Question

sketch a graph of \frac{(x - 1)^2}{4} - (y + 3)^2 = 1

Explanation:

Step1: Identify the conic section type

\[\frac{(x-1)^2}{4} - \frac{(y+3)^2}{1} = 1\]

Step2: Determine the center

\[(h, k) = (1, -3)\]

Step3: Find the semi-axes values

\[a = \sqrt{4} = 2,\quad b = \sqrt{1} = 1\]

Step4: Calculate the vertices

\[(h \pm a, k) \Rightarrow (3, -3)\text{ and }(-1, -3)\]

Step5: Determine the asymptotes

\[y + 3 = \pm \frac{1}{2}(x - 1)\]

Answer:

The graph is a horizontal hyperbola with:

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