Topic B: Lesson 16: Graphing quadratic equations from the vertex form
- [Instructor] 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. And this last form is what we're going to focus on in this video. This is sometimes known as vertex form and we're not gonna focus on how do you get from one of these other forms to a 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. 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, and it might be an upward opening parabola or a downward opening parabola. This one in particular is going to be an upward opening parabola, and so it might look something like this. It might look something like this right over here. And 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 three times x plus two 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. And 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 zero. So another way to think about it, it's only going to be additive to negative 27. So 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. And so, when will this equal zero? Well, it's going to be equal to zero when x plus two is going to be equal to zero. So you could just say, if you wanna find the x-coordinate of the vertex, well, for what x value does x plus two equal zero? And of course we can subtract two from both sides and you get x is equal to negative two, so we know that this x-coordinate right over here is negative two. And then what's the y-coordinate of the vertex? You could just say, "Hey, what is the minimum y "that this curve takes on?" Well, when x is equal to negative two, this whole thing is zero and y is equal to negative 27. Y is equal to negative 27, so this right over here is negative 27. And so the coordinates of the vertex here are negative two comma negative 27. And you are 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 two times x plus five, actually, let me make it x minus five. X minus five squared, and then let's say plus 10. Well here, this is gonna 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 two, so it's actually always gonna be non-positive. So this whole thing right over here is going to be less than or equal to zero for all x's, so it could only take away from the 10. So, where do we hit a maximum point? Well, we hit a maximum point when x minus five is equal to zero, when we're not taking anything away from the 10. And so, x minus five is equal to zero. Well, that of course is going to happen when x is equal to five, and that indeed is the x-coordinate for the vertex. And what's the y-coordinate for the vertex? Well, if x is equal to five and this thing is zero, you're not gonna be taking anything away from the 10 and so y is going to be equal to 10. And so the vertex here is x equals five, and I'm just gonna eyeball it, maybe it's right over here, x equals five. And y is equal to 10. If this is negative 27, this would be positive 27, 10 would be something like this. I'm not using the same scales for the x and y-axis, but there you have it. So it's five comma 10 and our curve is gonna 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 gonna make this up, we have y is equal to negative pi times x minus 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 zero, which is 2.8. And then if this is equal to zero, then this whole thing is going to be equal to zero 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.