No, the coordinates are vertex is (5, 6.5)

Thank you for the correction. Let's proceed with the correct vertex \((5, 6.5)\).

### Step 1: Identify the Vertex and Directrix
Given:
- The vertex of the parabola is at the point \((5, 6.5)\).
- The directrix is the horizontal line \(y = 8.5\).

### Step 2: Determine the Equation of the Parabola
The standard form of a parabola with a vertical axis of symmetry is:

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

Where \((h, k)\) is the vertex of the parabola. In this case, \((h, k) = (5, 6.5)\).

### Step 3: Calculate the Value of \(a\)
The distance between the vertex and the directrix is \(p\), and it is related to \(a\) by the equation:

\[
a = \frac{1}{4p}
\]

The distance between the vertex \((5, 6.5)\) and the directrix \(y = 8.5\) is:

\[
p = |8.5 - 6.5| = 2
\]

Thus,

\[
a = \frac{1}{4 \times 2} = \frac{1}{8}
\]

Since the parabola opens downwards (because the vertex is below the directrix), \(a\) will be negative:

\[
a = -\frac{1}{8}
\]

### Step 4: Write the Equation of the Parabola
Substituting the values of \(h = 5\), \(k = 6.5\), and \(a = -\frac{1}{8}\) into the standard form:

\[
(y - 6.5) = -\frac{1}{8}(x - 5)^2
\]

### Step 5: Determine the Focus and Axis of Symmetry
- **Focus**: The focus lies \(p = 2\) units below the vertex, at the point \((5, 4.5)\).
- **Axis of Symmetry**: The axis of symmetry is the vertical line that passes through the vertex, which is \(x = 5\).

### Final Answer
- **Equation of the parabola**: \((y - 6.5) = -\frac{1}{8}(x - 5)^2\)
- **Focus**: \((5, 4.5)\)
- **Axis of Symmetry**: \(x = 5\)

Would you like further details or explanations on any part of this problem? Here are some related questions to deepen your understanding:

1. How would the equation change if the vertex were shifted to another quadrant?
2. How do the coefficients in the parabola's equation affect its width and direction?
3. How would you derive the equation if the parabola opened to the right instead of downwards?
4. Can you find the coordinates of the directrix if the parabola opens upwards?
5. What is the geometric significance of the focus in the context of a parabola?

**Tip:** Understanding the relationship between the vertex, focus, and directrix is key to mastering parabolas and their equations.

Explore the equation of a parabola with its vertex at (5, 6.5) and directrix y = 8.5. Learn how to derive and understand the equation, calculate the focus, and identify the axis of symmetry. Suitable for high school students studying algebra and conic sections.

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