## End Behavior, Degree, and Leading Coefficient

*This post about end behavior, degree, and leading coefficient of a polynomial function is part of a series of posts to help you prepare for the Advanced Algebra and Functions part of the Accuplacer test.*

## Question

Consider the function

where *a* and *c* are integers and are constants and *c* is positive. The the graph *y* = *g*(*x*) goes toward negative infinity at both negative and positive extremes of *x*. Is *a *positive or negative? Is *c* even or odd?

## Solution

Let’s deal with *c* first.

### Sidebar: even and odd polynomial functions

The end behavior of a polynomial function with an even degree is in the same direction — either toward negative infinity or toward positive infinity — for both extremes of *x*. Think, for example, of . It goes up toward positive infinity as the absolute value of x increases, whether positive or negative. So do , , etc.

If you multiply any of those expressions by a leading coefficient of -1, or any negative number, then end behavior goes to negative infinity for both extremely negative and extremely positive values of *x*.

On the other hand, the end behavior of a polynomial with an odd degree is in opposite directions for extremely negative and extremely positive values of *x*. Think of , , etc.

So far, I’ve been talking about a monomial — or or *y* equals *x* to some other power. What happens in the more general case, when you get a polynomial with several terms, for example ?

When a polynomial contains terms of lower degree than the leading term, you get some noise, but end behavior is still determined by the highest-degree term. Note that as *x* gets very negative or very positive, the graph of starts to look more like the graph of .

### Back to the Question

The function in question, , has degree *c* + 1: The *c* comes from the power *c* on the term *x* + 4, and the +1 comes from the implied power of 1 on the term x – 3, for a total power of *x* equal to *c* + 1. Thus *c* + 1 is the degree of function *g*. And you’re told that this function’s end behavior is to approach negative infinity at both extremes of *x*. As shown above, a polynomial that goes in the same direction — either up or down — for both extremely positive and extremely negatives values of *x* has an even degree. That means *c*+1 is even, so ** c is odd**.

What about *a*? As shown above, a polynomial with even degree, like this one, that goes down at both extremes of *x* starts with a negative number. So *a* is negative.

This question is similar to question number 17 in the sample questions for the Accuplacer Advanced Algebra and Functions test.