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## Algebra (all content)

### Course: Algebra (all content)>Unit 10

Lesson 34: End behavior of polynomial functions

# End behavior of polynomials

Learn what the end behavior of a polynomial is, and how we can find it from the polynomial's equation.
In this lesson, you will learn what the "end behavior" of a polynomial is and how to analyze it from a graph or from a polynomial equation.

## What's "end behavior"?

The end behavior of a function $f$ describes the behavior of the graph of the function at the "ends" of the $x$-axis.
In other words, the end behavior of a function describes the trend of the graph if we look to the right end of the $x$-axis (as $x$ approaches $+\mathrm{\infty }$) and to the left end of the $x$-axis (as $x$ approaches $-\mathrm{\infty }$).
For example, consider this graph of the polynomial function $f$. Notice that as you move to the right on the $x$-axis, the graph of $f$ goes up. This means, as $x$ gets larger and larger, $f\left(x\right)$ gets larger and larger as well.
Mathematically, we write: as $x\to +\mathrm{\infty }$, $f\left(x\right)\to +\mathrm{\infty }$. (Say, "as $x$ approaches positive infinity, $f\left(x\right)$ approaches positive infinity.")
On the other end of the graph, as we move to the left along the $x$-axis (imagine $x$ approaching $-\mathrm{\infty }$), the graph of $f$ goes down. This means as $x$ gets more and more negative, $f\left(x\right)$ also gets more and more negative.
Mathematically, we write: as $x\to -\mathrm{\infty }$, $f\left(x\right)\to -\mathrm{\infty }$. (Say, "as $x$ approaches negative infinity, $f\left(x\right)$ approaches negative infinity.")

1) This is the graph of $y=g\left(x\right)$.
What is the end behavior of $g$?

## Determining end behavior algebraically

We can also determine the end behavior of a polynomial function from its equation. This is often helpful while trying to graph the function, as knowing the end behavior helps us visualize the graph at the "ends."
To determine the end behavior of a polynomial $f$ from its equation, we can think about the function values for large positive and large negative values of $x$.
Specifically, we answer the following two questions:
• As $x\to +\mathrm{\infty }$, what does $f\left(x\right)$ approach?
• As $x\to -\mathrm{\infty }$, what does $f\left(x\right)$ approach?

### Investigation: End behavior of monomials

Monomial functions are polynomials of the form $y=a{x}^{n}$ , where $a$ is a real number and $n$ is a nonnegative integer.
Let's algebraically examine the end behavior of several monomials and see if we can draw some conclusions.
2) Consider the monomial $f\left(x\right)={x}^{2}$.
For very large positive $x$ values, what best describes $f\left(x\right)$?

For very large negative $x$ values, what best describes $f\left(x\right)$?

3) Consider the monomial $g\left(x\right)=-3{x}^{2}$.
For very large positive $x$ values, what best describes $g\left(x\right)$?

For very large negative $x$ values, what best describes $g\left(x\right)$?

4) Consider the monomial $h\left(x\right)={x}^{3}$.
For very large positive $x$ values, what best describes $h\left(x\right)$?

For very large negative $x$ values, what best describes $h\left(x\right)$?

5) Consider the monomial $j\left(x\right)=-2{x}^{3}$.
For very large positive $x$ values, what best describes $j\left(x\right)$?

For very large negative $x$ values, what best describes $j\left(x\right)$?

### Concluding the investigation

Notice how the degree of the monomial $\left(n\right)$ and the leading coefficient $\left(a\right)$ affect the end behavior.
When $n$ is even, the behavior of the function at both "ends" is the same. The sign of the leading coefficient determines whether they both approach $+\mathrm{\infty }$ or whether they both approach $-\mathrm{\infty }$.
When $n$ is odd, the behavior of the function at both "ends" is opposite. The sign of the leading coefficient determines which one is $+\mathrm{\infty }$ and which one is $-\mathrm{\infty }$.
This is summarized in the table below.
End Behavior of Monomials: $f\left(x\right)=a{x}^{n}$
$n$ is even and $a>0$$n$ is even and $a<0$
As $x\to -\mathrm{\infty }$, $f\left(x\right)\to +\mathrm{\infty }$, and as $x\to +\mathrm{\infty }$, $f\left(x\right)\to +\mathrm{\infty }$.
As $x\to -\mathrm{\infty }$, $f\left(x\right)\to -\mathrm{\infty }$, and as $x\to +\mathrm{\infty }$, $f\left(x\right)\to -\mathrm{\infty }.$
$n$ is odd and $a>0$$n$ is odd and $a<0$
As $x\to -\mathrm{\infty }$, $f\left(x\right)\to -\mathrm{\infty }$, and as $x\to +\mathrm{\infty }$, $f\left(x\right)\to +\mathrm{\infty }$.
As $x\to -\mathrm{\infty }$, $f\left(x\right)\to +\mathrm{\infty }$, and as $x\to +\mathrm{\infty }$, $f\left(x\right)\to -\mathrm{\infty }.$

6) What is the end behavior of $g\left(x\right)=8{x}^{3}$?

### End behavior of polynomials

We now know how to find the end behavior of monomials. But what about polynomials that are not monomials? What about functions like $g\left(x\right)=-3{x}^{2}+7x$?
In general, the end behavior of a polynomial function is the same as the end behavior of its leading term, or the term with the largest exponent.
So the end behavior of $g\left(x\right)=-3{x}^{2}+7x$ is the same as the end behavior of the monomial $-3{x}^{2}$.
Since the degree of $-3{x}^{2}$ is even $\left(2\right)$ and the leading coefficient is negative $\left(-3\right)$, the end behavior of $g$ is: as $x\to -\mathrm{\infty }$, $g\left(x\right)\to -\mathrm{\infty }$, and as $x\to +\mathrm{\infty }$, $g\left(x\right)\to -\mathrm{\infty }$.

7) What is the end behavior of $f\left(x\right)=8{x}^{5}-7{x}^{2}+10x-1$?

8) What is the end behavior of $g\left(x\right)=-6{x}^{4}+8{x}^{3}+4{x}^{2}$?

## Why does the leading term determine the end behavior?

This is because the leading term has the greatest effect on function values for large values of $x$.
Let's explore this further by analyzing the function $g\left(x\right)=-3{x}^{2}+7x$ for large positive values of $x$.
As $x$ approaches $+\mathrm{\infty }$, we know that $-3{x}^{2}$ approaches $-\mathrm{\infty }$ and $7x$ approaches $+\mathrm{\infty }$.
But what is the end behavior of their sum? Let's plug in a few values of $x$ to figure this out.
$x$$-3{x}^{2}$$7x$$-3{x}^{2}+7x$
$1$$-3$$7$$4$
$10$$-300$$70$$-230$
$100$$-30,000$$700$$-29,300$
$1000$$-3,000,000$$7000$$-2,993,000$
Notice that as $x$ gets larger, the polynomial behaves like $-3{x}^{2}.$
But suppose the $x$ term had a little more weight. What would happen if instead of $7x$ we had $999x$?
$x$$-3{x}^{2}$$999x$$-3{x}^{2}+999x$
$10$$-300$$9,990$$9,690$
$100$$-30,000$$99,900$$69,900$
$1000$$-3,000,000$$999,000$$-2,001,000$
$10,000$$-300,000,000$$9,990,000$$-290,010,000$
Again, we see that for large values of $x$, the polynomial behaves like $-3{x}^{2}$. While a larger value of $x$ was needed to see the trend here, it is still the case.
In fact, no matter what the coefficient of $x$ is, for large enough values of $x$, $-3{x}^{2}$ will eventually take over!

## Challenge problems

9*) Which of the following could be the graph of $h\left(x\right)=-8{x}^{3}+7x-1$?
10*) What is the end behavior of $g\left(x\right)=\left(2-3x\right)\left(x+2{\right)}^{2}$?