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.

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!