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The given equation is:

\[
\frac{(x - 6)^2}{9} - \frac{(y + 8)^2}{4} = 1
\]

This is the equation of a hyperbola in standard form, specifically in the form:

\[
\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1
\]

### Breakdown:
- **Center**: The center of the hyperbola is \( (h, k) \). From the equation, we can identify that \( h = 6 \) and \( k = -8 \), meaning the center is at \( (6, -8) \).
- **Orientation**: Since the \( x \)-term comes first and the equation is in the form \(\frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1\), the hyperbola opens left and right.
- **Values of \(a\) and \(b\)**: 
    - \( a^2 = 9 \), so \( a = 3 \)
    - \( b^2 = 4 \), so \( b = 2 \)

This means the hyperbola has a horizontal transverse axis (along the x-axis) and a vertical conjugate axis (along the y-axis).

### Correct Answer:
- The center is \( (6, -8) \), \( a = 3 \), and \( b = 2 \).

Thus, the correct graph is the one with the choice:
**C(6, -8); a = 3; b = 2**, which is the **bottom-right option**.

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Would you like more details on hyperbolas or have further questions?

Here are five related questions for deeper understanding:
1. What is the general form of a hyperbola?
2. How do you distinguish between a hyperbola with horizontal vs. vertical axes?
3. What are the asymptotes of a hyperbola, and how are they calculated?
4. How can the foci of a hyperbola be determined from its equation?
5. What is the relationship between the values of \(a\), \(b\), and the hyperbola's eccentricity?

**Tip**: Always identify the center of a conic section first, as it helps in sketching the graph accurately!

Determine the correct graph for the hyperbola equation: (x - 6)^2 / 9 - (y + 8)^2 / 4 = 1

Learn how to identify the correct graph for the hyperbola given by the equation (x - 6)^2 / 9 - (y + 8)^2 / 4 = 1. This problem explores key concepts in analytic geometry, including the standard form of hyperbolas, identifying the center, and determining the transverse and conjugate axes. Suitable for high school students in Grades 9-12.

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