The equation of a parabola is 12y=(x−1)^2−48. Identify the vertex, focus, and directrix of the parabola.
To find the vertex of the parabola, we need to rewrite the equation in the form y = a(x-h)^2 + k.
12y = (x-1)^2 - 48
y = (1/12)(x-1)^2 - 4
This means the vertex is at (h, k) = (1, -4).
To find the focus of the parabola, we use the formula c = 1/(4a), which tells us how far the focus is from the vertex. In this case, a = 1/12, so c = 1/(4*1/12) = 3.
Since the parabola opens upwards, the focus is located 3 units above the vertex. So the focus is at (1, -1).
To find the directrix of the parabola, we use the formula y = k - a, where a = 1/12 and k = -4.
The directrix of the parabola is the horizontal line y = -4 - 1/12 = -49/12 or approximately -4.08.