Question 1175210

When you're graphing (or looking at a graph of) polynomials, it can help to already have an idea of what basic polynomial shapes look like. One of the aspects of this is "end behavior", and it's pretty easy. We'll look at some graphs, to find similarities and differences.

First, let's look at some polynomials of even degree with positive and negative leading coefficients:


<a href="https://ibb.co/nq2k7CY"><img src="https://i.ibb.co/nq2k7CY/unnamed.png" alt="unnamed" border="0"></a>

As you can see above, odd-degree polynomials have ends that head off in opposite directions. If they start "down" (entering the graphing "box" through the "bottom") and go "up" (leaving the graphing "box" through the "top"), they're positive polynomials, just like every positive cubic you've ever graphed. But If they start "up" and go "down", they're negative polynomials.

This behavior is true for all odd-degree polynomials.

simply:

for every polynomial {{{f(x)=ax^n}}} when {{{n}}} is {{{even}}} and {{{a}}} is {{{positive}}}, graph rises to the left and right


In particular, if the degree of a polynomial {{{f(x)}}} is {{{even}}} and the leading coefficient is {{{positive}}}, then {{{f(x)}}} → {{{infinity}}} as {{{x}}} → ± {{{infinity}}}.