The image you uploaded contains mathematical questions and a diagram related to the construction of a bridge over a river, modeled by a parabolic shape. I'll break down the questions and assist you step by step.

Here is a summary of the tasks:

1. Given Parabola Equation:
   The function provided is $p(x) = \frac{-5}{9} x^2 + 5$. This seems to represent the parabola for the bridge.

2. Question 2.1 asks to derive the general form of the quadratic function that fits the parabola over the river.

3. Question 2.1.1 asks you to derive the function in its general quadratic form, explaining the reasoning behind it.

4. Question 2.1.2 involves finding the equation of the parabola given three specific points.

Let's proceed with the calculations:

Step 1: General form of the quadratic function.

The general form of a quadratic function is:

$f(x) = ax^2 + bx + c$

From the given equation $p(x) = \frac{-5}{9} x^2 + 5$, you can notice that this already resembles a quadratic function in standard form with $a = \frac{-5}{9}$, $b = 0$, and $c = 5$. This function describes the parabolic shape of the bridge.

Would you like me to go further into the details for part 2.1, or explain any specific part of this process?

Follow-up questions:

1. How do you derive the coefficients for the general quadratic equation using points?
2. What is the method for determining the vertex form of a quadratic function from standard form?
3. How do the measurements of the bridge (height, width) relate to the parabola's parameters?
4. How can the quadratic model be used to predict traffic clearances under the bridge?
5. What are the practical considerations when choosing the right parabola for a bridge?

Tip: When working with quadratic equations, understanding the relationship between the coefficients and the shape of the parabola can help immensely. The vertex $(h, k)$ is crucial in determining the maximum or minimum point of the curve.