 It'll be useful to talk about transformations in function notation, so let's take a closer look at those. So we can transform the graph of y equals f of x in the same way that we transform the graph of any equation. But it's useful to remember to answer the question in the same language. And so if you start with an equation y equals something, you should answer with an equation y equals something. And if you start with a graph, you should answer with a graph. For example, let's say we want to write the equation of the graph reduced by shifting the graph of y equals f of x to the right two units, then down two units, then stretching vertically by a factor of three. So first we want to shift the graph to the right by two units. That's a horizontal translation. And since we're going to the right by two units, we can find the equation of the new graph by replacing x with x minus two. Next, we're going to shift the graph down by two units. That's a vertical translation. And since we're going down by two units, we can form the equation of the new graph by replacing y with y plus two. And then finally, we're stretching vertically by a factor of three. So that's a vertical stretch. And we can find the equation of the new graph by replacing y with y over three. Now this does give us an equation, but since we're given the original equation in the form y equals f of x, we should solve this equation for y. So solving gives us the equation of our new graph. What if we have a graph? Let's find the transformations necessary to obtain the graph of y equals f of x minus three, and then sketch our graph. So the problem here is identifying the transformations, and the important thing to remember is, it doesn't matter what you write at first, what matters is what you write at last. There's a limited number of transformations that we can do, so we can just go through them and see one that works. So I don't know, let's shift the graph vertically. If we shift the graph of y equals f of x vertically by k units, the new graph will have equation. And since we start with an equation y equals something, we should end with an equation y equals something, and we want the graph of y equals f of x minus three. That doesn't look like this. So it doesn't look like this is what we want. Now if you're a politician, you say that the reason that this didn't work is because of those obstructionist other guys. But if you're a sensible person, or a mathematician, you might try something else. If a vertical shift didn't work, how about a horizontal shift? If we shift the graph y equals f of x horizontally by h units, the new graph will have equation. And that looks exactly like what we want. So we'll shift the graph three units to the right, and that'll obtain the graph of y equals f of x minus three. Or how about something like this? Now there's many things you could do. Again, it doesn't matter what we write down at first. The only thing that matters is what we write down at last. And so here maybe we'll begin by shifting the graph of y equals f of x up by four units. If we do that, the equation of the new graph will be... Now keep in mind that our goal is to get this graph of y equals this thing. And so to make it easier to compare, let's solve each equation for y. So here we can solve this equation to get y equals... And that's not a bad start. We want y equals a whole bunch of stuff plus four, and we have y equals something plus four. So it looks like we're on the right track. Next, let's shift the graph to the right by three units. And this will give us a graph with equation. And again, that looks great. We want an x minus three inside the function. So we have an x minus three inside the function, and that's a good thing to have. Well, let's see. We've done a horizontal and vertical shift. Let's try stretching the graph a bit. So maybe we'll stretch the graph vertically by a factor of two. If we do that, then the graph of the new equation will be... Which we'll solve for y. And that's great. This is exactly what we want. We want y equals two times f of x minus three plus four. And we have y equals two times f of x minus three plus eight. Well, that's not what we want. So we should scrap this whole thing and go back to the beginning. Or we can try to fix it. The thing to recognize here is that the vertical translations are going to add or subtract a constant to the whole thing. So maybe I want to do another vertical shift at the end of this. In this case, I'll try a vertical shift by four units downward. And so I'll get an equation which we can solve for y to get, which is what we want. And since we've wrote down all of the transformations we needed, we can apply them to the graph of our original function to get the graph of the new function. So we'll take our graph and shift it up by four units. Then to the right by three units. Then stretch vertically by a factor of two. Then shift vertically downward by four units to get our final graph. An important question to always ask in mathematics and in life is could this have been done a different way, possibly even a better way? So we might try something different. What if we stretch vertically by a factor of two first, which will give us a graph with equation. Then shift to the right by three units, which gives us. And then shift vertically upward by four units. So if we stretch, shift right. Then shift upwards. We get the same graph we got before. Now we can use either set of transformations, but this one is a little bit easier, and so this suggests that if possible, do reflections and stretches first, then translations. So let's take a look at this. So let's try to do our reflections and stretches first. So if we reflect the graph across the x-axis, the new graph will have equation, which will solve for y, and see that it does have some similarity with what we're going for. If we then reflect across the y-axis, our new graph will have equation, which is starting to look good because we not only have the minus f, but we also have the minus x. If we shift our graph vertically upward by five units, we get a graph whose equation is, and again solving for y gives us something that looks even closer to what we want. Now let's shift the graph to the left by three units to get a graph whose equation is, where we've been careful to use parentheses so we're only replacing x with x plus three, and this gives us the equation that we want y equals five minus f of minus x minus three. Oh, so very close, but not quite. Well, let's be a politician and blame obstructionist for our problems. Or we could be a good human being, or a good mathematician, and try to fix them. And the thing that's worth noticing is that some things worked. So we reflected across the x-axis, the y-axis, and shifted vertically upward, and it was only with that last step that we didn't get what we wanted. So what if we go the other way and shift the graph to the right by three units, again making liberal use of parentheses, and upon simplification we get exactly what we want. And so this allows us to sketch our graph.