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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.

The function f(x) = −(x − 1)^2 + 8 models the path of a volleyball. What is the maximum height of the volleyball after it is served?

Learn how to find the maximum height of a volleyball using the quadratic function f(x) = −(x − 1)^2 + 8. This problem explores algebra concepts like the vertex form of a parabola, making it ideal for students in Grades 9-10.

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