1. **State the problem:** We are given the equation of a hyperbola: $$ (x - 1)^2 - \frac{(y + 2)^2}{9} = 1 $$ and asked to understand its properties and graph it. 2. **Identify the form:** This is a hyperbola centered at $(h, k) = (1, -2)$ with the form: $$ \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 $$ where $a^2 = 1$ and $b^2 = 9$. 3. **Key properties:** - Center: $(1, -2)$ - $a = \sqrt{1} = 1$ - $b = \sqrt{9} = 3$ - Since the $x$ term is positive and comes first, the hyperbola opens left and right (horizontal transverse axis). 4. **Vertices:** Located $a$ units left and right from the center along the $x$-axis: $$ (1 - 1, -2) = (0, -2) \quad \text{and} \quad (1 + 1, -2) = (2, -2) $$ 5. **Foci:** Located $c$ units from the center, where $c^2 = a^2 + b^2 = 1 + 9 = 10$, so $c = \sqrt{10}$. Foci coordinates: $$ (1 - \sqrt{10}, -2) \quad \text{and} \quad (1 + \sqrt{10}, -2) $$ 6. **Asymptotes:** Equations of asymptotes for horizontal hyperbola: $$ y = k \pm \frac{b}{a}(x - h) $$ Substitute values: $$ y = -2 \pm 3(x - 1) $$ which simplifies to: $$ y = -2 + 3(x - 1) = 3x - 5 $$ and $$ y = -2 - 3(x - 1) = -3x + 1 $$ 7. **Summary:** The hyperbola is centered at $(1, -2)$, opens left and right, with vertices at $(0, -2)$ and $(2, -2)$, foci at $(1 \pm \sqrt{10}, -2)$, and asymptotes $y = 3x - 5$ and $y = -3x + 1$. This matches the graph described with axes from -7 to 7 and the hyperbola centered at $(1, -2)$.