Which is the equation of a parabola with vertex (0, 0) and focus (0, 2)?

The equation of a parabola can be determined using the coordinates of its vertex and focus. In this case, the vertex of the parabola is at the origin (0, 0) and the focus is at (0, 2).

Since the focus is located above the vertex, we are dealing with a vertical parabola that opens upwards. The standard form of the equation for a parabola that opens upwards with a vertex at (h, k) is:

y = a(x - h)² + k

In our case, the vertex (h, k) is (0, 0), so the equation simplifies to:

y = ax²

Now, to find the value of ‘a’, we can use the distance from the vertex to the focus. The distance from the vertex (0, 0) to the focus (0, 2) is 2. For parabolas, this distance is given by the formula:

p = 1/(4a)

Here, ‘p’ is the distance from the vertex to the focus. We know that ‘p’ is 2, so:

2 = 1/(4a)

Solving for ‘a’, we get:

a = 1/(8)

Therefore, the equation of the parabola is:

y = (1/8)x²

This confirms that the equation of the parabola with vertex (0, 0) and focus (0, 2) is:

y = (1/8)x²