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.