The graph shown in the image is of a cubic function due to its S-shaped nature, which suggests that its highest degree term is $x^3$. Based on this, we can eliminate the options that involve quadratic equations.

The equation of the graph must also capture the intercepts. From the graph, it passes through the x-axis at points $x = -2$, $x = 0$, and $x = 2$. These x-intercepts suggest that the function has factors of $(x+2), x, (x-2)$, which leads to an expression like $y = x(x - 2)(x + 2)$, simplifying to:

$y = x(x^2 - 4) = x^3 - 4x$

This matches option D: $y = x^3 - 4x$.

Thus, the correct answer is D.

Would you like any further details on the process or graph interpretation?

Here are five related questions to deepen your understanding:

1. How can we distinguish between quadratic and cubic functions from their graphs?
2. What are the key features of a cubic function graph?
3. How can we derive the equation of a polynomial function from its graph?
4. What role do intercepts play in determining the factors of a function?
5. Can we determine if a cubic function has turning points by looking at the graph?

Tip: For cubic functions, look for the characteristic S-shape and check for the number of x-intercepts to identify potential factors.