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Sal rewrites a quadratic equation in vertex form and shows how it reveals the vertex of the corresponding parabola. Created by Sal Khan and Monterey Institute for Technology and Education.
Video transcript
We need to find the vertex and the axis of symmetry of this graph. The whole point of doing this problem is so that you understand what the vertex and axis of symmetry is. And just as a bit of a refresher, if a parabola looks like this, the vertex is the lowest point here, so this minimum point here, for an upward opening problem. If the parabola opens downward like this, the vertex is the topmost point right like that. It's the maximum point. And the axis of symmetry is the line that you could reflect the parabola around, and it's symmetric. So that's the axis of symmetry. That is a reflection of the left-hand side along that axis of symmetry. Same thing if it's a downward-opening parabola. And the general way of telling the difference between an upward-opening and a downward-opening parabola is that this will have a positive coefficient on the x squared term, and this will have a negative coefficient. And we'll see that in a little bit more detail. So let's just work on this. Now, in order to figure out the vertex, there's a quick and dirty formula, but I'm not going to do the formula here because the formula really tells you nothing about how you got it. But I'll show you how to apply the formula at the end of this video, if you see this on a math test and just want to do it really quickly. But we're going to do it the slow, intuitive way first. So let's think about how we can find either the maximum or the minimum point of this parabola. So the best way I can think of doing it is to complete the square. And it might seem like a very foreign concept right now, but let's just do it one step at a time. So I can rewrite this as y is equal to-- well, I can factor out a negative 2. It's equal to negative 2 times x squared minus 4x minus 4. And I'm going to put the minus 4 out here. And this is where I'm going to complete the square. Now, what I want to do is express the stuff in the parentheses as a sum of a perfect square and then some number over here. And I have x squared minus 4x. If I wanted this to be a perfect square, it would be a perfect square if I had a positive 4 over here. If I had a positive 4 over there, then this would be a perfect square. It would be x minus 2 squared. And I got the 4, because I said, well, I want whatever half of this number is, so half of negative 4 is negative 2. Let me square it. That'll give me a positive 4 right there. But I can't just add a 4 willy-nilly to one side of an equation. I either have to add it to the other side or I would have to then just subtract it. So here I haven't changed equation. I added 4 and then I subtracted 4. I just added zero to this little expression here, so it didn't change it. But what it does allow me to do is express this part right here as a perfect square. x squared minus 4x plus 4 is x minus 2 squared. It is x minus 2 squared. And then you have this negative 2 out front multiplying everything, and then you have a negative 4 minus negative 4, minus 8, just like that. So you have y is equal to negative 2 times this entire thing, and now we can multiply out the negative 2 again. So we can distribute it. Y is equal to negative 2 times x minus 2 squared. And then negative 2 times negative 8 is plus 16. Now, all I did is algebraically arrange this equation. But what this allows us to do is think about what the maximum or minimum point of this equation is. So let's just explore this a little bit. This quantity right here, x minus 2 squared, if you're squaring anything, this is always going to be a positive quantity. That right there is always positive. But it's being multiplied by a negative number. So if you look at the larger context, if you look at the always positive multiplied by the negative 2, that's going to be always negative. And the more positive that this number becomes when you multiply it by a negative, the more negative this entire expression becomes. So if you think about it, this is going to be a downward-opening parabola. We have a negative coefficient out here. And the maximum point on this downward-opening parabola is when this expression right here is as small as possible. If this gets any larger, it's just multiplied by a negative number, and then you subtract it from 16. So if this expression right here is 0, then we have our maximum y value, which is 16. So how do we get x is equal to 0 here? Well, the way to get x minus 2 equal to 0-- so let's just do it. x minus 2 is equal to 0, so that happens when x is equal to 2. So when x is equal to 2, this expression is 0. 0 times a negative number, it's all 0, and then y is equal to 16. This is our vertex, this is our maximum point. We just reasoned through it, just looking at the algebra, that the highest value this can take on is 16. As x moves away from 2 in the positive or negative direction, this quantity right here, it might be negative or positive, but when you square it, it's going to be positive. And when you multiply it by negative 2, it's going to become negative and it's going to subtract from 16. So our vertex right here is x is equal to 2. Actually, let's say each of these units are 2. So this is 2, 4, 6, 8, 10, 12, 14, 16. So my vertex is here. That is the absolute maximum point for this parabola. And its axis of symmetry is going to be along the line x is equal to 2, along the vertical line x is equal to 2. That is going to be its axis of symmetry. And now if we're just curious for a couple of other points, just because we want to plot this thing, we could say, well, what happens when x is equal to 0? That's an easy one. When x is equal to 0, y is equal to 8. So when x is equal to 0, we have 1, 2, 3, 4-- oh, well, these are 2. 2, 4, 6, 8. It's right there. This is an axis of symmetry. So when x is equal to 3, y is also going to be equal to 8. So this parabola is a really steep and narrow one that looks something like this, where this right here is the maximum point. Now I told you this is the slow and intuitive way to do the problem. If you wanted a quick and dirty way to figure out a vertex, there is a formula that you can derive it actually, doing this exact same process we just did, but the formula for the vertex, or the x-value of the vertex, or the axis of symmetry, is x is equal to negative b over 2a. So if we just apply this-- but, you know, this is just kind of mindless application of a formula. I wanted to show you the intuition why this formula even exists. But if you just mindlessly apply this, you'll get-- what's b here? So x is equal to negative-- b here is 8. 8 over 2 times a. a right here is a negative 2. 2 times negative 2. So what is that going to be equal to? It is negative 8 over negative 4, which is equal to 2, which is the exact same thing we got by reasoning it out. And when x is equal to 2, y is equal to 16. Same exact result there. That's the point 2 comma 16.