Polynomial functions
Without plotting a single point, how much of a polynomial’s graph can you know from its formula?
Where you are 6 inside
What this area is about
A polynomial function, such as f(x) = x^3 - 3x, adds up constant multiples of whole-number powers of x. Its graph is one smooth, unbroken curve, and a few facts about the formula fix much of its shape. The leading term decides where the two ends go. A polynomial of degree n \ge 1 has at most n real zeros and at most n - 1 turning points. The factors give the zeros and what the graph does at each one.
How these fit together
Results about shape. Degree, end behavior and shape reads the ends and the limits on zeros and turns from the degree. Zeros and multiplicity says that a zero of odd multiplicity crosses the axis and one of even multiplicity touches it and turns back.
A family up close. Cubic, quartic and higher degree functions sorts the shapes those graphs can take.
Procedures. Graphing polynomial functions puts zeros, multiplicity and end behavior into one sketch. Building a polynomial from its zeros runs the other way, from the zeros and one more point to the formula.
Modeling. Modeling with polynomial functions fits data that turns more often than a parabola can, or turns once off-center, and asks for which inputs the model means anything.
A path through it
Degree and end behavior, and zeros and multiplicity, first, in either order. Then higher degree shapes and graphing; building from zeros needs only multiplicity. Modeling comes after graphing.
Next: graphing rational functions, whose numerators and denominators are polynomials.