## 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

$\fn_jvn {\color{DarkBlue} g(x)=a(x-3)(x+4)^c}$

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 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  $\inline \fn_jvn {\color{DarkBlue} y=x^2}$. It goes up toward positive infinity as the absolute value of x increases, whether positive or negative. So do $\inline \fn_jvn y=x^4$, $\inline \fn_phv y=x^6$, etc.

 $\inline \fn_jvn {\color{DarkBlue} y=x^2}$$\inline \fn_jvn {\color{DarkBlue} y=x^2}$ $\inline \fn_jvn y=x^4$$\inline \fn_jvn y=x^4$ $\inline \fn_jvn y=x^6$$\inline \fn_jvn y=x^6$

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 $\inline \fn_jvn {\color{DarkBlue} y=x}$, $\inline \fn_jvn y=x^3$, etc.

 $\inline \fn_jvn {\color{DarkBlue} y=x}$$\inline \fn_jvn {\color{DarkBlue} y=x}$ $\inline \fn_jvn y=x^3$$\inline \fn_jvn y=x^3$ $\inline \fn_jvn y=x^5$$\inline \fn_jvn y=x^5$

So far, I’ve been talking about a monomial — $\inline \fn_jvn {\color{DarkBlue} y=x}$ or $\inline \fn_jvn {\color{DarkBlue} y=x^2}$ 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 $\inline \fn_jvn y=x^5+4x^4-2x^2+1$?

 $\inline \fn_jvn y=x^5+4x^4-2x^2+1$$\inline \fn_jvn y=x^5+4x^4-2x^2+1$

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 $\inline \fn_jvn y=x^5+4x^4-2x^2+1$ starts to look more like the graph of $\inline \fn_jvn y=x^5$.

### Back to the Question

The function in question, $\inline \fn_jvn g(x)=a(x-3)(x+4)^c$, 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.