Introduction to vertex form of a quadratic

It might not be obvious when you look at these three equations, but they're the exact same equation. They've just been algebraically manipulated. They are in different forms. This is the equation and sometimes called standard form for a quadratic. This is the quadratic in factored form. Notice this has been factored right over here. This last form is what we're going to focus on in this video. This is sometimes known as vertex form.

We're not going to focus on how do you get from one of these other forms to vertex form in this video; we'll do that in future videos. But what we're going to do is appreciate why this is called vertex form. Now to start, let's just remind ourselves what a vertex is. So, as you might remember from other videos, if we have a quadratic, if we're graphing y is equal to some quadratic expression in terms of x, the graph of that will be a parabola.

It might be an upward opening parabola or a downward opening parabola. So, this one in particular is going to be an upward opening parabola, and so it might look something like this. So, it might look something, something like this right over here. For an upward opening parabola like this, the vertex is this point right over here. You could view it as this minimum point. You have your x coordinate of the vertex right over there, and you have your y coordinate of the vertex right over here.

Now, the reason why this is called vertex form is it's fairly straightforward to pick out the coordinates of this vertex from this form. How do we do that? Well, to do that, we just have to appreciate the structure that's in this expression. Let me just rewrite it again. We have y is equal to 3 times (x + 2) squared minus 27. The important thing to realize is that this part of the expression is never going to be negative. No matter what you have here, if you square it, you're never going to get a negative value.

So, this is never going to be negative, and we're multiplying it by a positive right over here. This whole thing right over here is going to be greater than or equal to 0. So, another way to think about it is that it's only going to be additive to negative 27. Your minimum point for this curve right over here, for your parabola, is going to happen when this expression is equal to zero, when you're not adding anything to negative 27.

When will this equal zero? Well, it's going to be equal to zero when (x + 2) is going to be equal to 0. So, you could just say if you want to find the x coordinate of the vertex, well for what x value does (x + 2) equal 0? And, of course, we can subtract 2 from both sides, and you get x is equal to negative 2. So, we know that this x coordinate right over here is negative 2.

Then, what's the y-coordinate of the vertex? You could say, "Hey, what is the minimum y that this curve takes on?" Well, when x is equal to negative 2, this whole thing is 0, and y is equal to negative 27. So, this right over here is negative 27, and so the coordinates of the vertex here are (-2, -27). You were able to pick that out just by looking at the quadratic in vertex form.

Now, let's get a few more examples under our belt so that we can really get good at picking out the vertex when a quadratic is written in vertex form. So let’s say let’s pick a scenario where we have a downward opening parabola where y is equal to, let’s just say, negative 2 times (x - 5) squared plus 10. Well, here, this is going to be downward opening, and let's appreciate why that is.

So here, this part is still always going to be non-negative, but it's being multiplied by a negative 2. So it's actually always going to be non-positive. This whole thing right over here is going to be less than or equal to zero for all x's. So it can only take away from the 10. Where do we hit a maximum point? Well, we hit a maximum point when (x - 5) is equal to 0 and we're not taking anything away from the 10.

And so (x - 5) is equal to 0. Well, that, of course, is going to happen when x is equal to 5, and that indeed is the x coordinate for the vertex. What’s the y coordinate for the vertex? Well, if x is equal to 5, this thing is zero, and you're not going to be taking anything away from the ten. So, y is going to be equal to ten, and so the vertex here is (5, 10).

I’m just gonna eyeball it; maybe it’s right over here. x equals 5 and y equals 10. If this is negative 27, this would be positive 27. 10 would be something like this, not using the same scales for the x and y axis, but there you have it. So it's (5, 10) and our curve is going to look something; it's going to look something like this. I don't know exactly where it intersects the x-axis, but it's going to be a downward opening parabola.

Let's do one more example just so that we get really fluent at identifying the vertex from vertex form. So let’s say, I’m just going to make this up, we have y is equal to negative π times (x - 2.8) squared plus 7.1. What is the vertex of the parabola here? Well, the x coordinate is going to be the x value that makes this equal to 0, which is 2.8. And then if this is equal to 0, then this whole thing is going to be equal to 0, and y is going to be 7.1.

So now you hopefully appreciate why this is called vertex form; it's quite straightforward to pick out the vertex when you have something written in this way.