Find the equation of the directrix of the parabola $y = x^2.$
I recall that a parabola is the set of points that are equidistant from a focus and a directrix, which are a point and a line, respectively, that define the parabola. The standard form of a parabola with a vertical axis of symmetry is $y = a(x - h)^2 + k,$ where $(h, k)$ is the vertex, $a$ determines the width and direction of the parabola, and the focus is $(h, k + \frac{1}{4a}).$ The directrix is a horizontal line with the equation $y = k - \frac{1}{4a}.$ In this problem, the parabola is given as $y = x^2,$ which means that $a = 1, h = 0,$ and $k = 0.$ Therefore, the vertex is $(0, 0),$ the focus is $(0, \frac{1}{4}),$ and the directrix is $y = -\frac{1}{4}.$ 