The equation of a parabola
is 12y = (x- 1) - 48.
Identify the vertex, focus, and directrix of the parabola.
To determine the vertex, focus, and directrix of the parabola, we first need to put the equation in standard form.
Rearranging the given equation:
12y = (x - 1)^2 - 48
y = (1/12)(x - 1)^2 - 4
Now the equation is in the form y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Comparing the equation to standard form, we can see that h = 1 and k = -4. Therefore, the vertex of the parabola is V(1, -4).
Now we need to find the focus of the parabola. The focus of a parabola in the form y = a(x - h)^2 + k is located at the point (h, k + 1/(4a)). In this case, a = 1/12, h = 1, and k = -4.
Therefore, the focus of the parabola is located at F(1, -4 + 1/(4(1/12))) = F(1, -3.75).
Lastly, we need to find the directrix of the parabola. The directrix is a horizontal line located at a distance of 1/(4a) units below the vertex. In this case, a = 1/12.
Therefore, the directrix of the parabola is a horizontal line located at a distance of 1/(4(1/12)) units below the vertex, or 3 units below the vertex. The equation of the directrix is y = -4 - 3 = y = -7.