To effectively sketch the graph of the function $y = x^3 - x$, it is essential to identify key points, observe symmetry, and analyze the behavior of the graph as $x$ approaches large values.

1. Finding the Roots: To locate the roots, we set $y = 0$. Solving the equation $x^3 - x = 0$ yields:

From this, we find the roots $x = 0$, $x = 1$, and $x = -1$. These points represent where the graph intersects the x-axis.

2. Determining Turning Points: Next, we need to find the turning points by calculating the first derivative of the function:

Setting the first derivative to zero, we solve:

This gives us the solutions:

To classify these turning points, we calculate the second derivative:

Evaluating the second derivative at the turning points:

• At $x = \frac{1}{\sqrt{3}}$:

This indicates a local minimum.

• At $x = -\frac{1}{\sqrt{3}}$:

This indicates a local maximum.

3. Evaluating the Function at Turning Points: Now, we compute the values of $y$ at these turning points:

• For the local minimum:

• For the local maximum:

4. Analyzing Symmetry: The graph exhibits symmetry about the origin, which can be confirmed by the property $y(-x) = -y(x)$. This indicates that the function is odd.

5. Examining End Behavior: Finally, we analyze the end behavior of the function. As $x$ approaches infinity ($x \to \infty$), $y$ also approaches infinity ($y \to \infty$). Conversely, as $x$ approaches negative infinity ($x \to -\infty$), $y$ approaches negative infinity ($y \to -\infty$). This means the graph extends infinitely in both directions.

6. Sketching the Graph: With the roots, turning points, and symmetry established, you can proceed to plot these key points. Ensure the curve passes through the roots at $(0, 0)$, $(1, 0)$, and $(-1, 0)$ and follows the identified behavior, including the local maximum at $\left(-\frac{1}{\sqrt{3}}, -\frac{2}{3\sqrt{3}}\right)$ and the local minimum at $\left(\frac{1}{\sqrt{3}}, \frac{2}{3\sqrt{3}}\right)$. The final sketch will be a smooth curve that reflects all the analyzed characteristics.