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Finding the vertex of a parabola in standard form

Sal rewrites the equation y=-5x^2-20x+15 in vertex form (by completing the square) in order to identify the vertex of the corresponding parabola. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • mr pink red style avatar for user Neera Kapoor
    why is it that to find a vertex you must do -b/2a? is there a separate video on it?
    (43 votes)
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    • cacteye blue style avatar for user Jerry Nilsson
      A parabola is defined as
      𝑦 = 𝑎𝑥² + 𝑏𝑥 + 𝑐 for 𝑎 ≠ 0

      By factoring out 𝑎 and completing the square, we get
      𝑦 = 𝑎(𝑥² + (𝑏 ∕ 𝑎)𝑥) + 𝑐 =
      = 𝑎(𝑥 + 𝑏 ∕ (2𝑎))² + 𝑐 − 𝑏² ∕ (4𝑎)

      With ℎ = −𝑏 ∕ (2𝑎) and 𝑘 = 𝑐 − 𝑏² ∕ (4𝑎) we get
      𝑦 = 𝑎(𝑥 − ℎ)² + 𝑘

      (𝑥 − ℎ)² ≥ 0 for all 𝑥
      So the parabola will have a vertex when (𝑥 − ℎ)² = 0 ⇔ 𝑥 = ℎ ⇒ 𝑦 = 𝑘

      𝑎 > 0 ⇒ (ℎ, 𝑘) is the minimum point.
      𝑎 < 0 ⇒ (ℎ, 𝑘) is the maximum point.
      (40 votes)
  • leaf grey style avatar for user cmaryk12296
    Is there a video about vertex form?
    (10 votes)
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    • leaf blue style avatar for user Matthew Daly
      Not specifically, from the looks of things. When Sal gets into talking about graphing quadratic equations he talks about how to calculate the vertex. On the other hand, there are several exercises in the practice section about vertex form, so the hints there give a good sense of how to proceed.
      (14 votes)
  • duskpin ultimate style avatar for user kcharyjumayev
    In which video do they teach about formula -b/2a
    (14 votes)
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  • blobby green style avatar for user half.korean1
    Why does x+4 have to = 0?
    (7 votes)
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    • male robot johnny style avatar for user Adam Doyle
      Because then you will have a y coordinate for a given x. When x-4 = 0 (i.e. when x =4) you are left with just y=21 in the equation: because
      4-4=0
      0^2=0
      -3(0)=0
      This leaves the equation looking like y=0+21
      Then you know that when x=4 that y=21. Then you have solved for x and y.
      If you want to think about it a different way you could use y=f(x). Then f(4)=21. Some people might find the f(x) way easier to understand.
      (14 votes)
  • leaf blue style avatar for user William Carrier
    @ he mentions it's mentioned in multiple videos but I made a search and watched a few videos trying to find it in vain. Would someone kindly reply with URLs or page titles of the videos he referred to?
    (4 votes)
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    • blobby green style avatar for user Gad
      I don't know there those videos are, but I think is quite easy to realize it. See: the x coordinate of the vertex is the average point between the two roots of the quadratic (the two points where the graph of the parabola intersects with the horizonal axis. So if we know that the formula for one of those roots, as per the quadratic formula, is -b+square root of -bsquared-4ac, all divided by 2a, and the formula for the other root i the same but with a minus sign on the numerator instead (this is, -b - squareroot of bsquared -4ac, all divided by 2a). Then the average between those two roots is obtained by adding those two formulas and dividing by two.

      So: x coordinate of the vertex as an average point between the two roots->

      ((-b+sqrt(bsquared-4ac))/2a + (-b-sqrt(bsquared-4ac))/2a) / 2 =

      (-b/2a + (sqrt(bsquared-4ac))/2a -b/2a -(sqrt(bsquared-4ac))/2a) / 2=

      (2*-b/2a) /2 =
      -b/2a

      See how sqrt(bsquared-4ac)/2a and -sqrt(bsquared-4ac)/21 get cancelled out of the expression.

      So the key seems to be
      1) the x coordinate of the vertex is equal to the average point between the two roots of the parabola -the points where the parabola intersects the horizontal axis

      2) the roots of the parabola can be found via the quadratic formula.

      So applying the arithmetic average formula (a+b)/2 where a is -b+sqrt(bsquared-4ac)/2a and b is -b-sqrt(bsquared-4ac)/a gives -b/2a as solution for x coordinate of vertex. This formula also works if the parabola has only one root. In that case, the vertex would lie on the horizontal axis.-
      (7 votes)
  • starky ultimate style avatar for user Matthew
    Can someone explain why the x-vertex formula is -b/2a?
    (4 votes)
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    • stelly blue style avatar for user Kim Seidel
      We know we can find the x-intercepts of the parabola by using the quadratic formula.
      1st x-intercept: x = [-b + sqrt(b^2-4ac)]/(2a)
      2nd x-intercept: x = [-b - sqrt(b^2-4ac)]/(2a)

      The x-value of the vertex is located midway between these 2 points. If you average the two x-intercepts, you get their midpoint. This means you add the 2 points then divide by 2.
      ADD 1st: [-b + sqrt(b^2-4ac)]/(2a) + [-b - sqrt(b^2-4ac)]/(2a)
      -- Notice, we already have a common denominator, so we add the numerators.
      -- Also notice, the square roots add to 0 because one is positive and the other negative.
      -- This leaves: (-b-b)/(2a) = (-2b)/(2a) = -b/a

      DIVIDE BY 2: -b/a divided by 2 = -b/a * 1/2 = -b/(2a)

      Hope this helps.
      (6 votes)
  • marcimus pink style avatar for user ส็็็็็็็็็็็็็็็็็็็ sholmes ส้้้้้้้้้้้้้้้้้้้้
    At how does Sal get x=4? Wouldn't the expression -3(x-4)^2 have to equal - 21 for the whole equation to equal zero?
    (5 votes)
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  • blobby green style avatar for user Rico Jomer
    Why is x vertex equal to -b/2a ?
    (3 votes)
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  • hopper cool style avatar for user Ian
    This video is not about the equation y=-3x^2+24x-27
    It is about completing the square to solve 4x^2+40x-300=0
    Can anyone help me?
    The transcript is going but it is different words!
    (3 votes)
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  • leaf green style avatar for user Aisha Nusrat
    How can we find the domain and range after compeleting the square form?
    (0 votes)
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    • spunky sam blue style avatar for user Igal Sapir
      The Domain of a function is the group of all the x values allowed when calculating the expression.
      In this exercise all x values can be used so the domain is the group of all the Real numbers.
      Examples to functions that would limit the domain would contain operations like:
      Division - Because division by 0 is not allowed
      Square root - Because Square root of a negative number is not a real number
      As you can see there are no such operations in this exercise.
      The Range of a function is the group of all the y values that result from calculating the function for all the x values allowed (the Domain).
      As Sal explains in the last part of the video when you bring the parabola to its vertex form it is easier to see the Range.
      The free coefficient, i.e., the C in the video, is either the minimum or the maximum point of the Range.
      The sign of the leading coefficient, i.e., the A in the video, determines whether it is the minimum or the maximum.
      If A>0 the parabola open upwards (we call it smiling :-) and all other values of y will be greater than C, i.e., C is minimum and the Range is y>=C
      If A<0 the parabola open downwards (we call it weeping :-) and all other values of y will be smaller than C, i.e., C is maximum and the Range is y<=C
      In this exercise A is (-3) and it is negative, so 21 is the maximum and the Range is y<=21
      Hope it helps :-)
      (14 votes)

Video transcript

I have an equation right here. It's a second degree equation. It's a quadratic. And I know its graph is going to be a parabola. Just as a review, that means it looks something like this or it looks something like that. Because the coefficient on the x squared term here is positive, I know it's going to be an upward opening parabola. And I am curious about the vertex of this parabola. And if I have an upward opening parabola, the vertex is going to be the minimum point. If I had a downward opening parabola, then the vertex would be the maximum point. So I'm really trying to find the x value. I don't know actually where this does intersect the x-axis or if it does it all. But I want to find the x value where this function takes on a minimum value. Now, there's many ways to find a vertex. Probably the easiest, there's a formula for it. And we talk about where that comes from in multiple videos, where the vertex of a parabola or the x-coordinate of the vertex of the parabola. So the x-coordinate of the vertex is just equal to negative b over 2a. And the negative b, you're just talking about the coefficient, or b is the coefficient on the first degree term, is on the coefficient on the x term. And a is the coefficient on the x squared term. So this is going to be equal to b is negative 20. So it's negative 20 over 2 times 5. Well, this is going to be equal to positive 20 over 10, which is equal to 2. And so to find the y value of the vertex, we just substitute back into the equation. The y value is going to be 5 times 2 squared minus 20 times 2 plus 15, which is equal to let's see. This is 5 times 4, which is 20, minus 40, which is negative 20, plus 15 is negative 5. So just like that, we're able to figure out the coordinate. This coordinate right over here is the point 2, negative 5. Now it's not so satisfying just to plug and chug a formula like this. And we'll see where this comes from when you look at the quadratic formula. This is the first term. It's the x value that's halfway in between the roots. So that's one way to think about it. But another way to do it, and this probably will be of more lasting help for you in your life, because you might forget this formula. It's really just try to re-manipulate this equation so you can spot its minimum point. And we're going to do that by completing the square. So let me rewrite that. And what I'll do is out of these first two terms, I'll factor out a 5, because I want to complete a square here and I'm going to leave this 15 out to the right, because I'm going to have to manipulate that as well. So it is 5 times x squared minus 4x. And then I have this 15 out here. And I want to write this as a perfect square. And we just have to remind ourselves that if I have x plus a squared, that's going to be x squared plus 2ax plus a squared. So if I want to turn something that looks like this, 2ax, into a perfect square, I just have to take half of this coefficient and square it and add it right over here in order to make it look like that. So I'm going to do that right over here. So if I take half of negative 4, that's negative 2. If I square it, that is going to be positive 4. I have to be very careful here. I can't just willy nilly add a positive 4 here. I have equality here. If they were equal before adding the 4, then they're not going to be equal after adding the 4. So I have to do proper accounting here. I either have to add 4 to both sides or I should be careful. I have to add the same amount to both sides or subtract the same amount again. Now, the reason why I was careful there is I didn't just add 4 to the right hand side of the equation. Remember, the 4 is getting multiplied by 5. I have added 20 to the right hand side of the equation. So if I want to make this balance out, if I want the equality to still be true, I either have to now add 20 to y or I have to subtract 20 from the right hand side. So I'll do that. I'll subtract 20 from the right hand side. So I added 5 times 4. If you were to distribute this, you'll see that. I could have literally, up here, said hey, I'm adding 20 and I'm subtracting 20. This is the exact same thing that I did over here. If you distribute the 5, it becomes 5x squared minus 20x plus 20 plus 15 minus 20. Exactly what's up here. The whole point of this is that now I can write this in an interesting way. I could write this as y is equal to 5 times x minus 2 squared, and then 15 minus 20 is minus 5. So the whole point of this is now to be able to inspect this. When does this equation hit a minimum value? Well, we know that this term right over here is always going to be non-negative. Or we could say it's always going to be greater than or equal to 0. This whole thing is going to hit a minimum value when this term is equal to 0 or when x equals 2. When x equals 2, we're going to hit a minimum value. And when x equals 2, what happens? Well, this whole term is 0 and y is equal to negative 5. The vertex is 2, negative 5.