 So let's take a look at quadratic graphs in some detail. So the graph y equals x squared, the basic quadratic, has several distinguishing features. We know that the shape of the graph, this particular shape, is parabolic. The figure is actually a parabola. And we also note that the graph here falls until we reach a lowest point here. That's going to be called our vertex. And then after that point, the graph rises. So now any transformation of this graph is going to slide it around horizontally. Vertically, it'll stretch it out, it'll do some various things. But this vertex, this particular behavior of the vertex, is going to be unchanged. The graph will fall until it hits it and rise after. Or if I flip this graph across the x-axis, it'll rise until it hits the vertex and then fall thereafter. So the vertex is actually a fairly important point. It's where the graph changes direction. So let's take a look at a quadratic graph. We know this is a quadratic because we have this x squared term. y equals minus 2x squared plus 12x minus 14. Now the easiest way to graph this is to change it, to transform it, into some transformation of the basic graph f of x equals x squared. So I need to figure out how I'm going to shift it horizontally, how I'm going to scale it, how I'm going to shift it vertically, and so on. So to do that, we'll complete the square. So now I want to complete the square, and the only thing I've ever completed the square on has been an x squared plus other stuff. So I need to do something with this minus 2, and again the constant minus 14 is going to cause some complication. So we'll split off that constant and we'll factor for the rest of the terms, we'll factor out a minus 2. So I factor out the minus 2, that's minus 2, that leaves me an x squared. Minus 2 from here gives me a minus 6x. The constant I'll put off to the side and ignore for a little bit. So now I have something that looks like I could complete the square of x squared minus 6x. And so I had the first two terms of a perfect square. The next term that I need is going to be plus 9. And again, I don't want to change the equality, so I need to subtract 9 at the same time. So I'm going to add and subtract 9. And now I have a perfect square and some other stuff. So I'll simplify, I'll do a little bit of algebraic manipulation here. This is the perfect square x minus 3 squared. The minus 9 gets carried along for the right. I'll expand this out slightly. This is minus 2 times all this. Minus 2 times minus 9, that's plus 18. And the minus 14 still carried along for the right. And I'll do that final bit of algebraic simplification. So now I have my perfect square and my leftover terms. Now, this allows me to look at the graph. This graph, y equals all of this. Well, this is my graph, y equals x squared. I'm going to shift it to the right by three units. I'm going to stretch it vertically by a factor of 2. Flip it across the x-axis. And then I'm going to shift it vertically by four units. So there's my graph, y equals minus 2x minus 3 squared plus 4. Now, the important thing to recognize here is the vertex undergoes exactly the same transformations. So I'm going to shift that vertex to the right by four units. And that'll put me at 3-0. I'm going to stretch the graph vertically by a factor of 2. Well, that doesn't actually change where the vertex is. I'm going to reflect the graph across the x-axis. Again, that doesn't change the vertex. And then I'm going to shift the graph vertically by four units. And so my graph vertex now moves up to the location 3-4. So now I know where the vertex is located. It's located at 3-4. And again, that vertex is going to be where the graph changes directions. And if I look at the graph, I see that the graph is rising until I hit x equals 3. And then it's falling afterwards. So the graph is rising for x less than 3. And it's falling for x greater than 3.