The given function is:

\[
f(x) = -5(x - 8)^2
\]

### Analysis of the Function

1. **General Form**:
   This is a quadratic function in vertex form:
   \[
   f(x) = a(x - h)^2 + k
   \]
   Here, \( a = -5 \), \( h = 8 \), and \( k = 0 \).

2. **Vertex**:
   The vertex of the parabola is at \( (h, k) = (8, 0) \).

3. **Direction of Opening**:
   Since \( a = -5 \), the parabola opens downward because the coefficient of the squared term is negative.

4. **Steepness**:
   The absolute value of \( a \) (here, \( |a| = 5 \)) determines the steepness of the parabola. A larger value of \( |a| \) indicates a steeper graph.

5. **Axis of Symmetry**:
   The axis of symmetry is the vertical line passing through the vertex, given by:
   \[
   x = h = 8
   \]

6. **Maximum Value**:
   Since the parabola opens downward, the vertex represents the maximum value of the function:
   \[
   f(8) = -5(8 - 8)^2 = 0
   \]
   Thus, the maximum value is \( 0 \) at \( x = 8 \).

### Key Points:
1. **Vertex**: \( (8, 0) \)
2. **Axis of Symmetry**: \( x = 8 \)
3. **Direction**: Downward
4. **Maximum Value**: \( f(x) = 0 \)
5. **Domain and Range**:
   - **Domain**: \( (-\infty, \infty) \)
   - **Range**: \( (-\infty, 0] \)

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### Related Questions:
1. What is the y-intercept of the function \( f(x) = -5(x - 8)^2 \)?
2. How do we calculate specific values of \( f(x) \) for given \( x \)-values?
3. How does changing the value of \( a \) affect the parabola’s shape?
4. What is the effect of shifting \( h \) and \( k \) in the vertex form of a quadratic equation?
5. Can this function be rewritten in standard form \( ax^2 + bx + c \)? If yes, how?

### Tip:
When analyzing quadratic functions, always identify the vertex and axis of symmetry first—they are key to understanding the function's behavior.

Learn about the quadratic function f(x) = -5(x - 8)^2, including its vertex, axis of symmetry, direction, maximum value, and domain and range. This page provides a detailed step-by-step analysis of the function, making it ideal for students in Grades 8-10.

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