The equation \((y-4)^2 = 8(x-3)\) represents a parabola. To understand its properties, we can rewrite it in a more standard form.

### Step 1: Identify the Parabola's Standard Form
The standard form of a parabola that opens horizontally is:
\[
(y - k)^2 = 4p(x - h)
\]
where \((h, k)\) is the vertex of the parabola, and \(p\) is the distance from the vertex to the focus (the parabola opens to the right if \(p > 0\) or to the left if \(p < 0\)).

### Step 2: Compare with the Given Equation
Given equation:
\[
(y-4)^2 = 8(x-3)
\]

Here, \(k = 4\), \(h = 3\), and \(4p = 8\). Thus:
\[
p = \frac{8}{4} = 2
\]

### Step 3: Interpret the Results
- **Vertex**: The vertex of the parabola is at \((3, 4)\).
- **Focus**: Since \(p = 2\), the focus is \(2\) units to the right of the vertex. Therefore, the focus is at \((3 + 2, 4) = (5, 4)\).
- **Directrix**: The directrix is a vertical line \(2\) units to the left of the vertex. Thus, the directrix is \(x = 3 - 2 = 1\).
- **Orientation**: Since the equation is in the form \((y-k)^2 = 4p(x-h)\), this parabola opens to the right.

### Summary:
- **Vertex**: \((3, 4)\)
- **Focus**: \((5, 4)\)
- **Directrix**: \(x = 1\)
- **Opening**: To the right

Would you like more details or have any questions about this? Here are some related questions you might find helpful:

1. How can I find the length of the latus rectum for this parabola?
2. What is the equation of the parabola if it opens downward instead of to the right?
3. How can I convert this parabola equation into a function of \(x\)?
4. How would the graph of this equation change if the coefficient 8 was negative?
5. Can you explain the geometric significance of the focus and directrix for this parabola?

**Tip:** Understanding the standard forms of conic sections helps in quickly identifying the key features like the vertex, focus, and directrix.

Explore the properties of the parabola described by the equation (y-4)^2 = 8(x-3). Learn about its vertex, focus, directrix, and orientation. Suitable for advanced high school students.

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