Question

graph the hyperbola described by the equation:

\\\frac{(x - 3)^2}{16} - \frac{(y + 2)^2}{25} = 1\\

Explanation:

Step1: Identify the center

$$(h, k) = (3, -2)$$

Step2: Find semi-axes lengths

$$a = \sqrt{16} = 4,\quad b = \sqrt{25} = 5$$

Step3: Determine the vertices

$$(h \pm a, k) = (3 \pm 4, -2) \implies (-1, -2)\text{ and }(7, -2)$$

Step4: Determine the co-vertices

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

Step5: Find the asymptotes

$$y - k = \pm \frac{b}{a}(x - h) \implies y + 2 = \pm \frac{5}{4}(x - 3)$$

Answer:

The hyperbola is a horizontal hyperbola with the following key features:

• Center: \((3, -2)\)
• Vertices: \((-1, -2)\) and \((7, -2)\)
• Co-vertices: \((3, -7)\) and \((3, 3)\)
• Asymptotes: \(y + 2 = \pm \frac{5}{4}(x - 3)\)