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The function given is:

\[
f(x) = -(x - 8)^2 + 21
\]

This is in vertex form, which is:

\[
f(x) = a(x - h)^2 + k
\]

where \((h, k)\) represents the vertex.

### Identifying the vertex:
From the equation \(f(x) = -(x - 8)^2 + 21\), we can compare it with the vertex form and see that:
- \(h = 8\)
- \(k = 21\)

Thus, the vertex is \((8, 21)\).

### Axis of symmetry:
The axis of symmetry is the vertical line that passes through the vertex, and its equation is \(x = h\). Therefore, the axis of symmetry is:
\[
x = 8
\]

### Maximum or minimum value:
Since the coefficient of \((x - 8)^2\) is negative (the equation has a leading \(-\)), the parabola opens downwards, meaning the vertex is a **maximum** point.

The maximum value of the function is the \(y\)-coordinate of the vertex, which is:
\[
\text{Maximum value} = 21
\]

### Domain and range:
- The domain of any quadratic function is all real numbers, so the domain is:
\[
\text{Domain}: (-\infty, \infty)
\]
- Since the parabola opens downward and has a maximum value of 21, the range is:
\[
\text{Range}: (-\infty, 21]
\]

Would you like more details or further explanations?

Here are five related questions to practice:
1. How do you determine whether the vertex represents a maximum or minimum?
2. What is the impact of changing the value of \(h\) in the vertex form equation?
3. How would the graph change if the coefficient of \((x - 8)^2\) was positive?
4. Can you find the y-intercept of this function?
5. How can you find the x-intercepts of this quadratic function?

**Tip**: In vertex form, the sign of the coefficient \(a\) in \(a(x-h)^2 + k\) determines if the parabola opens upwards or downwards.

Identify the vertex, the axis of symmetry, the maximum or minimum value, and the domain and range of the function: f(x) = -(x - 8)^2 + 21

Learn how to identify the vertex, axis of symmetry, maximum value, and the domain and range of the quadratic function f(x) = -(x - 8)^2 + 21. This solution covers key algebraic concepts such as vertex form and the properties of parabolas. Suitable for Grades 8-10.

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