Thanks to those who helped with my question: y+3=-1/12(x-1)^2. The only thing I am still unsure of is how to write the equation for the directrix. Do I just simply write 0=3-3, (the values plugged in for y=k-p). Thanks again for your help!!
Whoops and one thing I forgot. Is it allowed for a parobala to open upward and have the directrix intercept it? Since p is greater than zero, it would have to open upward, but then the directrix would intercept the parobala (the directrix follows the x axis, and is at zero y)
p = Negative. (I called p a but same difference)
I showed you that this parabola opens downward in my explanation of the axis of symmetry at x = 1
the directrix is a horizontal line (m=0)
y = 0 x + 0
or
y = 0
The form is
(x-h)^2 = 4 a (y-k)
here
(x-1)^2 = -12 (y+3)
so 4 a = -12
a = -3
vertex to focus = a = -3
so focus at -3-3 = -6 so (1,-6)
directrix is horizontal line a units above vertex so it is the line y = 0
To write the equation of the directrix, you need to understand the standard form of a parabolic equation. The standard form of a parabola equation is given by:
(y - k) = 4p(x - h)^2
In this equation, (h, k) represents the vertex of the parabola, and p represents the distance between the vertex and the focus point.
Now, let's go through the process step by step:
Step 1: Convert the given equation, y + 3 = -1/12(x - 1)^2, to the standard form by isolating y.
Start by subtracting 3 from both sides of the equation:
y = -1/12(x - 1)^2 - 3
Step 2: Identify the vertex.
From the standard form, we can see that the vertex is at the point (h, k). In this case, h = 1 and k = -3.
So, the vertex is (1, -3).
Step 3: Determine the distance between the vertex and the focus (which is equal to p).
For a parabola in the standard form, p is the constant multiplying the expression (x - h)^2. In this case, p = -1/12.
Step 4: Write the equation of the directrix.
The directrix is a horizontal line that is parallel to the x-axis and equidistant to the focus and the vertex. Since the parabola opens downward, the directrix will be p units below the vertex.
The equation of the directrix is given by the equation:
y = k + p
Substituting the values, we get:
y = -3 + (-1/12)
Simplifying this equation, we get:
y = -3 - 1/12
So, the equation of the directrix is:
y = -37/12
Therefore, the equation of the directrix for the given parabola is y = -37/12.