 This video is going to talk about quadratic functions. We want to look at the characteristics of a quadratic graph. That's what this section is mainly going to be talking about. So we know that we have x-intercepts and they're the zeros of the functions. Sometimes these are nice to find and sometimes they're not. If it's factorable, then it's nice. If it's not factorable, then sometimes you get decimals and it's too hard to just look at the equation to find the x-intercepts. The y-intercept is always the constant of the function. So that's just c. The n-behavior talks about when Tweet opens up or opens down. And this is a quadratic so both ends will go the same because it's a degree 2. Vertex is hk when we're doing transformations. We talked about it being a times x-minus h quantity squared plus k. If I'm in the form of ax squared plus bx plus c, how do I find the vertex then? And we have what we call the vertex formula. And the vertex formula helps you to figure out what your x and your y are. So the x of the vertex is going to be negative b over 2a. That's the formula. And then the y of that vertex is going to be f of negative b over 2a. In other words, just plug and check your negative b over 2a. And the other thing that we didn't talk about that isn't listed here, but the a is going to tell us the direction it opens. So if it's a positive a, it's going to open up. And if it's a negative a, it's going to open down. And then finally we have the axis of symmetry, which is the vertical line through the vertex. Well if it's a vertical line, that means it's an x-equal equation. And it goes through the vertex so it must be equal to the x-value of our vertex. So it's equal to either negative b over 2a or h, whichever one you happen to know. So let's try it. Graph the function using the quadratic characteristics for the x-intercepts. If I look at this one, I need factors of 19. They're going to add up to 10 and 19 is a prime number. So we're not going to be able to find those x-intercepts. So they're not nice. I'll just put it that way. We're not going to graph those because they're not nice. The y-intercept though, we know that's going to be 0 negative 19. So 0, we'll call this down here negative 19. That's the y-intercept. And then the vertex, well remember this isn't in the vertex form. We have to do negative b over 2a. And if we do negative b, that's going to be negative 10 over 2 times a, which is negative 1. So the x is going to be equal to negative 10 over negative 2 or just 5. And then we plug in chug 5. So the opposite of 5 squared plus 10 times 5 minus 19. And if we plug all that in there, we're going to find out that we have y is equal to 6. So we have 5 and 6. So this is my vertex. Now this negative here tells me that it's going to open down. And I can see that if I have a vertex here and go through this point, sure enough it's going to go down. So I would have this part I could graph. But I know the vertex, everything is just symmetrical on the other side. So it would look something like this. And the axis of symmetry, oh by the way this was 5, 6. And the axis of symmetry then is going to be x is equal to that 5, the x of my vertex. So here we have a problem that they're asking us to complete the square and to put it in transformation form, or vertex form. And then we can graph. To remember how to complete the square, you have 2x squared plus 12x. And then remember you want to put the constant on the other side, so minus 5. But then we talked about briefly, but I didn't really do one before, that you have to have x squared. So you have to divide everything by 2. So this is x squared here plus 6x plus some number is equal to negative 5 halves plus some number. Then take half of the middle term, 6 over 2 is equal to 3. And that's actually going to be what's inside our binomial squared. And then we take that 3 and we square it and that tells us what we add to both sides. So plus 9 and 9 is 18 halves since I have it in fractions here. So that's going to be equal to 13 halves. But I need everything on one side equal to 0 to be able to use the vertex form. And I want to clear the fraction first. So I have 2 times x plus 3 quantity squared equal to the 13. So I would really have, let's go this way, 2 times x plus 3 quantity squared minus 13 equal to 0. So that tells me that my vertex is going to be opposite of this, negative 3 and then negative 13. So negative 3 and we'll call this negative 13 down here. So negative 3, negative 13 is my vertex. It opens up. It's got a 2 in front of it. So that means it's going to be skinnier than a normal one like this one. Be skinnier than that one would have been. We know the y-intercept is equal to 0, 5 because of the constant. And if I try to factor, factors of 10, they're going to add up to 12. This isn't factorable though. But I have enough points here that, because I know it's a quadratic. So I had the vertex and I can come up here and get my y-intercept and then make it symmetrical across that and make it a little skinnier. It's not asking me for specific points. It just wants to know that it's graphed. So it's a skinny graph because of the 2. It's a vertex, negative 3, negative 13 after I completed the square. And I knew that my y-intercept from the original formula equation was 0, 5. So that gave me a point of reference. So now we have this problem here. And it tells us that the vertex is negative 2, 1. And if I put it in transformation form or vertex form, then I know it's going to be x plus 2 since it's a negative 2 and plus 1. But I still need to find my a. So to find a, I'm just going to find a nice point on my graph. And a nice point on my graph would be this point right here, which is the point 0, 2. So remember that this is x and this is y. And also remember that f of x is equal to y. So that means that my y, f of x, is 2 is equal to a, which I don't know, times 0 plus 2 quantity squared plus 1. But this is 2 is equal to 2 squared, which would be 4a plus 1. So track the one from both sides. So 1 is equal to 4a and divide by the 4 and we find out that a is equal to 1 fourth. So our equation is really f of x is equal to 1 fourth x plus 2 quantity squared plus 1. Negative 2, 1 is my vertex from here. 1 fourth means it's going to be compressed or squashed or smushed, as we said. So it's going to be a little bit wider graph and that's exactly what we have. Now we have to find the equation completely by ourselves. So first thing we want to know, h is equal to 1 if we went over 1. k is equal to 2 because we went up 2 and we need to find a. And we should be able to find a point on our graph somewhere. So we have the point 0, 0 or we have the 2, 0 right here. So we have y is equal to a times x minus h quantity squared plus k. y we said is, that means x is 2 here, y is 0. So y is 0, we don't know what a is. x is 2 minus my h, which is 1 quantity squared plus k, which is 2. So 0 is equal to 2 minus 1 is 1, 1 squared is 1a plus 2. To track the 2 I have a negative 2 equal to 1a or just plain old a. And you can see that it's a negative so it opens down and it's a negative 2. 2 means that it's going to be stretched and let's verify that our graph opens down and is stretched. It definitely opens down so my negative I agree with and it's kind of skinny so I would agree that it would be a stretched graph. So our equation is going to be y is equal to negative 2 times x minus 1 quantity squared plus my k, which is plus 2.