what is an equation of a parabola with the given vertex and focus? Vertex: (-2,5); focus: (-2,6)
a parabola with focus at (0,p) and directrix at y = -p is
x^2 = 4py
The vertex is midway between the focus and directrix, so your parabola has p = 1
x^2 = 4y
Now shift things so the vertex is at (-2,5) and you have
(x+2)^2 = 4(y-5)
(x-h)^2 = 4p(y-k), (h,k) = (x,y)
x = -2
4p = 4
p = 1
h = -2
k = 5
(x+2)^2 = 4(y-5)
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To find the equation of a parabola with a given vertex and focus, you can use the formula:
(x - h)^2 = 4p(y - k)
where (h, k) is the vertex and p is the distance between the vertex and the focus.
In this case, the vertex is (-2,5) and the focus is (-2,6).
Step 1: Determine the value of p, which is the distance between the vertex and the focus.
Since the y-coordinate of the vertex and the focus differ by 1, we can see that p = 1.
Step 2: Substitute the values of h, k, and p into the equation.
(x - (-2))^2 = 4(1)(y - 5)
Simplifying the equation, we have:
(x + 2)^2 = 4(y - 5)
So, the equation of the parabola with the given vertex and focus is:
(x + 2)^2 = 4(y - 5)