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Let's go through the tasks one by one.

### Given Information:
We have a quadratic function representing the ball's trajectory:

\[
f_a(x) = ax^2 + x + 2.4
\]

where \(a \in \mathbb{R}\) and the domain is \(D_f = [0; 4.025]\), representing the horizontal distance of the ball in meters. The function gives the ball's height above the ground. The basketball hoop is at a horizontal distance of \(4.025\) meters from the throw point, and the height of the hoop is \(3.05\) meters.

---

### **1. Justify that the function \( f_a \) models all shots where the ball is thrown at a 45° angle from the release point \((0, 2.4)\):**

The ball is released at a height of \(2.4\) meters, and the trajectory is given by the quadratic function. A throw at an angle of 45° implies that the derivative of the function at \(x = 0\) should give a slope of 1 (since \(\tan(45^\circ) = 1\)).

We find the derivative:

\[
f'_a(x) = 2ax + 1
\]

At \(x = 0\):

\[
f'_a(0) = 2a(0) + 1 = 1
\]

So, the derivative confirms that the slope is 1 at \(x = 0\), matching the 45° throw angle. Therefore, the function correctly models the shots where the release is at 45°.

---

### **2. Draw the graphs for \(a = -0.5, -0.3, -0.1, 0.1, 0.3\), and justify why \(a\) must be negative:**

Since \(a\) determines the curvature of the parabola, positive values of \(a\) make the parabola open upwards, which doesn't fit the physical situation because the ball must follow a downward trajectory. Thus, \(a\) must be negative to model a downward curve (representing the effect of gravity on the ball).

- For \(a = -0.5, -0.3, -0.1\), the parabola will open downwards.
- For \(a = 0.1, 0.3\), the parabola opens upwards, which is not physically realistic for this context.

You would need to graph these functions using software or plotting tools to visually confirm this. The negative \(a\) values create trajectories where the ball falls after reaching a peak, modeling a realistic shot.

---

### **3. Approximate a value of \(a\) such that the ball passes through the center of the hoop:**

The hoop is at a height of \(3.05\) meters and a horizontal distance of \(4.025\) meters. We need to find \(a\) such that:

\[
f_a(4.025) = 3.05
\]

Using the equation \(f_a(x) = ax^2 + x + 2.4\), we substitute \(x = 4.025\) and \(f_a(4.025) = 3.05\):

\[
a(4.025)^2 + 4.025 + 2.4 = 3.05
\]

Now, solving for \(a\):

\[
a(16.201025) + 4.025 + 2.4 = 3.05
\]
\[
a(16.201025) = 3.05 - 4.025 - 2.4
\]
\[
a(16.201025) = -3.375
\]
\[
a = \frac{-3.375}{16.201025} \approx -0.2083
\]

So, approximately, \(a \approx -0.208\).

---

### Recap and Further Questions:

1. The function \( f_a(x) \) models the shot at 45° correctly because the derivative at the release point gives a slope of 1.
2. A graph of the function for different values of \(a\) can be drawn, showing that negative \(a\) results in a downward trajectory.
3. The value of \(a \approx -0.208\) ensures that the ball passes through the center of the hoop at a height of 3.05 m.

### Would you like further details on any part of the solution? Here are five follow-up questions:

1. How does the quadratic term \( ax^2 \) affect the trajectory of the ball for various values of \(a\)?
2. Why is the derivative important for understanding the angle of the throw?
3. Can we model other angles using the same method? What would change in the function?
4. How could air resistance or spin be included in this model?
5. What methods can we use to plot the graphs of these functions and visually compare them?

### Tip:
Always double-check the units and values in real-world physics problems, as small errors in assumptions or conversions can lead to significant differences in the results.

Löse die Aufgaben vollständig: Die Freiwurflinie beim Basketball ist 4,025 m vom Mittelpunkt des Korbrings entfernt (horizontal gemessen), der Korbring hängt 3,05 m über dem Boden. Die Flugbahn kann mithilfe einer Funktion f_a(x) = ax^2 + x + 2,4 modelliert werden.

Solve quadratic function problems modeling basketball shot trajectories. Learn how to justify shot angles, graph functions, and find the correct parameter for a 45° angle shot, covering algebraic concepts like derivatives and trajectory modeling.

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