This exercise involves mathematical modeling of two basketball throws using quadratic functions, representing the height of the ball as a function of the distance to the basket.

Here are the main points and the questions you're working on:

Functions:

1. First throw (function f): $f(x) = -0.5x^2 + 2.25x + 2$
2. Second throw (function g): $g(x) = -0.8x^2 + 3.45x + 2$

These functions represent the height of the basketball $h$ in meters as a function of the horizontal distance $d$ in meters from the player.

The graph shows the paths of both throws and asks you to find the corresponding values and behaviors.

Key Questions:

1. Part 1: Associating curves with corresponding functions. You need to identify which curve corresponds to function $f$ (the first throw) and which corresponds to function $g$ (the second throw). The graph shows two parabolas, and the question asks you to match each curve with the correct function.

2. Part 2: Determining coordinates. You need to determine the coordinates of the point where the basket is located, symbolized by point P on the graph. This is the point where the ball reaches the basket’s height.

3. Part 3: Ball remaining at 3 meters. For the first throw (function $f$), you are asked to determine the distance from the ground at which the ball remains at 3 meters in height. This is a question about finding when the height $h(x) = 3$ for the first throw.

4. Part 4: Tangent lines. You are tasked with finding the tangent lines at certain points:

   • For the second throw, you need to determine the coefficient of the tangent at the point where the abscissa $x = 0$ (when the ball is launched).
   • For the first throw, you must determine the coefficient of the tangent at the point where $x = 2$.

Next Steps:

If you want help with any of these specific parts (like calculating coordinates, solving for distances, or determining tangents), feel free to ask! Would you like assistance with one of the questions or a particular part of the exercise?

To continue:

1. Would you like to solve for the coordinates of point P?
2. Would you like to calculate the distance when the ball is at 3 meters?
3. Need help finding the tangent lines for the given functions?
4. Do you want a breakdown of how to solve for the tangent's coefficients?
5. Would you like to explore the graph further and determine more details?

Tip: Start by solving the quadratic equations for the values you need (like $h = 3$) to get more insight into the ball’s path!