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.