### Instruction: Find the focus of the the parabola $y = x^2.$.### Response: To find the focus of a parabola, I need to recall the standard form of a parabola, which is $y = a(x - h)^2 + k,$ where $(h, k)$ is the vertex and $a$ determines the width and direction of the parabola.
In this case, the given equation is already in standard form, with $a = 1, h = 0,$ and $k = 0.$
So the vertex is $(0, 0).$
I also need to remember the formula for the distance from the vertex to the focus, which is $\frac{1}{4a}.$
Since $a = 1,$ the distance from the vertex to the focus is $\frac{1}{4}.$
Now I need to decide whether the focus is above or below the vertex.
Since the parabola opens upward, the focus must be above the vertex.
Therefore, to find the coordinates of the focus, I need to add $\frac{1}{4}$ to the y-coordinate of the vertex, and keep the x-coordinate the same.