https://file.aiquickdraw.com/mathbot/file/17280859495744lhdrzqg.jpg

To determine the quadratic function based on the graph provided, let's go through the following steps:

1. **Identify the vertex**: The vertex of the parabola is clearly shown at the point \( (3, -10) \).
   
2. **Determine the form of the quadratic equation**: The general form of a quadratic equation is \( f(x) = a(x - h)^2 + k \), where:
   - \( (h, k) \) is the vertex of the parabola,
   - \( a \) is a constant that determines the direction and width of the parabola.

   From the graph, the vertex is \( (h, k) = (3, -10) \), so we can substitute these values into the equation, giving us:
   \[
   f(x) = a(x - 3)^2 - 10
   \]

3. **Determine the value of \( a \)**: To find \( a \), we need another point on the graph. From the image, a clear point on the parabola is \( (0, 0) \) (where the parabola passes through the origin).

   We can substitute this point \( (0, 0) \) into the equation to solve for \( a \):
   \[
   0 = a(0 - 3)^2 - 10
   \]
   \[
   0 = a(9) - 10
   \]
   \[
   10 = 9a
   \]
   \[
   a = \frac{10}{9}
   \]

4. **Final equation**: Now that we have \( a = \frac{10}{9} \), the equation becomes:
   \[
   f(x) = \frac{10}{9}(x - 3)^2 - 10
   \]

This is the quadratic function that describes the given graph.

Let me know if you'd like further details or if anything is unclear.

### Related questions:
1. How does the value of \( a \) affect the width and direction of a parabola?
2. How can you derive the quadratic equation from three points on a graph?
3. What is the significance of the vertex form compared to the standard form of a quadratic function?
4. How does shifting a parabola horizontally or vertically affect its equation?
5. How can you find the axis of symmetry from the equation of a parabola?

### Tip:
The vertex form of a quadratic equation is useful when you know the vertex of the parabola, making it easier to write the equation directly.

Determine the quadratic function whose graph is given below.

Learn how to determine the quadratic function based on a given graph with the vertex at (3, -10) and the point (0, 0). The problem demonstrates using the vertex form of a quadratic equation and solving for the parameter 'a'. Suitable for students in Grades 9-11.

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