 All right, so let's take a look at the graphs of quadratic functions. So a quadratic graph is the graph of a surprised quadratic function, and these look like, well, the simplest of these quadratic functions is f of x equals x squared. So that is a function that involves a square term, and my functions can get much more complicated. f of x equals one-third x negative squared plus 12x minus 17, and so on. So there's another quadratic function, and if I want to graph a quadratic function, what I can do is I can pick lots of random points and hope that we pick the right ones to get the shape of the graph, or we might use our ability to find graph transformations and see if I can take my nice simple graph y equals x squared here and transform it into something else. Now, if you're only ever going to plot one quadratic function in your entire lifetime, plotting random points is actually more efficient. But if you're going to graph more than one quadratic function, a little bit of analysis goes a long way towards making things much easier. So let's think about what this graph of y equals x squared looks like. So I might start by checking out the symmetry of this graph. So I'll check it out. If I have some point x, y on the graph, then y equals x squared, so I have to regard y equals x squared. This is a true statement. So now I want to replace x with negative x, and so that gives me y equals negative x squared. After some difficult algebra, we have the conclusion negative x squared is x squared. So this equation becomes y equals x squared, and because we can regard this equation as true, this equation also true, and that tells me that negative x, y, is also on the graph. Which means that the graph is symmetric about the y-axis. Likewise, if I replace y with negative y, I get negative y equals x squared. Well, if y is equal to x squared, negative y can't be equal to x squared, and so I know this second statement is false, and so the graph is not symmetric about the x-axis. And finally, if I replace x with negative x and y with negative y, I get another false statement, negative y equals x squared, and so the graph is not symmetric about the origin either. Well, the next thing we should do, let's go ahead and find those x and y intercepts. So the x-intercept is going to be the place on the graph where x is equal to zero. So since y equals x squared, then the y-coordinate is going to be y equals zero squared, and so my y-intercept is going to be zero, zero. The x-intercept is going to be the place on the graph where y is equal to zero. So since y equals x squared, then the x-coordinates are going to be the solutions to the equation zero equals x squared, and I'll solve that. Fantastically difficult equation, x equals zero, and so the x-intercept is zero, zero, which also happens to be the y-intercept. This graph goes through the origin. Now I can plot a couple of points. So first I'll plot the x-intercept, which is also the y-intercept, so that's going to be there. If x equals one, y equals one squared, so again the graph is y equals x squared, the point one one is on the graph, up there someplace. By symmetry, because I know the graph is symmetric about the y-axis, I know that the point negative one one is also on the graph. I have another point that's on the opposite. It's on the left side of the y-axis. So I'll plot that point. Let's get at least another, let's get a few more points. If x equals two, y equals two squared, and that's equal to four. So I know the point two four is also on the graph. And again, because of the symmetry, because the graph is symmetric about the y-axis, I also know the point negative two four is also located on the graph. And so that's enough to give us a rough sketch of the graph, so I'll connect the dots and smooth things out. My graph looks something like that. Well, what about any other quadratic? Well, let's take a look at the graph of y equals negative two, x minus three squared plus four. So I should view these as a transformation of the graph of f of x equals x squared. So I'll start with that graph because I know what that looks like. And I'll apply a sequence of transformations to produce the graph that we want. So order of operation says do stuff inside parentheses first. So the first thing I want to do is I want to find f of x minus three. So that's going to be x minus three squared. And this is a transformation of the graph of y equals f of x, x minus three is going to be shifting the graph to the right by three units. So I'll slide things over a little bit. Let's see. Order of operation says next thing I do is take care of the negative. So I'll take a look at, sorry, I'll take care of the two. And so my graph of two f of x minus three, that's two x minus three squared, is going to be the graph of y equals f of x minus three, this, and I'm going to stretch it out vertically by a factor of two. So I'm going to stretch it out that way. I'm going to get something like that. Now I do have that negative to take care of. So I'll go negative two f of x minus three. That's negative two x minus three squared. I'm getting close to what I want. And that's going to be the graph of two f of x minus three. That negative reflects me across the x-axis. So I'll be reflected across the x-axis. And then finally add four. The last thing we do is we add four. And so negative two f of x minus three plus four, that's this, which is exactly what we're looking for, that adding four shifts my graph vertically by four units. And I slide up to there and there's my final graph.