1. The problem is to understand and analyze the function $y = x^2$. 2. This is a quadratic function, which generally has the form $y = ax^2 + bx + c$. Here, $a=1$, $b=0$, and $c=0$. 3. Important properties of quadratic functions include the vertex, axis of symmetry, and intercepts. 4. For $y = x^2$, the vertex is at the origin $(0,0)$ because the function is in standard form with no shifts. 5. The axis of symmetry is the vertical line $x=0$. 6. The y-intercept is found by evaluating $y$ at $x=0$: $y = 0^2 = 0$. 7. The x-intercepts are found by solving $x^2 = 0$, which gives $x=0$. 8. The graph is a parabola opening upwards because $a=1 > 0$. 9. The function is always non-negative, $y \\geq 0$ for all real $x$. 10. This function is symmetric about the y-axis. Final answer: The function $y = x^2$ is a parabola with vertex at $(0,0)$, axis of symmetry $x=0$, and intercepts at $(0,0)$.