The problem provided asks for the transformation applied to the graph of $f(x) = x^3$ to get the graph of the function $g(x) = (x + 3)^3$.

The key observation here is that the transformation is within the argument of the function. Specifically, adding $+3$ inside the function $(x + 3)$ causes a horizontal shift to the left by 3 units.

This happens because adding a constant $h$ inside a function $f(x)$ in the form $f(x + h)$ shifts the graph horizontally by $-h$ units. Therefore, $(x + 3)^3$ shifts the graph 3 units to the left.

Hence, the correct answer is H. Shift the graph 3 units to the left.

Let me know if you'd like any further explanation or have questions!

Here are 5 follow-up questions related to this topic:

1. What happens when the function $f(x) = x^3$ is transformed into $g(x) = (x - 2)^3$?
2. How would the graph of $f(x) = x^3$ change if it were transformed to $g(x) = x^3 + 5$?
3. What is the effect of multiplying the function $f(x) = x^3$ by a constant, say $g(x) = 2x^3$?
4. How do vertical shifts differ from horizontal shifts in terms of function transformations?
5. Can you describe the transformation for the function $f(x) = x^3$ if it becomes $g(x) = -x^3$?

Tip: Always check whether the transformation is happening inside the function (affecting $x$) for horizontal shifts, or outside the function (affecting the whole expression) for vertical shifts.