Question

find the vertex, focus, and directrix for the parabola \\((x - 1)^2 = 4(y + 3)\\)

vertex:
focus:
equation of the directrix:

Explanation:

Step 1: Identify standard form

Compare the given equation with standard form \((x-h)^2 = 4p(y-k)\).

$$(x-1)^2 = 4(y+3)$$

Step 2: Determine the vertex

Find \(h\) and \(k\) from the equation.

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

Step 3: Find value of p

Solve for \(p\) using the coefficient of the linear term.

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

Step 4: Calculate the focus

Add \(p\) to the \(y\)-coordinate of the vertex.

$$(h, k+p) = (1, -3+1) = (1, -2)$$

Step 5: Find the directrix

Subtract \(p\) from the \(y\)-coordinate of the vertex.

$$y = k - p \implies y = -3 - 1 = -4$$

Answer:

Vertex: \((1, -3)\)
Focus: \((1, -2)\)
Equation of the directrix: \(y = -4\)