Find the equation of the parabola determined by the given information:
vertex at the origin, focus at (0,3)
To find the equation of a parabola given its vertex and focus, you can use the general equation of a parabola that is vertically oriented. The equation is as follows:
(x - h)^2 = 4p(y - k)
Where (h, k) represents the vertex of the parabola, and p represents the distance between the vertex and the focus.
Given that the vertex is at the origin (0, 0) and the focus is at (0, 3), we can substitute these values into the equation:
(x - 0)^2 = 4p(y - 0)
Simplifying,
x^2 = 4py
Now we need to find the value of p, which represents the distance between the vertex and the focus. In this case, the distance from the vertex to the focus is 3, so p = 3.
Substituting the value of p into the equation, we get:
x^2 = 4(3)y
Simplifying further,
x^2 = 12y
Therefore, the equation of the parabola determined by the vertex at the origin and the focus at (0, 3) is x^2 = 12y.