Polynomial Functions & Graphs

Zeros, multiplicity, and end behavior from the formula.

The idea

The graph of a polynomial is a single unbroken curve, with no jumps, gaps, or vertical asymptotes, so two things settle its shape: where it meets the $x$-axis, and what it does far out at either end.

The meeting points are the real zeros, the inputs with $P(x) = 0$. In factored form we can read them off, since a product is zero exactly when one of its factors is: a factor $(x - r)$ places a zero at $x = r$. The number of times that factor repeats is the zero's multiplicity, and it decides what the curve does there, because near $x = r$ the repeated factor $(x - r)^{m}$ controls the sign of the whole product. An odd power changes sign as $x$ passes $r$, so the curve crosses the axis; an even power keeps its sign, so the curve touches the axis and turns back.

Far from the origin the highest power dwarfs the rest, so the end behavior is that of the leading term $a_{n}x^{n}$ alone. An even degree sends both ends the same way — up when $a_{n} > 0$, down when $a_{n} < 0$ — and an odd degree sends them opposite ways. The zeros, their multiplicities, and the two ends are enough to sketch the curve.

Ways to work on it

Not sure where to start? Take the ten-question placement test.