Question

graph: \\((y + 2)^2 = -4(x - 1)\\)

Explanation:

Step1: Identify the standard form

$$(y - k)^2 = 4p(x - h)$$

Step2: Determine the vertex

$$(h, k) = (1, -2)$$

Step3: Find the value of p

$$4p = -4 \implies p = -1$$

Step4: Determine the focus

$$(h + p, k) = (1 - 1, -2) = (0, -2)$$

Step5: Determine the directrix

$$x = h - p \implies x = 1 - (-1) \implies x = 2$$

Step6: Find key points

$$x = 0 \implies (y + 2)^2 = 4 \implies y = -1 \text{ or } y = -3$$

Answer:

The graph of the equation \((y + 2)^2 = -4(x - 1)\) is a horizontal parabola opening to the left with its vertex at \((1, -2)\), focus at \((0, -2)\), directrix \(x = 2\), and passing through the points \((0, -1)\), \((0, -3)\), \((-3, 0)\), and \((-3, -4)\).