To find the equation of the directrix, as well as the coordinates of the vertex and the focus of the given equation, let's work step by step:
Step 1: Identify the standard form of the equation
The equation you provided, y + 3 = -(1/12)(x - 1)^2, is in vertex form: y = a(x - h)^2 + k. Here, (h, k) represents the vertex of the parabola.
Step 2: Identify the vertex
Comparing the given equation with the standard form, we can see that h = 1 and k = -3. Therefore, the vertex of the parabola is at (1, -3).
Step 3: Find the value of "a"
From the given equation, we can see that the coefficient 'a' is -(1/12). The value of 'a' determines the shape and direction of the parabola.
Step 4: Calculate the directrix
The directrix of a parabola is a straight line perpendicular to the axis of symmetry. Its equation is given by:
y = k - a
Substituting the values of k and a, we get:
y = -3 - (-(1/12))
Simplifying further:
y = -3 + 1/12
y = -35/12
Therefore, the equation of the directrix is y = -35/12.
Step 5: Calculate the focus
The distance between the vertex and the focus is denoted by 'p' and is given by p = 1 / (4a). In this case, a = -(1/12), so:
p = 1 / (4 * -(1/12))
Simplifying further:
p = -3
Since the parabola opens downwards, the focus will be located above the vertex. So, to get the coordinates of the focus, we move p units above the vertex.
For the given equation, the coordinates of the focus are (1, -3 - 3) = (1, -6).
So, to summarize:
- The vertex is at (1, -3).
- The equation of the directrix is y = -35/12.
- The focus is at (1, -6).