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Intro to end behavior of polynomials

Sal explains what "end behavior" is and what affects the end behavior of polynomial functions. Created by Sal Khan.

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  • blobby green style avatar for user Mina
    Sal keeps saying "really really negative" and "less negative" "more positive. What does that mean?
    (16 votes)
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  • starky seedling style avatar for user evsciuto05
    how I do this in real-time?
    (8 votes)
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    • aqualine seed style avatar for user Navy Gravy
      You look at the highest exponent and check the sign of the leading coefficient. If the exponent is even or odd, that will show whether or not the ends will be together or not. Next if the leading coefficient is positive or negative then you will know whether or not the ends are together or not. Remember: odd - the ends are not together and even - the ends are together.
      (1 vote)
  • spunky sam green style avatar for user 21jclever
    I'm trying to do the practice questions on here but it's multiple-choice. And the answers look very different from what the dude is talking about in this video. How does the infinite symbol involve this stuff?
    (9 votes)
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  • piceratops ultimate style avatar for user Kevin George Joe
    at sal talks about the middle of the polynomial function doing some "funky stuff" in the middle. is there a video or a link that you can share with me so as to get some insight on this middle behavior?
    (6 votes)
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  • purple pi purple style avatar for user Mohamed Abdelhamid
    wouldn't it be helpful if you started with linear equations since they are easier to remember and they also follow the same pattern.
    (5 votes)
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  • blobby green style avatar for user zachmoran18
    Could anyone explain to me how to find the zeros of polynomial functions? I know the zeros indicate a change in direction, but I'm not exactly clear on how to find them. Possible explanation or video explanation would help a ton! Thank you
    (4 votes)
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  • mr pink red style avatar for user starryowl
    what actually is end behaviour? Is it just the middle part of the graph or the whole thing?
    (2 votes)
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    • piceratops tree style avatar for user VVCephei
      End behavior tells you what the value of a function will eventually become. For example, if you were to try and plot the graph of a function f(x) = x^4 - 1000000*x^2 , you're going to get a negative value for any small x, and you may think to yourself - "oh well, guess this function will always output negative values.". But that's not so. x^4 will inevitably become greater than 1000000*x^2 at some point, and the outputs of the function will become positive. And as x become larger and larger, the function will be approaching +inf. That's what end behavior is.
      (3 votes)
  • blobby green style avatar for user Aysha S
    If the degree of a polynomial function is odd or even what is the behavior of the function at + or - Infinity
    (3 votes)
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  • blobby green style avatar for user sanjanakabad13
    This doesn't make any sense still. At 2.48, what does he mean by if a < 0, we have the opposite? Does that mean that if a<0 that values decrease as a increases or does that mean that as a decreases, other values also decrease?
    And do the exponential powers have something to do with the values increasing and decreasing?
    (3 votes)
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  • duskpin ultimate style avatar for user Maralmaa
    In why function a>0 is started by negative when a is greater than zero a>0? If a is >0 won't it be positive?
    (3 votes)
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Video transcript

What I want to do in this video is talk a little bit about polynomial end behavior. And this is really just talking about what happens to a polynomial if as x becomes really large or really, really, really negative. For example, we're familiar with quadratic polynomials where y is equal to ax squared plus bx plus c. We know that if a is greater than 0, this is going to be an upward opening parabola of some kind. So it's going to look something like that, the graph of this equation, or of this function, you could say. And if a is less than 0, it's going to be a downward opening parabola. We've spent less time with third degree polynomials, but we've also seen those a little bit. So for example, if you have the third degree polynomial, y is equal to ax to the third plus bx squared plus cx plus d, if a is greater than 0-- I don't want to use that brown color. If a is greater than 0, when x is really, really, really negative, this whole thing is going to be really, really, really negative. And then it's going to increase as x becomes less negative. It's going do something-- it might do a little bit of funky stuff in between. But then as x becomes more and more and more positive, it will become more and more and more positive as well. So it might look something like this when a is greater than 0. But what about when a is less than 0? Well then, just like here, we would flip it. We would flip it so that if a is less than 0, when x is really negative, you're going to multiply that times a negative a and you're going to get a positive value. So it's going to look something like this. And then it's going to go like this. It might do a little bit of this type of business in between. But then its end behavior, it starts decreasing again. It starts decreasing. So when we talk about end behavior, we're talking about the idea of what is this function? What does this polynomial do as x becomes really, really, really, really positive and as x becomes really, really, really, really negative? And kind of fully recognizing that some weird things might be happening in the middle. But we just want to think about what happens at extreme values of x. Now obviously for the second degree polynomial nothing really weird happens in the middle. But for a third degree polynomial, we sort of see that some interesting things can start happening in the middle. But the end behavior for third degree polynomial is that if a is greater than 0-- we're starting really small, really low values-- and as a becomes positive, we get to really high values. If a is less than 0 we have the opposite. And these are kind of the two prototypes for polynomials. Because from there we can start thinking about any degree polynomial. So let's just think about the situation of a fourth degree polynomial. So let's say y is equal to ax to the fourth power plus bx to the third plus cx squared plus dx plus-- I don't want to write e because e has other meanings in mathematics. I'll say plus-- I'm really running out of letters here. I'll just use f, although this isn't the function f. This is just a constant f right over here. So let's just think about what this might look like. Let's think about its end behavior, and we could think about it relative to a second degree polynomial. So its end behavior, if x is really, really, really, really negative, x to the fourth is still going to be positive. And if a is greater than 0 when x is really, really, really negative, we're going to have really, really positive values, just like a second degree. And when x is really positive, same thing. x to the fourth is going to be positive, times a is still going to be positive. So its end behavior is going to look very similar to a second degree polynomial. Now, it might do-- in fact it probably will do some funky stuff in between. It might do something that looks kind of like that in between. But we care about the end behavior. I guess you could call the stuff that I've dotted lined in the middle, this is called the non-end behavior, the middle behavior. This will obviously be different than a second degree polynomial. But what happens at the ends will be the same. And the reason why, when you square something, or you raise something to the fourth power, you raise anything to any even power for a very large-- as long as a is greater than 0, for very large positive values, you're going to get positive values. And for very large negative values, you're going to get very large positive values. You take a negative number, raise it to the fourth power, or the second power, you're going to get a positive value. Likewise, if a is less than 0, you're going to have very similar end behavior to this case. For a polynomial where the highest degree term is even-- so this is a is less than 0-- your end behavior when a is really, really, really, really negative, this thing is going to be really, really, really positive. We're going to be multiplying it times a negative, so it's going to be really, really, really, really negative. So it'll look like this. And likewise, when x is really, really, really positive, you get the same thing. Because you're going to be multiplying a positive times a, which is negative, and in between it might be doing something like that. But its end behavior, you see, is very similar to a second degree polynomial. So if you ignore this, its end behavior is very similar. Now the same is true for a fifth degree if you were to compare it to a third degree. And the overall idea here is what happens to this value when we get really large x's or really small x's? Are we taking it to an even power? In which case for either really negative values or really positive values we're going to get positive values. And then it depends what our coefficient a is. Or are we taking it to an odd power? So the general idea-- and actually let me just do a fifth degree, just to make the clear. So if I had something of the form y is equal to ax to the fifth plus bx to the fourth plus-- and it just went all the way-- I won't even have to write it. This thing, if a is greater than 0, would look something like this. Its end behavior is very similar to a third degree polynomial where a is greater than 0. At the end it would do this. Now, it might do kind of some craziness like this. I have to get this right. So 1, 2, 3-- it might do some craziness like this in between. But then for really large x's it will look the same as ax to the third when a is greater than 0. So once again, very, very similar end behavior when a is greater than 0, and very similar end behavior when a is less than 0. It would look like this. At the ends at a negative value it will be positive because this part is going to be really negative. But then it's going to be multiplied by a negative to get a positive. And for really positive values of x, it will be negative. Because once again, this a term is going to be negative. And then what it does in between-- at least for the sake of this video-- we're not really thinking about. So the big takeaway here-- and this is kind of a little bit of a drum roll here, we're talking about end behavior-- if you're looking at an even degree polynomial it's going to have end behavior like a second degree polynomial. If you ignore what happens in the middle, what happens at really negative values of x and really positive values of x is going to be very similar to a second degree polynomial. And if your degree is odd, you're going to have very similar end behavior to a third degree polynomial. You might do all sorts of craziness in the middle, but given for a given a, whether it's greater than 0 or less than 0, you will have end behavior like this, or end behavior like that.