rite a quadratic function in the vertex form f left parenthesis x right parenthesis equals a left parenthesis x minus h right parenthesis squared plus k such that the graph of the function opens down and has a vertex at left parenthesis 8 comma negative 9 right parenthesis and passes through point left parenthesis 5 comma negative 18 right parenthesis.

To write a quadratic function in the vertex form \( f(x) = a(x - h)^2 + k \), we need to find the value of \( a \), given that the vertex \((h, k)\) is \((8, -9)\) and the graph passes through the point \((5, -18)\).

### Step-by-Step Solution:

1. **Identify the given information:**
   - Vertex \((h, k) = (8, -9)\)
   - Point on the graph \((x, f(x)) = (5, -18)\)

2. **Substitute the vertex into the vertex form:**
   \[
   f(x) = a(x - 8)^2 - 9
   \]

3. **Use the point \((5, -18)\) to find \(a\):**
   Substitute \(x = 5\) and \(f(x) = -18\) into the equation:
   \[
   -18 = a(5 - 8)^2 - 9
   \]

4. **Simplify the equation:**
   \[
   -18 = a(-3)^2 - 9
   \]
   \[
   -18 = a(9) - 9
   \]

5. **Solve for \(a\):**
   \[
   -18 + 9 = 9a
   \]
   \[
   -9 = 9a
   \]
   \[
   a = -1
   \]

6. **Write the quadratic function in vertex form:**
   \[
   f(x) = -1(x - 8)^2 - 9
   \]
   or
   \[
   f(x) = -(x - 8)^2 - 9
   \]

### Conclusion:
The quadratic function in vertex form that opens down, has a vertex at \((8, -9)\), and passes through the point \((5, -18)\) is:

\[
f(x) = -(x - 8)^2 - 9
\]

### Would you like more details or have any questions?

### Related Questions:
1. How do you convert a quadratic function from vertex form to standard form?
2. What are the characteristics of a parabola that opens downward?
3. How do you find the axis of symmetry for a quadratic function in vertex form?
4. How can you find the roots of a quadratic function given in vertex form?
5. How does changing the value of \(a\) in the vertex form affect the graph of the parabola?

### Tip:
For a quadratic function in vertex form, the sign of \(a\) determines whether the parabola opens upwards (\(a > 0\)) or downwards (\(a < 0\)).

Learn how to write a quadratic function in vertex form f(x) = a(x - h)^2 + k where the graph opens downward, with a vertex at (8, -9) and passing through the point (5, -18). Step-by-step solution provided.

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