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The question asks you to select the correct quadratic equation that models the height of a basketball based on the given table of distance and height. Then, you need to use the equation to find the maximum height of the basketball.

### Table:
- Distance (x) in feet: 0, 2, 6, 9, 12, 13, 15
- Height (f(x)) in feet: 4, 8.4, 12.1, 13.2, 12, 10.5, 9.8

You are provided with two quadratic equations to choose from:
1. \( f(x) = -0.118x^2 + 2.112x + 4.215 \)
2. \( f(x) = -0.117x^2 + 2.112x + 4.215 \)

To determine the maximum height of the basketball, we need to find the vertex of the quadratic function. The vertex of a parabola \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:
\[
x = -\frac{b}{2a}
\]
This will give us the distance at which the maximum height occurs. After finding this x-value, we substitute it into the equation to find the corresponding height.

### Step-by-Step Process:
1. **Identify the coefficients**:
   - \( a = -0.118 \) or \( -0.117 \)
   - \( b = 2.112 \)

2. **Find the vertex** for both options:
   \[
   x = -\frac{2.112}{2 \times -0.118} = 8.949 \text{ feet (for the first equation)}
   \]
   \[
   x = -\frac{2.112}{2 \times -0.117} = 9.02 \text{ feet (for the second equation)}
   \]

3. **Substitute the x-value back into the equation**:
   Using the first equation, substitute \( x = 8.949 \):
   \[
   f(8.949) = -0.118(8.949)^2 + 2.112(8.949) + 4.215
   \]
   The approximate height will be 13.665 feet.

Thus, the correct equation is **\( f(x) = -0.118x^2 + 2.112x + 4.215 \)** and the maximum height is approximately **13.665 feet**.

Would you like further details on any step or have any questions?

### Relative Questions:
1. How do you determine the vertex of any quadratic function?
2. What role does the sign of the leading coefficient play in determining the shape of a parabola?
3. Why is the vertex important when analyzing quadratic functions in real-world contexts?
4. How can you verify that your solution is correct by graphing?
5. Can quadratic regression be used to derive the same equation from the data?

**Tip:** The x-coordinate of the vertex of a quadratic function is always the point where the maximum or minimum value occurs, depending on whether the parabola opens downwards or upwards.

Amery recorded the height and distance of a basketball when shooting a free throw, and you need to choose the quadratic equation for the function. Then, use the equation to find the approximate maximum height of the basketball.

This problem involves using a quadratic equation to model the height of a basketball as a function of distance. It covers key algebraic concepts like finding the vertex to determine the maximum height. Suitable for high school students studying quadratic functions.

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