The end behavior of a polynomial function is determined by the term with the highest degree.

In this case, the term with the highest degree is 3x^6. As the value of x moves away from zero to the right or to the left, the value of 3x^6 becomes increasingly larger, meaning that the graph of the polynomial function will increase without bound in both the positive and negative directions. Therefore, the end behavior of the graph of the polynomial function f(x) = 3x^6 + 30x^5 + 75x^4 is that it increases without bound in both the positive and negative directions.