>> We know that graphs are related to equations and we know that if we make a change in an equation it'll result in a change in the graph. Well it's also the case that we can imagine graphs of certain kinds of equations. Now we already know that with linear equations if we think about a linear equation written in a certain form, that is in the slope intercept form, then we can imagine its behavior. We can imagine where it goes through the coordinate plane and how steep it is and that kind of thing. Well we're going to extend that idea we're going to consider some more complicated equations and more complicated graphs here. And it'll turn out that we'll be able to easily imagine these graphs and we'll easily be able to imagine the changes in graphs as we make subtle changes to those equations. All right let's begin with a rather degenerate quadratic equations. F of X is X squared. When we graph this we write it in this form, Y equals X squared, here's the table of values associated with that equation. If I plot the points and construct the graph, I get this parabolic curve. Okay now let's begin there, let's make a little change. We're going to take the F of X is X squared, the X squared business and subtract 3. We would write this then as Y equals X squared minus 3. Here's a table of values associated with that equation. Now just take a look at the table of values, it'll give us a pretty good idea of what's going to happen to the graph. Now this column of X values is identical to this column of X values. But look at the Y values here. 0149. Over here, we have negative 3, negative 21, and 6. Now if you compare these two you'll notice that for each Y value these are three smaller than those were. Now think about Y as a vertical distance. The vertical distances here are just three smaller across the board than all of the vertical distances over there. So it'll be no wonder that the graph associated with this equation is just three down from the previous graph. Now we can think of it conceptually like this as well that again, Y represents vertical distances. So the Y equals X squared that we had over there, you see, this is just a set of vertical distances, you see, given by or described by this. And now this Y is those other, those vertical distances we had over here, minus 3. So subtract 3 for all of those and the graph falls 3 on the coordinate plane. All right, I'm kind of over explaining the situation but it's a very important idea for a particular shift in the graph. Now if I plot these points, let's see, 0 goes to negative 3, and then plus or minus 1 goes to negative 2 and then plus or minus 2 goes to 1 and then plus or minus 3 goes to 6. All right, so this graph sort of predictably looks like this. It's a shift downward. Now here's what I want you to understand conceptually, that we have a prevailing operation here squaring and with our other families of equations and their graphs we're going to have a prevailing operation, we're going to be talking about absolute value and cubing and square rooting. Okay. So for each degenerate kind of equation and graph we have a prevailing operation. And what we should notice here is that we're adding or subtracting, in this case subtracting, after we perform the prevailing operation, that is we square X and then we subtract 3. So the graph falls 3. If we were adding after the prevailing operation of squaring then the graph would rise on the coordinate plane for the same reasons that we talked about here. Now let's talk about the notion of adding or subtracting before performing the prevailing operation. That is suppose we consider an age function, which is X minus 3 then square, you see, so we're subtracting 3 before performing the squaring. Okay. We would use this form to make a table of values, this turns out to be the table, I'm not using the plus minus here because we don't have an X distinct, a little X squared giving us the same value for plus or minus values that would be squared. Okay. Anyway let's plot these points and just see what happens. 0 goes to 9, 1 to 4, 2 to 1, 3 to 0, 4 to 1, 5 to 4, 6 to 9, graph looks like this. Now notice the shift and all we're talking about so far is shifts that are up down left and right.