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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 plus, infinity) and to the left end of the x-axis (as x approaches minus, infinity).
A polynomial labeled y equals f of x is graphed on an x y coordinate plane. The graph curves up from left to right passing through the negative x-axis side, curving down through the origin, and curving back up through the positive x-axis. A horizontal arrow points to the right labeled x gets more positive. A vertical arrow points up labeled f of x gets more positive.
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 parenthesis, x, right parenthesis gets larger and larger as well.
Mathematically, we write: as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity. (Say, "as x approaches positive infinity, f, left parenthesis, x, right parenthesis approaches positive infinity.")
A polynomial labeled y equals f of x is graphed on an x y coordinate plane. The graph curves up from left to right passing through the negative x-axis side, curving down through the origin, and curving back up through the positive x-axis. A horizontal arrow points to the left labeled x gets more negative. A vertical arrow points down labeled f of x gets more negative.
On the other end of the graph, as we move to the left along the x-axis (imagine x approaching minus, infinity), the graph of f goes down. This means as x gets more and more negative, f, left parenthesis, x, right parenthesis also gets more and more negative.
Mathematically, we write: as x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity. (Say, "as x approaches negative infinity, f, left parenthesis, x, right parenthesis approaches negative infinity.")

1) This is the graph of y, equals, g, left parenthesis, x, right parenthesis.
A polynomial is graphed on an x y coordinate plane. The graph curves down from left to right passing through the negative x-axis side and curving back up through the negative x-axis. It curves down through the positive x-axis.
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, right arrow, plus, infinity, what does f, left parenthesis, x, right parenthesis approach?
• As x, right arrow, minus, infinity, what does f, left parenthesis, x, right parenthesis approach?

### Investigation: End behavior of monomials

Monomial functions are polynomials of the form y, equals, a, x, start superscript, n, end superscript , 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 parenthesis, x, right parenthesis, equals, x, squared.
For very large positive x values, what best describes f, left parenthesis, x, right parenthesis?

For very large negative x values, what best describes f, left parenthesis, x, right parenthesis?

3) Consider the monomial g, left parenthesis, x, right parenthesis, equals, minus, 3, x, squared.
For very large positive x values, what best describes g, left parenthesis, x, right parenthesis?

For very large negative x values, what best describes g, left parenthesis, x, right parenthesis?

4) Consider the monomial h, left parenthesis, x, right parenthesis, equals, x, cubed.
For very large positive x values, what best describes h, left parenthesis, x, right parenthesis?

For very large negative x values, what best describes h, left parenthesis, x, right parenthesis?

5) Consider the monomial j, left parenthesis, x, right parenthesis, equals, minus, 2, x, cubed.
For very large positive x values, what best describes j, left parenthesis, x, right parenthesis?

For very large negative x values, what best describes j, left parenthesis, x, right parenthesis?

### Concluding the investigation

Notice how the degree of the monomial left parenthesis, start color #11accd, n, end color #11accd, right parenthesis and the leading coefficient left parenthesis, start color #1fab54, a, end color #1fab54, right parenthesis 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 plus, infinity or whether they both approach minus, infinity.
When n is odd, the behavior of the function at both "ends" is opposite. The sign of the leading coefficient determines which one is plus, infinity and which one is minus, infinity.
This is summarized in the table below.
End Behavior of Monomials: f, left parenthesis, x, right parenthesis, equals, start color #1fab54, a, end color #1fab54, x, start superscript, start color #11accd, n, end color #11accd, end superscript
start color #11accd, n, end color #11accd is even and start color #1fab54, a, end color #1fab54, is greater than, 0start color #11accd, n, end color #11accd is even and start color #1fab54, a, end color #1fab54, is less than, 0
As x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity, and as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity.
A parabola is graphed on an x y coordinate plane. The graph curves down from left to right touching the origin before curving back up. The top part of both sides of the parabola are solid. The middle of the parabola is dashed.
As x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, and as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, point
A parabola is graphed on an x y coordinate plane. The graph curves up from left to right touching the origin before curving back down. The bottom part of both sides of the parabola are solid. The middle of the parabola is dashed.
start color #11accd, n, end color #11accd is odd and start color #1fab54, a, end color #1fab54, is greater than, 0
start color #11accd, n, end color #11accd is odd and start color #1fab54, a, end color #1fab54, is less than, 0
---
As x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, and as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity.
A cubic function is graphed on an x y coordinate plane. The graph curves up from left to right passing through the origin before curving up again. The bottom part and the top part of the graph are solid while the middle part of the graph is dashed.
As x, right arrow, minus, infinity, f, left parenthesis, x, right parenthesis, right arrow, plus, infinity, and as x, right arrow, plus, infinity, f, left parenthesis, x, right parenthesis, right arrow, minus, infinity, point
A cubic function is graphed on an x y coordinate plane. The graph curves down from left to right passing through the origin before curving down again. The top part and the bottom part of the graph are solid while the middle part of the graph is dashed.

6) What is the end behavior of g, left parenthesis, x, right parenthesis, equals, 8, x, cubed?

### 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 parenthesis, x, right parenthesis, equals, minus, 3, x, squared, plus, 7, x?
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 parenthesis, x, right parenthesis, equals, minus, 3, x, squared, plus, 7, x is the same as the end behavior of the monomial minus, 3, x, squared.
Since the degree of start color #1fab54, minus, 3, end color #1fab54, x, start superscript, start color #11accd, 2, end color #11accd, end superscript is even left parenthesis, start color #11accd, 2, end color #11accd, right parenthesis and the leading coefficient is negative left parenthesis, start color #1fab54, minus, 3, end color #1fab54, right parenthesis, the end behavior of g is: as x, right arrow, minus, infinity, g, left parenthesis, x, right parenthesis, right arrow, minus, infinity, and as x, right arrow, plus, infinity, g, left parenthesis, x, right parenthesis, right arrow, minus, infinity.

7) What is the end behavior of f, left parenthesis, x, right parenthesis, equals, 8, x, start superscript, 5, end superscript, minus, 7, x, squared, plus, 10, x, minus, 1?

8) What is the end behavior of g, left parenthesis, x, right parenthesis, equals, minus, 6, x, start superscript, 4, end superscript, plus, 8, x, cubed, plus, 4, x, squared?

## 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 parenthesis, x, right parenthesis, equals, minus, 3, x, squared, plus, 7, x for large positive values of x.
As x approaches plus, infinity, we know that minus, 3, x, squared approaches minus, infinity and 7, x approaches plus, infinity.
But what is the end behavior of their sum? Let's plug in a few values of x to figure this out.
xminus, 3, x, squared7, xminus, 3, x, squared, plus, 7, x
1minus, 374
10minus, 30070minus, 230
100minus, 30, comma, 000700minus, 29, comma, 300
1000start color #ca337c, minus, 3, comma, 000, comma, 000, end color #ca337c7000start color #ca337c, minus, 2, comma, 993, comma, 000, end color #ca337c
Notice that as x gets larger, the polynomial behaves like minus, 3, x, squared, point
But suppose the x term had a little more weight. What would happen if instead of 7, x we had 999, x?
xminus, 3, x, squared999, xminus, 3, x, squared, plus, 999, x
10minus, 3009, comma, 9909, comma, 690
100minus, 30, comma, 00099, comma, 90069, comma, 900
1000minus, 3, comma, 000, comma, 000999, comma, 000minus, 2, comma, 001, comma, 000
10, comma, 000start color #ca337c, minus, 300, comma, 000, comma, 000, end color #ca337c9, comma, 990, comma, 000start color #ca337c, minus, 290, comma, 010, comma, 000, end color #ca337c
Again, we see that for large values of x, the polynomial behaves like minus, 3, x, squared. 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, minus, 3, x, squared will eventually take over!

## Challenge problems

9*) Which of the following could be the graph of h, left parenthesis, x, right parenthesis, equals, minus, 8, x, cubed, plus, 7, x, minus, 1?