The given function for the path of the volleyball is:

$f(x) = - (x - 1)^2 + 8$

This function is in the form $f(x) = - (x - h)^2 + k$, which is the vertex form of a parabola that opens downwards (because of the negative sign in front of $(x - h)^2$). In this form:

• $h = 1$ is the x-coordinate of the vertex.
• $k = 8$ is the y-coordinate of the vertex.

Since the vertex represents the maximum point of the parabola (due to it opening downwards), the maximum height of the volleyball is the y-coordinate of the vertex, which is:

$k = 8 \text{ feet}$

Answer: The maximum height of the volleyball is 8 feet.

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you determine the vertex of a parabola given in vertex form?
2. Why does the negative sign in front of $(x - 1)^2$ indicate that the parabola opens downwards?
3. How can we find the x-coordinate where the maximum height occurs in other scenarios?
4. How would the equation change if the volleyball reached a maximum height of 10 feet instead?
5. What is the physical significance of the vertex in a projectile motion context?

Tip: For any quadratic function in vertex form, the vertex $(h, k)$ gives the maximum or minimum value of the function directly, depending on the direction of the parabola.