1. **State the problem:** We want to graph the function $y = x^2$. 2. **Formula and rules:** The function $y = x^2$ is a quadratic function, which graphs as a parabola opening upwards. 3. **Key points:** To graph it, calculate $y$ for several values of $x$: $$ \begin{aligned} x &= -2, & y &= (-2)^2 = 4 \\ x &= -1, & y &= (-1)^2 = 1 \\ x &= 0, & y &= 0^2 = 0 \\ x &= 1, & y &= 1^2 = 1 \\ x &= 2, & y &= 2^2 = 4 \end{aligned} $$ 4. **Plot points:** Plot the points $(-2,4)$, $(-1,1)$, $(0,0)$, $(1,1)$, and $(2,4)$ on the coordinate plane. 5. **Shape:** Connect these points with a smooth curve forming a U-shaped parabola. 6. **Vertex:** The vertex is at $(0,0)$, the minimum point. 7. **Axis of symmetry:** The vertical line $x=0$ is the axis of symmetry. This is how you graph $y = x^2$ step-by-step.