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Comparing maximum points of quadratic functions

Given several quadratic functions represented in different forms, Sal finds the one with the lowest maximum value. Created by Sal Khan.

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  • blobby green style avatar for user levi sleight
    i have literally no clue whatsoever what that last bit where he "completed the square" was. can anyone help me with this?
    (13 votes)
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  • blobby green style avatar for user Mark Cuoccio
    Sal lost me in the last 30 seconds of the video. The negative: - (x - 3) seemed to disappear with no explanation. What happened, To me it seems that x-3 = 0 would produce x = 3, but the preceding negative would make x = -3 and y = 8. Is that what happened? Some of these videos assume that we can follow ideas with leaps of imagination that my current math knowledge will not permit. I need to see EACH STEP EXPLICITLY EXPLAINED!
    (6 votes)
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    • duskpin ultimate style avatar for user amydylee
      Haha, I'm answering this 7 years later...
      Anyway, x = 3 because in order to find the y-value, -(x-3)^2 has to equal 0. And the only way that's possible is if x = 3. If you plug in x = 3, then -(3-3)^2 = 0. I guess that equals -0, but -0 and 0 are the same thing. If x = -3 like you thought, then -(-3-3)^2 = -36, and that doesn't help to find the y-value.
      Hope this helped!
      (2 votes)
  • male robot hal style avatar for user 北島
    For the minimum/maximum, is the vertex the same thing?
    (4 votes)
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  • blobby green style avatar for user Levon Tarver
    Where did the 6x go when he converted it to a perfect square?
    (1 vote)
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  • purple pi purple style avatar for user Judith Jones
    I solved this problem without completing the square. I just tried out x=2 in f(x)
    f(2) =-4 +12 -1 =7
    If a point of f(x) is 7 then its maximum cannot be less than -1 so of the three functions the one with the lowest maximum is h
    (3 votes)
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  • blobby green style avatar for user Devika
    Why did Sal use -1 as the maximum value? (at ) There is also 4 (x-coordinate). Can't he just take the average of the two? But if he does, it becomes a crazy fraction. Can somebody please clear this up for me? Thank you so much.
    (2 votes)
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  • purple pi teal style avatar for user Avishek
    At the beginning of the video we're given a quadratic function g(x) for which only a table of x & y values are provided. My question is how do we find the quadratic equation of such a function for which only a few x & y values are given.
    (2 votes)
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    • aqualine ultimate style avatar for user Zander Hall
      Hello Avishek, in this problem we only need the max value, not the equation. There is an interesting property of quadratics in that, every quadratic equation,when graphed, has either a maximum or a minimum. What do i mean by this? well with the -x^2+6x-1 for example, the max is 8, and no matter what x-value we choose, we will never find a higher value. Hence, since the table for g(x) shows that g(0) =5 and the values of both adjacent x-values are less, we can conclude that 5 is the maximum value for g(x). As for finding the equation for g(x)? It could be approximated accurately, but it would take a good bit of guess work, or calculus. Hope this helps! Side Note: after some calculations, g(x) appears to be -(x^2)+5 though this is just what i came up with.
      (1 vote)
  • piceratops ultimate style avatar for user Kevin George Joe
    At Sal begins using completing the square method in order to find the lowest maximum value.
    Can I use X= -b/2a where X gives the lowest maximum value and b and a are the coefficients of x and x^2 respectively.
    Thanks.
    PS I did get the correct answer using this method. He showed it in finding the x coordinate of finding the vertex of a parabola.
    (1 vote)
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    • spunky sam blue style avatar for user Alfred
      By completing the square, Sal finds a quadratic (x - something)^2 that is shifted up or down by some other thing. X=-b/2a does not take into consideration the "c" term or constant. The quadratic shows explicitly the min or max value of this function.
      (3 votes)
  • orange juice squid orange style avatar for user sam.i.peka
    Are there any videos where we can do some exercises on getting the perfect square? I've seen it done on a few videos now but must have missed wherever that was.
    (1 vote)
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  • winston default style avatar for user TangJefferson11
    At , why do we take half of -6?
    (1 vote)
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    • mr pink green style avatar for user David Severin
      We are working with completing the square. It starts from the concept that (x + b) ^2 = x(x+b)+b(x+b) = x^2 + bx + bx + b^2 or x^2 + 2bx + b^2. Since the middle has to come from two middle terms, we divide by two. In the specific problem, we have to say that 2b = - 6, thus when we divide both sides, we get b = -3 and b^2 = (-3)^2 = 9.
      (2 votes)

Video transcript

Which quadratic has the lowest maximum value? So let's figure out the maximum value for each of these-- and they're defined in different ways-- and then see which one is the lowest. And I'll start with the easiest. So h of x. We can just graphically look at it, visually look at it, and say-- what's the maximum point? And the maximum point looks like it's right over here when x is equal to 4. And when x is equal to 4, y or h of x is equal to negative 1. So the maximum for h of x looks like it is negative 1. Now, what's the maximum for g of x? And they've given us some points here and here. Once again, we can just eyeball it, and say-- well, what's the maximum value they gave us? Well, 5 is the largest value. It happens when x is equal to 0. g of 0 is 5. So the maximum value here is 5. Now, f of x. They just give us an expression to define it. And so it's going take a little bit of work to figure out what the maximum value is. The easiest way to do that for a quadratic is to complete the square. And so let's do it. So we have f of x is equal to negative x squared plus 6x minus 1. I never like having this negative here. So I'm going to factor it out. This is the same thing as negative times x squared minus 6x and plus 1. And I'm going to write the plus 1 out here because I'm fixing to complete the square. Now, just as a review of completing the square, we essentially want to add and subtract the same number so that part of this expression is a perfect square. And to figure out what number we want to add and subtract, we look at the coefficient on the x term. It's a negative 6. You take half of that. That's negative 3. And you square it. Negative 3 squared is 9. Now, we can't just add a 9. That would change the actual value of the expression. We have to add a 9 and subtract a 9. And you might say-- well, why are we adding and subtracting the same thing if it doesn't change the value of the expression? And the whole point is so that we can get this first part of the expression to represent a perfect square. This x squared minus 6x plus 9 is x minus 3 squared. So I can rewrite that part as x minus 3 squared and then minus 9-- or negative 9-- plus 1 is negative 8. Let me do that in a different color so we can keep track of things. So this part right over here is negative 8. And we still have the negative out front. And so we can rewrite this as-- if we distribute the negative sign-- negative x minus 3 squared plus 8. Now, let's think about what the maximum value is. And to understand the maximum value, we have to interpret this negative x minus 3 squared. Well, x minus 3 squared-- before we think about the negative-- that is always going to be a positive value. Or it's always going to be non-negative. But then, when we make it negative, it's always going to be non-positive. Think about it. If x is equal to 3, this thing is going to be 0. And you take the negative of that, it's going to be 0. x is anything else, x is anything other than 3, this part of the expression is going to be positive. But then, you have a minus sign. So you're going to subtract that positive value from 8. So this actually has a maximum value when this first term right over here is 0. The only thing that this part of the expression could do is subtract from the 8. If you want to get a maximum value, this should be equal to 0. This equals 0 when x is equal to 3. When x is equal to 3, this is 0. And our function hits its maximum value of 8. So this has a max-- let me do that in a color that you can actually read-- this has a max value of 8. So which has the lowest maximum value? h of x.