What is the equation of a parabola with its vertex at the origin (0, 0) and focus at the point (0, 2)?

The equation of a parabola can be derived based on its geometric properties. In this case, we have a parabola with a vertex located at the origin (0, 0) and a focus located at the point (0, 2).

Firstly, since the focus is positioned above the vertex along the y-axis, this indicates that the parabola opens upwards. The standard form of a vertical parabola’s equation can be written as:

y = a(x - h)² + k

In this equation, (h, k) is the vertex of the parabola, which in our case translates to (0, 0). Therefore, we can simplify our equation to:

y = ax²

Next, we need to determine the value of ‘a’. The distance from the vertex to the focus is important in identifying ‘a’. This distance is known as ‘p’. In our scenario, the focus is 2 units above the vertex, thus:

p = 2

For parabolas opening upwards, the relationship between ‘a’ and ‘p’ is defined as:

a = rac{1}{4p}

Substituting the value of ‘p’ into this equation yields:

a = rac{1}{4 	imes 2} = rac{1}{8}

Now we can substitute ‘a’ back into our parabola’s equation:

y = rac{1}{8}x²

Therefore, the equation of the parabola with its vertex at (0, 0) and its focus at (0, 2) is:

y = rac{1}{8}x².

This parabola will open upwards, with its vertex at the origin and focus correctly positioned at (0, 2).