Final answer:
The standard form equation of the parabola with vertex at the origin and focus at (0,-1/20) is y = -5x², as it opens downwards due to the negative y-coordinate of the focus.
Step-by-step explanation:
To write a standard form equation of a parabola with its vertex at the origin and focus at (0,-1/20), we recall that the standard form of a parabola with the vertex at the origin is y = ax² when the parabola opens upwards or downwards, and x = ay² when it opens to the left or to the right.
Since the focus is at (0,-1/20), and the vertex is at the origin, the parabola opens upwards if the focus is above the vertex or downwards if the focus is below the vertex. In this case, since the focus has a negative y-coordinate, the parabola opens downwards. Therefore, we use the downward-opening parabola equation y = ax².
The distance between the vertex and the focus is the absolute value of the y-coordinate of the focus, |0 - (-1/20)| = 1/20. This distance is also known as the focal length, denoted by 'p'. Therefore, we have p = 1/20. For a downward-opening parabola, a is negative and we have a = -1/(4p). Substituting p, we get a = -1/(4×(1/20)) = -20/4 = -5. Thus, the equation of the parabola is y = -5x².