Question

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

Explanation:

Step1: Identify the conic section

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

Step2: Determine the center

$$(h, k) = (1, -3)$$

Step3: Find the semi-axes

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

Step4: Locate the vertices

$$(1 \pm 2, -3) \Rightarrow (3, -3)\text{ and }(-1, -3)$$

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