To find the standard form of the equation of a parabola, we start by identifying its orientation based on the position of the focus and vertex. In this case, the vertex is at the origin (0, 0) and the focus is at (0, 5).
Since the focus is located above the vertex, this indicates that the parabola opens upwards. The standard form of the equation for a parabola that opens upwards is:
y = ax^2
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, 5) is 5 units. The value of p (the distance from the vertex to the focus) is thus 5.
In the standard form, we can also express the parabola using the formula:
y = (1/(4p))x^2
Substituting p with 5, we get:
y = (1/(4*5))x^2
Simplifying this yields:
y = (1/20)x^2
Therefore, the standard form of the equation of the parabola with vertex at the origin and focus at (0, 5) is:
y = (1/20)x^2