the equation of a parabola is 12y = (x-1) ^2-48 identify the vertex, focus, and directrix of the parabola.

the equation of a vertically opening parabola is

(x - h) ² = 4p (y - k)

where (h, k) are the coordinates of the vertex and p is the distance of the focus and directrix from the vertex. If 4p is positive parabola opens up and if 4p is negative parabola opens down

rearrange 12y = (x - 1) ² - 48 into this form

(x - 1) ² = 12y + 48

(x - 1) ² = 12 (y + 4) ← in standard form

Vertex = (1, - 4)

4p = 12 ⇒ p = 3 ⇒ parabola opens upwards

the focus is above the vertex and the directrix is below the vertex in a vertically opening up parabola

Focus = (1, - 4 + 3) = (1, - 1)

Directrix has equation y = - 4 - 3 ⇒ y = - 7