Question

graph the hyperbola described by the equation:

\\\frac{(y + 2)^2}{25} - \frac{(x + 2)^2}{1} = 1\\

Explanation:

Step1: Identify the center

$$(h, k) = (-2, -2)$$

Step2: Find semi-axes values

$$a^2 = 25 \implies a = 5,\quad b^2 = 1 \implies b = 1$$

Step3: Determine the vertices

$$(h, k \pm a) \implies (-2, -2 \pm 5) \implies (-2, 3)\text{ and }(-2, -7)$$

Step4: Find asymptote equations

$$y - k = \pm \frac{a}{b}(x - h) \implies y + 2 = \pm 5(x + 2)$$

Answer:

The hyperbola is a vertical hyperbola with:

• Center: \((-2, -2)\)
• Vertices: \((-2, 3)\) and \((-2, -7)\)
• Asymptotes: \(y + 2 = \pm 5(x + 2)\)