>> In this section we'll be talking about quadratic functions. Quadratic functions are just one branch of an overall family of polynomial functions, some of which we have studied in the past. Let's put the polynomial functions in perspective just for a moment, and then we'll get on with the matter at hand. These are polynomial functions. They are called polynomial functions because the expression used to describe them are polynomials. Now notice the notation over here. It's customary in mathematics for letters at the front end of the alphabet to stand for real numbers; and letters at the back end of the alphabet like X, Y and Z to stand for variables. So here A, B, C, and D stand for real numbers, X is the variable involved. Now the difference among these polynomial functions is in the degree of the polynomial expression. This is a 0 degree expression, 1st degree, 2nd degree, 3rd degree and so on. We have studied 0 degree polynomial functions if you will; their just constant functions, it would be like F of X is equal to 2. And we know all about the graph of Y equals 2, it would be a horizontal line going through Y equals 2. We've also studied 1st degree polynomial functions, linear equations you see; and we know that in this form if we see a polynomial function in this form we can read its slope and Y intercepts so we can describe the behavior of the graph rather easily. Well now we're talking about quadratic functions and that's this rascal right here, 2nd degree functions, and later we'll be talking about 3rd degree polynomial functions and beyond. Well, we've actually seen polynomial functions of the 2nd degree; that is, quadratic functions before and we saw them when we talked about transformations. Now remember at that time, transformations can either be rigid or non-rigid. Rigid transformations are changes in graphs that kind of move left, right, up, down or maybe we have a reflection of one kind of another. Non-rigid transformations are transformations that really change the shape of the graph. It causes the graph to either be sort of compressed or thinner or maybe expanded like this, kind of fatter a little bit in the curves. Well, let's review that just a little bit. We looked at situations where we might have an equation like this and we can think of this as F of X equals this expression or Y equals this expression, it doesn't make any difference. But recall at the time that we were identifying a prevailing operation and in this case it's squaring, and we know that the squaring operation leads to a graph which is a parabolic curve. All right now. Our various transformations arise by virtue of the various numbers that we see here. When we have an addition or subtraction of the variable before the squaring operation, then the transformation, it's rigid, is left or right but in a direction we don't expect. If we see X minus 3, then the graph actually moves plus 3 on the coordinate plane. Now this is the graph of Y equals X squared. When we talk about X minus 3 squared, then this graph moves 3 to the right. Now let's just identify the position of the vertex, the change in the vertex. So the vertex would go from here 1, 2, 3, to the right and the rest of the graph would be like this. All right now what about the 2? Well the 2 means that the graph will be relatively thinner, you see, just a little bit thinner like this because we're sending those vertical distances up the coordinate plan twice as fast as they were before, that's one way to look at it. Now the negative on 2 means we want the opposite of all of these vertical distances, and so we have a reflection then in the X axis. So now we have a parabolic curve that opens downward and is relatively thin, and then minus 1 sends the whole graph down 1 on the coordinate plane. So it kind of looks like this. Okay so that was the idea.