Final answer:
After substituting -x for x in the polynomial f(x) = x^3 + 11x^2 + 39x + 45, it's determined that the graph of the polynomial has neither y-axis symmetry nor origin symmetry.
Step-by-step explanation:
To determine whether the graph of the polynomial f(x) = x^3 + 11x^2 + 39x + 45 has y-axis symmetry, origin symmetry, or neither, we need to look at the function and its behavior when we substitute -x for x.
Y-axis symmetry is present if f(-x) = f(x), which would indicate that the graph is mirrored across the y-axis. For the given polynomial, if we substitute -x into the function, we get: f(-x) = (-x)^3 + 11(-x)^2 + 39(-x) + 45, which simplifies to -x^3 + 11x^2 - 39x + 45. This is not the same as the original function, so there is no y-axis symmetry.
To check for origin symmetry, we must have f(-x) = -f(x). Applying -x in the function we obtained -x^3 + 11x^2 - 39x + 45, which needs to be the negative of the original function for origin symmetry to hold. Multiplying the original function by -1, we get -x^3 - 11x^2 - 39x - 45, which again doesn't match our substitution result, showing there is no origin symmetry either.
Therefore, the graph of the given polynomial has neither y-axis symmetry nor origin symmetry.