What determines the rise and fall of a polynomial

Graphs of polynomials either "rise to the right" or they "fall to the right", and they either "rise to the left" or they "fall to the left." ... The behavior of a polynomial graph as x goes to infinity or negative infinity is determined by the leading coefficient, which is the coefficient of the highest degree term.

Off topic but if I ask a question will someone answer soon or will it take a few days?

Questions are answered by other KA users in their spare time. So, there is no predictable time frame to get a response. Many questions get answered in a day or so.

I'm still so confused, this is making no sense to me, can someone explain it to me simply? this is Hard. Thanks! :D

All polynomials with even degrees will have a the same end behavior as x approaches -∞ and ∞. If the value of the coefficient of the term with the greatest degree is positive then that means that the end behavior to ∞ on both sides. If the coefficient is negative, now the end behavior on both sides will be -∞. If the polynomials degree is odd, then the end behavior will be different on both sides. If the leading coefficient is positive then the end behavior will be -∞ as x approaches -∞ and ∞ as x approaches ∞. Notice this is from bottom left to top right. If the leading coefficient is negative, the function will now be from top left to bottom right. So its end behavior will be ∞ as x approaches -∞ and -∞ as x approaches ∞. Hope this helps!

So the leading term is the term with the greatest exponent always right?

Yes. It would be best to put the terms of the polynomial in order from greatest exponent to least exponent before you evaluate the behavior

How to not fail? how to do?

Go through the lessons, understand the content, and show your mastery on tests

How are the key features and behaviors of polynomial functions changed by the introduction of the independent variable in the denominator (dividing by x)?

For polynomials without a constant term, dividing by x will make a new polynomial, with a degree of n-1, that is undefined at 0. For example, x³+2x will become x²+2 for x≠0. With a constant term, things become a little more interesting, because the new function actually isn't a polynomial anymore. If we divided x²+2 by x, now we have x+(2/x), which has an asymptote at 0. In terms of end behavior, it also will change when you divide by x, because the degree of the polynomial is going from even to odd or odd to even with every division, but the leading coefficient stays the same.

The infinity symbol throws me off and I don't think I was ever taught the formula with an infinity symbol. I need so much help with this. I thought that the leading coefficient and the degrees determine if the ends of the graph is... up & down, down & up, up & up, down & down. Thank you for trying to help me understand.

Well, let's start with a positive leading coefficient and an even degree. This would be the graph of x^2, which is up & up, correct? That means that when x increases, y increases. And when x decreases, y still increases. You can rewrite up & up as x→+∞, f(x)→+∞ & x→-∞, f(x)→+∞. Same logic goes for the other behaviors.

What if you have a funtion like f(x)=-3^x? How would you describe the left ends behaviour?

FYI... you do not have a polynomial function. You have an exponential function. So, you might want to check out the videos on that topic. Related to your specific question... Try some numbers to see what happens. -3^0 = -1 -3^1 = -3 -3^2 = -9 -3^3 = 27 ...etc... Keep trying some numbers to get a sense of the end behavior.

Main content

Algebra 2

Course: Algebra 2 > Unit 5

Lesson 3: End behavior of polynomials

End behavior of polynomials

Google Classroom

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

+ \infty

) and to the left end of the

x

-axis (as

x

approaches

- \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 (x)

gets larger and larger as well.

Mathematically, we write: as

x \to + \infty

f (x) \to + \infty

. (Say, "as

x

approaches positive infinity,

f (x)

approaches positive infinity.")

On the other end of the graph, as we move to the left along the

x

-axis (imagine

x

approaching

- \infty

), the graph of

f

goes down. This means as

x

gets more and more negative,

f (x)

also gets more and more negative.

Mathematically, we write: as

x \to - \infty

f (x) \to - \infty

. (Say, "as

x

approaches negative infinity,

f (x)

approaches negative infinity.")

Check your understanding

1) This is the graph of

y = g (x)

What is the end behavior of $g$ ‍?

(Choice A)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice B)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.
(Choice C)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice D)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.

[I need help!]

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 $x \to + \infty$ ‍, what does $f (x)$ ‍ approach?
As $x \to - \infty$ ‍, what does $f (x)$ ‍ 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 (x) = x^{2}

For very large positive $x$ ‍ values, what best describes $f (x)$ ‍?

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

[I need help!]

3) Consider the monomial

g (x) = - 3 x^{2}

For very large positive $x$ ‍ values, what best describes $g (x)$ ‍?

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

[I need help!]

4) Consider the monomial

h (x) = x^{3}

For very large positive $x$ ‍ values, what best describes $h (x)$ ‍?

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

[I need help!]

5) Consider the monomial

j (x) = - 2 x^{3}

For very large positive $x$ ‍ values, what best describes $j (x)$ ‍?

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

[I need help!]

Concluding the investigation

Notice how the degree of the monomial

(n)

and the leading coefficient

(a)

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

+ \infty

or whether they both approach

- \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

+ \infty

and which one is

- \infty

This is summarized in the table below.

**End Behavior of Monomials: $f (x) = a x^{n}$ ‍**
$n$ ‍ is even and $a > 0$ ‍	$n$ ‍ is even and $a < 0$ ‍
As $x \to - \infty$ ‍, $f (x) \to + \infty$ ‍, and as $x \to + \infty$ ‍, $f (x) \to + \infty$ ‍. 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 $x \to - \infty$ ‍, $f (x) \to - \infty$ ‍, and as $x \to + \infty$ ‍, $f (x) \to - \infty .$ ‍ 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.

$n$ ‍ is odd and $a > 0$ ‍	$n$ ‍ is odd and $a < 0$ ‍
As $x \to - \infty$ ‍, $f (x) \to - \infty$ ‍, and as $x \to + \infty$ ‍, $f (x) \to + \infty$ ‍. 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 $x \to - \infty$ ‍, $f (x) \to + \infty$ ‍, and as $x \to + \infty$ ‍, $f (x) \to - \infty .$ ‍ 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.

Check your understanding

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

(Choice A)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice B)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.
(Choice C)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice D)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.

[I need help!]

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 (x) = - 3 x^{2} + 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 (x) = - 3 x^{2} + 7 x

is the same as the end behavior of the monomial

- 3 x^{2}

Since the degree of

- 3 x^{2}

is even

(2)

and the leading coefficient is negative

(- 3)

, the end behavior of

g

is: as

x \to - \infty

g (x) \to - \infty

, and as

x \to + \infty

g (x) \to - \infty

Check your understanding

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

(Choice A)
As $x \to + \infty$ ‍, $f (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $f (x) \to + \infty$ ‍.
(Choice B)
As $x \to + \infty$ ‍, $f (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $f (x) \to - \infty$ ‍.
(Choice C)
As $x \to + \infty$ ‍, $f (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $f (x) \to + \infty$ ‍.
(Choice D)
As $x \to + \infty$ ‍, $f (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $f (x) \to - \infty$ ‍.

[I need help!]

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

(Choice A)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice B)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.
(Choice C)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice D)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.

[I need help!]

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 (x) = - 3 x^{2} + 7 x

for large positive values of

x

x

approaches

+ \infty

, we know that

- 3 x^{2}

approaches

- \infty

and

7 x

approaches

+ \infty

But what is the end behavior of their sum? Let's plug in a few values of

x

to figure this out.

$x$ ‍	$- 3 x^{2}$ ‍	$7 x$ ‍	$- 3 x^{2} + 7 x$ ‍
$1$ ‍	$- 3$ ‍	$7$ ‍	$4$ ‍
$10$ ‍	$- 300$ ‍	$70$ ‍	$- 230$ ‍
$100$ ‍	$- 30,000$ ‍	$700$ ‍	$- 29,300$ ‍
$1000$ ‍	$- 3,000,000$ ‍	$7000$ ‍	$- 2,993,000$ ‍

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

7 x

we had

999 x

$x$ ‍	$- 3 x^{2}$ ‍	$999 x$ ‍	$- 3 x^{2} + 999 x$ ‍
$10$ ‍	$- 300$ ‍	$9,990$ ‍	$9,690$ ‍
$100$ ‍	$- 30,000$ ‍	$99,900$ ‍	$69,900$ ‍
$1000$ ‍	$- 3,000,000$ ‍	$999,000$ ‍	$- 2,001,000$ ‍
$10,000$ ‍	$- 300,000,000$ ‍	$9,990,000$ ‍	$- 290,010,000$ ‍

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 $h (x) = - 8 x^{3} + 7 x - 1$ ‍?

[I need help!]

10*) What is the end behavior of $g (x) = (2 - 3 x) (x + 2)^{2}$ ‍?

(Choice A)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice B)
As $x \to + \infty$ ‍, $g (x) \to + \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.
(Choice C)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to + \infty$ ‍.
(Choice D)
As $x \to + \infty$ ‍, $g (x) \to - \infty$ ‍, and as $x \to - \infty$ ‍, $g (x) \to - \infty$ ‍.

[I need help!]

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kyle.davenport
Posted 7 years ago. Direct link to kyle.davenport's post “What determines the rise ...”
What determines the rise and fall of a polynomial
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(11 votes)
Answer
- Lara ALjameel
  Posted 6 years ago. Direct link to Lara ALjameel's post “Graphs of polynomials eit...”
  Graphs of polynomials either "rise to the right" or they "fall to the right", and they either "rise to the left" or they "fall to the left." ... The behavior of a polynomial graph as x goes to infinity or negative infinity is determined by the leading coefficient, which is the coefficient of the highest degree term.
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𝘽𝘼𝙏𝙈𝘼𝙉
Posted 3 years ago. Direct link to 𝘽𝘼𝙏𝙈𝘼𝙉's post “I'm still so confused, th...”
I'm still so confused, this is making no sense to me, can someone explain it to me simply? this is Hard. Thanks! :D
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(6 votes)
Answer
- obiwan kenobi
  Posted 3 years ago. Direct link to obiwan kenobi's post “All polynomials with even...”
  All polynomials with even degrees will have a the same end behavior as x approaches -∞ and ∞. If the value of the coefficient of the term with the greatest degree is positive then that means that the end behavior to ∞ on both sides. If the coefficient is negative, now the end behavior on both sides will be -∞.
  
  If the polynomials degree is odd, then the end behavior will be different on both sides. If the leading coefficient is positive then the end behavior will be -∞ as x approaches -∞ and ∞ as x approaches ∞. Notice this is from bottom left to top right. If the leading coefficient is negative, the function will now be from top left to bottom right. So its end behavior will be ∞ as x approaches -∞ and -∞ as x approaches ∞. Hope this helps!
  Comment on obiwan kenobi's post “All polynomials with even...”
  (20 votes)
Mellivora capensis
Posted 7 years ago. Direct link to Mellivora capensis's post “So the leading term is th...”
So the leading term is the term with the greatest exponent always right?
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(5 votes)
Answer
- Wayne Clemensen
  Posted 3 years ago. Direct link to Wayne Clemensen's post “Yes. It would be best to ...”
  Yes. It would be best to put the terms of the polynomial in order from greatest exponent to least exponent before you evaluate the behavior
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  (8 votes)
335697
Posted 2 years ago. Direct link to 335697's post “Off topic but if I ask a ...”
Off topic but if I ask a question will someone answer soon or will it take a few days?
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(8 votes)
Answer
- Kim Seidel
  Posted 2 years ago. Direct link to Kim Seidel's post “Questions are answered by...”
  Questions are answered by other KA users in their spare time. So, there is no predictable time frame to get a response. Many questions get answered in a day or so.
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  (9 votes)
jenniebug1120
Posted 6 years ago. Direct link to jenniebug1120's post “What if you have a funtio...”
What if you have a funtion like f(x)=-3^x? How would you describe the left ends behaviour?
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(1 vote)
Answer
- Kim Seidel
  Posted 6 years ago. Direct link to Kim Seidel's post “FYI... you do not have a ...”
  FYI... you do not have a polynomial function. You have an exponential function. So, you might want to check out the videos on that topic.
  
  Related to your specific question... Try some numbers to see what happens.
  -3^0 = -1
  -3^1 = -3
  -3^2 = -9
  -3^3 = 27
  ...etc...
  Keep trying some numbers to get a sense of the end behavior.
  Button navigates to signup page
  (10 votes)
Tori Herrera
Posted 4 years ago. Direct link to Tori Herrera's post “How are the key features ...”
How are the key features and behaviors of polynomial functions changed by the introduction of the independent variable in the denominator (dividing by x)?
Button navigates to signup pageComment on Tori Herrera's post “How are the key features ...”
(3 votes)
Answer
- Seth
  Posted 4 years ago. Direct link to Seth's post “For polynomials without a...”
  For polynomials without a constant term, dividing by x will make a new polynomial, with a degree of n-1, that is undefined at 0. For example, x³+2x will become x²+2 for x≠0. With a constant term, things become a little more interesting, because the new function actually isn't a polynomial anymore. If we divided x²+2 by x, now we have x+(2/x), which has an asymptote at 0. In terms of end behavior, it also will change when you divide by x, because the degree of the polynomial is going from even to odd or odd to even with every division, but the leading coefficient stays the same.
  Button navigates to signup page
  (5 votes)
perez, dakota
Posted a year ago. Direct link to perez, dakota's post “How to not fail? how to d...”
How to not fail? how to do?
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- chasephuayoung88
  Posted 6 months ago. Direct link to chasephuayoung88's post “Go through the lessons, u...”
  Go through the lessons, understand the content, and show your mastery on tests
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  (2 votes)
Katelyn Clark
Posted 6 years ago. Direct link to Katelyn Clark's post “The infinity symbol throw...”
The infinity symbol throws me off and I don't think I was ever taught the formula with an infinity symbol. I need so much help with this. I thought that the leading coefficient and the degrees determine if the ends of the graph is... up & down, down & up, up & up, down & down. Thank you for trying to help me understand.
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(2 votes)
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- Raymond
  Posted 6 years ago. Direct link to Raymond's post “Well, let's start with a ...”
  Well, let's start with a positive leading coefficient and an even degree. This would be the graph of x^2, which is up & up, correct?
  
  That means that when x increases, y increases. And when x decreases, y still increases.
  You can rewrite up & up as x→+∞, f(x)→+∞ & x→-∞, f(x)→+∞.
  Same logic goes for the other behaviors.
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  (6 votes)
SOULAIMAN986
Posted 4 years ago. Direct link to SOULAIMAN986's post “In the last question when...”
In the last question when I click I need help and it’s simplifying the equation where did 4x come from?
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Advait Jay
Posted 4 months ago. Direct link to Advait Jay's post “Why is it so hard to find...”
Why is it so hard to find out end behaviors in polynomials with a degree higher than 2??
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