Final answer:
The statement about cubic polynomial functions rising to the left and falling to the right is false; the behavior of a cubic function's graph depends on the sign and magnitude of its coefficients, particularly the leading coefficient.
Step-by-step explanation:
The statement that a graph of a cubic polynomial function that rises to the left must necessarily fall to the right is false. While this behavior is common for many cubic functions, it's not a definitive rule. Cubic functions are defined by the equation f(x) =
+ cx + d, where a, b, c, and d are constants, and a ≠ 0. The sign and magnitude of these constants, especially the leading coefficient 'a', influence the end behavior of the function.
For example, if 'a' is positive, the right end of the cubic function's graph will rise, and depending on the signs and magnitudes of the other coefficients, the left end might also rise or fall. Similarly, if 'a' is negative, the right end will fall, and the left end's behavior can vary. Cubic graphs can also exhibit turning points where the function's direction changes, which is dictated by the local minima and maxima.
Therefore, while many cubic polynomials may display the behavior of rising to the left and falling to the right, others with different coefficients could have a completely different end behavior or shape, such as rising on both ends or falling on both ends. Two-dimensional (x-y) graphing tools and learning about graphing polynomials can provide visual understanding of these concepts.