Polynomial functions follow the form p(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a1x + a0, where aₙ is non-zero, giving the function its degree.
Grasping how these functions behave requires understanding their end behavior, local extrema, and points of inflection.
The end behavior is determined by the leading term aₙxⁿ: if n is even, the function's arms point in the same direction; if n is odd, they point in opposite directions.
Local maxima and minima occur where the function switches from increasing to decreasing or vice versa, and between any two distinct real zeros, there's at least one extremum.
Points of inflection happen where the graph changes concavity, from concave up to concave down or the other way around.
A common error is assuming even-degree polynomials always have a global extremum; they do, but you must verify if it's a maximum or minimum.
Misreading these behaviors leads to incorrect predictions about the function's graph.