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.