Question

How do the coefficients of a polynomial affects its end behavior?

Answer 1

For even degree polynomials, a positive leading coefficient implies `y->+infty` as `x->pm infty`, while a negative leading coefficient implies `y->-infty` as `x->pm infty`. For odd degree polynomials, a positive leading coefficient implies `y->+infty` as `x->+ infty` and `y->-infty` as `x->-infty`, while a negative leading coefficient implies `y->-infty` as `x->+ infty` and `y->+infty` as `x->-infty`.

Explanation:

A (real) polynomial of (integer) degree `n` is a function of the form `p(x)=a_{n}x^{n}+a_{n-1}x^[n-1}+a_{n-2}x^{n-2}+cdots+a_{2}x^2+a_{1}x+a_{0}`, where `a_{n} != 0` (otherwise it wouldn't be degree `n`), and all the other `a`'s are arbitrary real numbers (and they can be zero).

If `n` is even, then `a_{n}>0` implies that `y->+infty` as `x->pm infty` and `a_{n}<0` implies `y->-infty` as `x->pm infty`.

If `n` is odd, then `a_{n}>0` implies that `y->+infty` as `x->+ infty` and `y->-infty` as `x->-infty` and `a_{n}<0` implies that `y->-infty` as `x->+ infty` and `y->+infty` as `x->-infty`.

The values of the other coefficients are irrelevant for determining the end behavior.