

Learn how to write the equation of a quadratic function from a graph. This guide covers key algebraic concepts including factored and standard forms, analyzing roots, and the symmetry of parabolas. Suitable for students in Grades 8-10.

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