How do you sketch the graph of y = x^3 - x?

To sketch the graph of \( y = x^3 - x \), identify key points, symmetry, and behaviour at large \( x \).

First, find the roots of the equation by setting \( y = 0 \). Solving \( x^3 - x = 0 \) gives \( x(x^2 - 1) = 0 \), so \( x = 0 \), \( x = 1 \), and \( x = -1 \). These are the points where the graph crosses the x-axis.

Next, determine the turning points by finding the first derivative, \( y' = 3x^2 - 1 \), and setting it to zero. Solving \( 3x^2 - 1 = 0 \) gives \( x = \pm \frac{1}{\sqrt{3}} \). To classify these points, find the second derivative, \( y'' = 6x \). At \( x = \frac{1}{\sqrt{3}} \), \( y'' > 0 \), indicating a local minimum. At \( x = -\frac{1}{\sqrt{3}} \), \( y'' < 0 \), indicating a local maximum.

Evaluate the function at these points: \( y\left(\frac{1}{\sqrt{3}}\right) = \frac{2}{3\sqrt{3}} \) and \( y\left(-\frac{1}{\sqrt{3}}\right) = -\frac{2}{3\sqrt{3}} \).

The graph is symmetric about the origin because \( y(-x) = -y(x) \), indicating odd symmetry.

Finally, analyse the end behaviour. As \( x \to \infty \) or \( x \to -\infty \), \( y \to \infty \) or \( y \to -\infty \), respectively, showing the graph extends to infinity in both directions.

Plot the roots, turning points, and sketch the curve, ensuring it passes through these points and follows the identified behaviour.