What is the focus and directrix of the parabola y=x^2

First the vertex has to be found. Since there is nothing being added to the y or the x, the vertex is at (0,0).

Next, we have to identify the orientation of the parabola. The squared is on the x and the x is positive, so this is an upward-opening parabola. 

The coefficient of the un-squared side (x^2) is always 4p, so in this case 4p = 1. When we solve for p we get p = 1/4

Finally, the focus is always inside of the parabola, that is, inside the concavity or the curve. Since this is facing upwards, the focus must be higher than the vertex. So we add 1/4 to the y-value of the vertex and we get the focus to be at (0,1/4).

The directrix will always be the opposite direction, so we subtract 1/4 from the y-value of the vertex, and we get the directrix to be y = -1/4 (remember, the directrix is a line, not a point).