given that a parabola has a directix of =-3.25 and focus of (0,-2.75) find the vertex
I assume you mean directix of y=-3.25
WEll, vertex is half way between focus and directrix.
vertex (0,-3)
0,-3
To find the vertex of a parabola given the directrix and focus, you can use the formula:
Vertex = (h, k)
where h is the x-coordinate and k is the y-coordinate of the vertex.
First, determine the equation of the parabola based on the given directrix and focus. The formula for the equation of a parabola with a vertical axis is:
(y - k)^2 = 4p(x - h)
where (h, k) is the vertex and p is the distance between the vertex and the focus (which is also the distance between the vertex and the directrix).
In this case, the y-coordinate of the focus is -2.75, which is also the y-coordinate of the vertex (k = -2.75). The x-coordinate of the vertex (h) is the midpoint between the x-coordinate of the focus and the x-coordinate of the directrix.
The given directrix is x = -3.25, so the x-coordinate of the directrix is -3.25.
Now, compute the x-coordinate of the vertex (h):
h = (x-coordinate of the focus + x-coordinate of the directrix) / 2
h = (0 + (-3.25)) / 2
h = -3.25 / 2
h = -1.625
Thus, we have the vertex V(-1.625, -2.75).