Using the following information for the Vertex and Directrix, write the standard form equation for the parabola with what is given below.
Vertex: (-2,1)
Directrix: x=1
The horizontal parabola
y^2 = 4px has directrix p units from the vertex. So, since our directrix is 3 units from the vertex, we start with
y^2 = 12x
But, that's with a vertex of (0,0). So, our parabola is
(y-1)^2 = 12(x+2)
But, that opens to the right. Our vertex is to the left of the directrix, so we wind up with
(y-1)^2 = -12(x+2)
Verify that at
http://www.wolframalpha.com/input/?i=parabola+%28y-1%29%5E2+%3D+-12%28x%2B2%29
To write the standard form equation for the parabola given the vertex and the directrix, we can use the following steps:
Step 1: Identify the coordinates of the vertex. In this case, the vertex is given as (-2,1).
Step 2: Identify the equation for the directrix. In this case, the directrix is given as x=1.
Step 3: Determine the orientation of the parabola. The orientation can be either vertical or horizontal, depending on whether the directrix is vertical or horizontal. Since the directrix is x=1 (a vertical line), the parabola will have a vertical orientation.
Step 4: Use the vertex form of the standard equation for a vertically-oriented parabola, which is given by:
(x - h)^2 = 4p(y - k)
Where (h, k) are the coordinates of the vertex, and p is the distance between the vertex and the focus point on the parabola.
Step 5: Calculate the value of p. The distance between the vertex and the directrix is equal to the distance between the vertex and the focus point, so p can be calculated using the formula:
p = (Distance between vertex and directrix) / 2
In this case, the distance between the vertex and the directrix is 1 unit (since the directrix is x=1 and the x-coordinate of the vertex is -2). Therefore, p = 1 / 2 = 0.5.
Step 6: Substitute the values of (h, k) and p into the vertex form equation. In this case, (h, k) = (-2, 1) and p = 0.5. Thus, the equation becomes:
(x + 2)^2 = 4(0.5)(y - 1)
Simplifying the equation further would give the standard form equation for the parabola:
(x + 2)^2 = 2(y - 1)