[AUDIO LOGO] - Hey, everyone, I'm Ms. Milkosky, the mathematician. And today we are going to find the vertex of the parabola, or of a parabola. So I have one right here-- y equals x squared plus 10x plus 5. So as you can see, it's in standard form. And here is a question I have for you. How do we find the vertex of a parabola when it's given in standard form? 

So let's come over here and write what the formula is for that. So the vertex is located at-- let me do this in a different color-- the opposite of b over 2a comma f of the opposite of b over 2a. Oops. Wait. I need to close the parentheses there. All right. 

So a lot of students get confused. When they see opposite of b over 2a, they generally understand, but f of the opposite of b over 2a can be confusing for people sometimes. So what I recommend doing is, if you have it in the form y equals x squared plus 10x plus 5, like this, replace the y with an f of x and then equals x squared plus 10x plus 5 because that will help you understand better what you're doing. 

Basically what you're doing is you're finding the x-coordinate of the vertex. And then whatever number you get, you're plugging it into the function for x. So literally, it's just, this is your x-coordinate of the vertex. This is your y. And you just find x and then plug it back into the function to get y. So this means f of whatever. If I said f of 3, you would plug 3 in for x, and out would pop your y value. 

So let's give this a shot. The first thing you need to do is label your a, your b, and your c. And back when I was in the classroom, I legitimately gave-- I don't know if it was a point per one or if it was a point if they did all three or maybe a half a point for each one. I can't remember. But basically, I required that my students write what each one is. 

So a is the coefficient of the x-squared term. So in this case, because there's nothing out front, that means it's 1. We could place a 1 here, but we don't write that. It's assumed. So a equals 1. b is the coefficient of the x term, of the linear term. So that's going to be 10. 

And c is your constant. So a, b, c because we know we can write it also as ax squared plus bx plus c. So we just need to know those values. I'm just going to erase that so we have more room. All right. 

So let's start with finding the x-coordinate of the vertex. So we know it's the opposite of b over 2x. So I'm just going to write-- so the x-coordinate, "coord," we'll just call it-- and that's going to be the opposite of b over 2a. So b is 10. So the opposite of b is negative 10 over 2 times a, which is 1. 

So then that's just going to be negative 10 over 2. And negative 10 divided by 2 is negative 5. Perfect. So now we can come over here and say that our x-coordinate-- so we could write x-coord-- I don't know. You could just write x equals, I suppose, but it's the x-coordinate. 

So we'll come over here, and we're going to say, OK, so the x-coordinate is negative 5. Beautiful. Now, if we want to know what the y-coordinate is, remember, we just want to know, what's y when x equals negative 5? That's literally what we're doing. It's f of negative 5. 

So we come over. I'll use green. Maybe I'll draw a line here. Visually, I think that'll be better. OK, so now we want the y-coordinate. I'll call it "coord," again. So that's just going to be f of negative 5. 

Now, you've been given stuff like that since probably middle school, where it's just, evaluate the function for the given value, f of negative 5, and you just plug it in. So we come up here, and we say, OK, so this is f of negative 5. That means, wherever I see an x, I'm putting in a negative 5. 

So negative 5 squared, because I've got x squared, plus 10 times negative 5 plus 5. OK, negative 5 squared-- that's going to be positive 25. And then 10 times-- so then plus, and then 10 times negative 5 is negative 50 and then plus 5. 

So remember, our shorthand is we tend not to write plus negative, so we just write it as subtraction, so 25 minus 50 plus 5. So 25 negative 50 is negative 25 and then plus 5, so that goes-- I know some students do little arrows to show that that's what they did, and I think that's great. 

So negative 25-- sorry-- positive 25 plus negative 50, or minus 50, is negative 25 and then plus 5, and that's going to be negative 20. Perfect. So our y-coordinate is negative 20. Awesome. 

So there you go. So it was as simple as that. We just know that it's the opposite of b over 2a, f of the opposite of b over 2a. So basically, we find the x-coordinate. We plug it back into the function, that value back into the function to get the y-coordinate. And then we're all set. We've got the vertex. 

Hey, everyone. I'm Ms. Milkosky, the mathematician, and I hope you found that last video helpful. If you have any questions or if there are any topics you'd like to see me cover in future videos, definitely let me know in the comments below. Thank you again for watching. I hope you have a wonderful day. And I'm going to leave you with the most important piece of advice I can give you, which is, when you're doing mathematics, always ask why.