The function $f(x) = x^2$ is a parabola that opens upwards. Let’s break down how to sketch the graph and analyze its properties.

---

Steps to Sketch the Graph:

1. Plot Key Points:
   • The vertex is at $(0, 0)$ (minimum point).
   • Select additional points to plot: $(-2, 4), (-1, 1), (0, 0), (1, 1), (2, 4)$.
2. Symmetry:
   • The graph is symmetric about the $y$-axis.
3. Draw the Curve:
   • Connect the points with a smooth curve, ensuring the "U" shape.

---

Properties of $f(x) = x^2$:

1. Domain: $(-\infty, \infty)$ (all real numbers).
2. Range: $[0, \infty)$ (values start at 0 and increase).
3. Symmetry: The graph is symmetric about the $y$-axis.
4. Vertex: $(0, 0)$, which is the minimum point.
5. Intercepts:
   • $y$-intercept: $(0, 0)$.
   • $x$-intercept: $(0, 0)$ (double root).
6. Behavior:
   • As $x \to \infty$, $f(x) \to \infty$.
   • As $x \to -\infty$, $f(x) \to \infty$.

---

Would you like me to sketch the graph for you visually or provide further details?

---

Questions for Further Exploration:

1. How does the graph of $f(x) = x^2$ compare to $f(x) = ax^2 + bx + c$?
2. What transformations occur if the function is changed to $f(x) = (x - h)^2 + k$?
3. How do we calculate the axis of symmetry for any quadratic function?
4. What is the effect of adding or subtracting a constant $c$ to the function $f(x) = x^2$?
5. How do quadratic functions apply in real-world scenarios, such as physics or economics?

Tip:

When sketching functions, always start by identifying key features like intercepts, vertex, and symmetry to simplify the process.