parabola with vertex at (3,-2) and a directreix of x=2. how do i wrtie this as an equation....PLEASE HELPPPPPPPP
To write the equation of a parabola with the vertex at (h, k) and the directrix given as a vertical line x = a, you can use the formula:
4p(y - k) = (x - h)^2
where:
- p is the distance between the vertex and the focus (or from the vertex to the directrix)
- (h, k) are the coordinates of the vertex
In this case, the vertex is (3, -2) and the directrix is x = 2. To determine the value of p and write the equation, we need to find the distance between the vertex and the directrix.
The directrix x = 2 is a vertical line, and it is parallel to the y-axis. Since the parabola opens upwards or downwards, the distance from the vertex to the directrix is the vertical distance.
The vertex is at (3, -2), and the directrix is the line x = 2. So, the distance between the vertex and the directrix is 2 units.
Now let's plug in the values into the equation:
4p(y - k) = (x - h)^2
4p(y - (-2)) = (x - 3)^2
4p(y + 2) = (x - 3)^2
We know that p = the distance between the vertex and the directrix, which is 2 units.
4(2)(y + 2) = (x - 3)^2
8(y + 2) = (x - 3)^2
Therefore, the equation of the parabola is 8(y + 2) = (x - 3)^2.