Which of the following is an equivalent form of the equation of the graph shown in the xy plane above, from which the coordinates of vertex A can be identified as constants in the equation? Okay, so in these questions, you got to be careful why you got to be careful, just because I said some but not really, why you got to be careful, because a lot of the answer choices can be reasonable equations that describe the parabola, but only one of them are in a form that will help you identify the vertex of the parabola. Okay. So which one is that now you probably have a guess. But if you're here, then you probably have no clue. So, this is how we do it.

First of all, first step. Keep in mind that the best way to find out the vertex of a parabola Literally through rewriting the equation in what we call the vertex form. So what's the vertex form? Well, that's just writing it in a y is equal to a times x minus h squared plus k, where h and k are the point of the vertex. So this is literally why and you can really just know it, why we call this the vertex form because it's just pick out these what value for h and k are, and that will be the point for your vertex. Excellent.

Okay. So how could we change this? Because this is the equation of the parabola, how can we get it into vertex form? Okay, so if you bear with me, I will show you how to do it. Or what you can do is just pause the video. Think about it.

Work hard, sweat a little bit, and then come back and play this and I'll wait for you. Go ahead. I'm waiting. Okay, did you try it? No, I do. didn't try it, I know you did it.

Okay. So let's take a look at how we're going to do this. So let's rewrite this down here. y is equal to x squared minus two x minus 15. So what you want to do is complete the square. So you want to complete the square for this part right here.

Now, if you don't remember what completing the square is, we have absolutely no idea what I'm spouting out, you can watch one of our earlier videos describing in detail what completing the square is. But just as a quick refresher, how you'd want to complete the square for this is you want to take this value right here, which we call B, and divided by two and square it. Whatever that value is, you will add it to our equation down here. Okay, now watch, watch how we do this. This might not make too much sense right now, but let's take a look at how it goes down. So we get y is equal to All right, so our B in this case What our b is negative two, okay, that's what we mean by B, the B's just whatever the coefficient is in front of the x. okay not to find the x squared and not the negative 15.

It's whatever it's in front of the x. So you have the negative two divided by two. Okay, I'm gonna write that right here divided by two, and square. So negative two divided by two is negative one, you square that, you'll get one. So we're going to add that one to this equation. So we will get x squared minus two x plus one.

And we'll throw some parentheses, because I like to throw parentheses minus 15. But notice how by adding that one, I just messed up this whole equation. It's not the equation it was before. So just to make sure I didn't anger the math gods, I'm going to add a or rather subtract a one right? just added a one here, and I subtracted a one. That's just pretty much zero.

Okay, so, in reality, I didn't actually change anything here, I just added a one and I subtract it again. But even though I added nothing, this makes a huge difference because completing the square depends on this. So afterwards how we're going to change this is you can rewrite this into x minus one squared minus now the negative 15 minus one gives me negative 16. Now, do you believe me that x minus one squared is equal to x squared minus two x plus one? Of course you don't. I know you're doubting me.

So for all the doubters and the haters out there, this is how we do it. So x minus one times x minus one just so you can go to sleep at night, comfortably, and knowing that this actually makes sense. x times x is x squared. Then negative one times x is minus x, then x times negative one is minus x. And then negative one times negative one is plus one, positive one. And how do we simplify that?

Well, we get x squared. Combine the negative x's, we get negative two x, we add the one. All right, this one goes out to all the doubters. That is exactly the same thing. Cool. So now we've established that this is the same thing as that, which is the point of completing the square, then?

Well, we're done. Because what is this guy? This guy is our vertex form. And what is our point? Well, I'll point is, remember, h k, in this case, our H is one, and our K is minus 16. or minus 16. So that point doesn't matter.

But what matters is the best form which one looks like it. Your boy D. That is it. Go on. Have a nice drink. You answered the question correctly.