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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).
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.")
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.")

Check your understanding

1) This is the graph of y, equals, g, left parenthesis, x, right parenthesis.
What is the end behavior of g?
Choose 1 answer:
Choose 1 answer:

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?
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For very large negative x values, what best describes f, left parenthesis, x, right parenthesis?
Choose 1 answer:
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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?
Choose 1 answer:
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For very large negative x values, what best describes g, left parenthesis, x, right parenthesis?
Choose 1 answer:
Choose 1 answer:

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?
Choose 1 answer:
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For very large negative x values, what best describes h, left parenthesis, x, right parenthesis?
Choose 1 answer:
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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?
Choose 1 answer:
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For very large negative x values, what best describes j, left parenthesis, x, right parenthesis?
Choose 1 answer:
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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.
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
start color #11accd, n, end color #11accd is odd and start color #1fab54, a, end color #1fab54, is greater than, 0start 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.
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

Check your understanding

6) What is the end behavior of g, left parenthesis, x, right parenthesis, equals, 8, x, cubed?
Choose 1 answer:
Choose 1 answer:

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.

Check your understanding

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?
Choose 1 answer:
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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?
Choose 1 answer:
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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?
Choose 1 answer:
Choose 1 answer:

10*) What is the end behavior of g, left parenthesis, x, right parenthesis, equals, left parenthesis, 2, minus, 3, x, right parenthesis, left parenthesis, x, plus, 2, right parenthesis, squared?
Choose 1 answer:
Choose 1 answer:

Want to join the conversation?

  • blobby green style avatar for user kyle.davenport
    What determines the rise and fall of a polynomial
    (10 votes)
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    • blobby green style avatar for user Lara ALjameel
      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.
      (10 votes)
  • duskpin ultimate style avatar for user Mellivora capensis
    So the leading term is the term with the greatest exponent always right?
    (4 votes)
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  • blobby green style avatar for user 335697
    Off topic but if I ask a question will someone answer soon or will it take a few days?
    (5 votes)
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  • starky sapling style avatar for user Joseph SR
    I'm still so confused, this is making no sense to me, can someone explain it to me simply? this is Hard. Thanks! :D
    (2 votes)
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    • hopper cool style avatar for user obiwan kenobi
      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!
      (7 votes)
  • blobby green style avatar for user Tori  Herrera
    How are the key features and behaviors of polynomial functions changed by the introduction of the independent variable in the denominator (dividing by x)?
    (3 votes)
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    • blobby green style avatar for user Seth
      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.
      (4 votes)
  • blobby green style avatar for user jenniebug1120
    What if you have a funtion like f(x)=-3^x? How would you describe the left ends behaviour?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      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.
      (9 votes)
  • mr pants purple style avatar for user Katelyn Clark
    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.
    (2 votes)
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    • male robot hal style avatar for user Raymond
      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.
      (6 votes)
  • mr pants teal style avatar for user SOULAIMAN986
    In the last question when I click I need help and it’s simplifying the equation where did 4x come from?
    (0 votes)
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  • duskpin ultimate style avatar for user Sirius
    What are the end behaviors of sine/cosine functions?
    (3 votes)
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  • primosaur seed style avatar for user 999988024
    Hi, How do I describe an end behavior of an equation like this? This is an answer to an equation. How do I find the answer like this.

    As x decreases, y increases and
    as x increases, y increases.
    (2 votes)
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    • female robot grace style avatar for user loumast17
      End behavior is looking at the two extremes of x. As x gets closer to infinity and as x gets closer to negative infinity. Let's look at a simple example.

      For x^2 hopefully you know the shape is a big U. Now, as x gets larger, or in other words as x goes to infinity what happens? You can try numbers out. try f(10), f(100), f(1000) and so on. The numbers just keep increasing. This also shows if you look at the graph. A U just keeps going up as you head right. And of course heading right is making x increase. so as x increases y increases, since up is the direction of increasing y.

      Similarly, as you decrease x the U is also going up, or in other words as you head to the left, the direction of decreasing x, y also increases since it is also going up since it is U shapped.

      this fits the example you gave too.If we looked at another what happens though?

      with x^3 as you move right, or in other words as x increases, the line also goes up, or in other words y increases. If you go left, or if x decreases, the line goes down, so y decreases.

      Let me know if this didn't make sense or help.
      (3 votes)