Question

Describe the end behavior of polynomial graphs with odd and even degrees. Talk about positive and negative leading coefficients.

Answer 1

Answer with explanation:

The general polynomial function is given by:

$P(x)=a_nx^n+a_{n-1}x^{n-1}+.....+a_1x+a_0$

where $a_n\neq 0$ and it is the leading coefficient.

Also, if n is even then it is a even degree polynomial.

and if n is odd then it is a odd degree polynomial.

The end behavior of a polynomial is the behavior when x tends to infinity from both the sides.

i.e. when x → -∞ and when x → ∞

Now,

If n is even:

1)

$a_n\ \text{is positive}$

then if x → -∞  then P(x) → ∞

and if x → ∞ then P(x) → ∞

2)

$a_n\ \text{is negative}$

then if x → -∞  then P(x) → -∞

and if x → ∞ then P(x) → -∞

If n is odd:

1)

$a_n\ \text{is positive}$

then if x → -∞  then P(x) → -∞

and if x → ∞ then P(x) → ∞

2)

$a_n\ \text{is negative}$

then if x → -∞  then P(x) → ∞

and if x → ∞ then P(x) → -∞

Answer 2

A polynomial with a positive leading coefficient will always be increasing, as x approaches infinity. Consider a polynomial with a positive leading coefficient and is degree 'n', where 'n' is an integer greater than zero. Differentiating the function tells us whether the function is increasing or decreasing. Since the coefficient of the first term of the derivative is positive (since 'n' and the coefficient is positive), the limit as x approaches infinity is positive infinity, as the highest degree is all that needs to be evaluated. Since the limit is positive, the function increases indefinitely.
Even if you differentiate once more (assuming n > 1), we find that the leading term still has a positive coefficient. This means that the limit as x approaches infinity, is once again, positive. This indicates positive concavity, thus, supporting the argument.

The same argument can be made to claim that polynomials with a negative leading coefficient approach negative infinity, as x approaches infinity.