 Today we're going to discuss just a few things dealing with transformations. You should have already learned about the basic transformations which I just have summarized here. These are your transformations that affect the x values or horizontal transformations to the graph. So notice all of these transformations occur inside right by the x value of the function. And then these transformations here are all vertical transformations. So these are changes that will happen to the y value. So it will shift up or down or stretch up and down. But notice that these transformations all are outside of the function. And so they affect the y values. What we want to look at today is rather than just seeing one transformation happen at once, we want to see what would happen if you have multiple transformations and how the order that you apply them make a difference. So what we're going to look at first is this first example that you have on your paper. So if you notice this function here that's given, the first thing you always want to look at is you want to identify what the parent function is or what toolbox function you're dealing with. So if you notice here we have a cube root here at the top. So the original function that we would be dealing with would be the cube root of x. So that's the graph that we are going to start with. So take a minute if you need to pause the video and go ahead and look back at your toolbox functions and see if you can just graph that parent function on the graph below. Alright, on your graph you should have a graph that looks somewhat similar to the one that I have graphed below. You do always want to look at a few key points just because then those are the points you can use in order to transform this function. Now that I have the original graphed, now I'm going to look at what different transformations are actually going to happen to the graph that I have. So first I will look at inside by the x. Notice that we just have an x minus one. So we're looking at our horizontal transformations and we just have one here because you just have an x minus one. And you can look back if you need to at what that means but x minus one is going to mean that your graph is going to shift to the right one. So all of our values here are going to shift to the right one. And I will do that in just a second but first I'm going to go through our vertical transformations as well. So horizontally we just have one thing and so it doesn't really matter the order that you apply that in because there's just one so that's what we would start with. Now if we look at our vertical transformations here, the vertical transformations are anything that happens outside of the function itself. So we have the negative two and then we also have the plus one. So vertically we actually have a lot happening on this graph. So what we would do here is you want to remember that vertically what you're going to start with is you're going to start with any multiplication that you have with your function. So any stretching, compressing, or reflections, that's what you want to do first. And then you'll follow that with any shifting that you have. So any vertical shifting is always going to be the last thing that you do. So here we have that negative two. Now the negative two actually does two things to the graph. The two will stretch the graph but then the negative sign will reflect the graph. And it's going to reflect the graph over the x-axis because all of our y values are changing. So those two things can kind of be interchanged. You can reflect and then stretch or stretch and then reflect. It doesn't matter at all because they kind of go together. And then our second thing that we have is we have this plus one at the end and the plus one is going to shift the graph up two. Whoops, sorry, not up two. Up one. Now these are all of our transformations and I always do the horizontal transformations first and then the vertical transformations. Now what I'll try to do on here is I'll try to sketch just one by one what happens so that you can see it all. On the paper I've given you a little bit of space so you can use two different graphs if that's more helpful. And if this gets too messy I can always start a new one as well. But I'm going to start right here. So this is my first thing that I'm going to do. I'm going to shift the graph one to the right. So each dot that I had before, I'm shifting it one to the right. So here I'm going to be off just a little bit on there. But that at least gives me my general shape. And so this blue graph, I'll label it with a one, is what I get just after that first horizontal shift. Now that I've done that, I am going to do my first vertical change. So I'm going to separate it. I'm first going to stretch everything by two. So stretching by two will only change the y values and it's going to multiply all of the y values by two. So where I am right here at zero I'll stay at zero. But this is at two one. It's going to stretch or multiply the y value by two so it will go up to two two. And then down here it really was at nine two. Well that's going to stretch up from two to four. And if I go in reverse it was at zero negative one. It will stretch down to zero negative two. And then over here it was at negative seven, negative two. It stretches down to negative four. So all of those y values I'm multiplying by two and I get that second graph. I'll label that with a two down there. And that was this second thing, stretching everything by two. So that's a vertical stretch. Now I'm going to have to switch colors on us. I'll go to orange for this. But the reflection, all that does is it takes all of these y values and simply makes them the opposite value. So if I start in the middle again I stay at zero because zero of course the opposite is just zero. But when I'm up here at positive two I'm going to jump down and now go to negative two. When I'm here at positive four I'm going to jump down and be at negative four. And then here when I was at negative two I'm going to be at positive two. And when I was at negative four it will now be at positive four. So notice I'm only changing the y values because vertical changes only change the y values. Now I know that's a little bit messy but that is our third change. We're reflecting over the x axis. And now that we have that we have just one thing left. Our final step is going to be this shift to move everything up one. So that's our fourth graph. And I'm going to do it in purple here. And what we have to do is we have to shift this orange graph that we had just shift everything up one. So again I don't like to do every single point I just like to use the key points that I've been using because then I'll get the general shape. My orange graph came in a little bit messy but we can at least get the right idea here. And our final graph will be this purple graph. So I'll label that with a number four. So you'll notice if we had done the vertical transformations in the opposite order we would have gotten a completely different graph. So you have to make sure you pay attention to the order with those. Let's look at one more example in order to continue to practice this idea. Okay, our second example is this equation. Y equals negative one half quantity X plus one cubed minus two. So if you look at this again your first step is always to identify the parent function. And here you'll notice we have a cubed. And so our parent function is the X cubed function. Again, feel free to pause the video if you want to in order to come up with the original graph of your cubic function. But I'm going to graph that right away so that we know what we're referring to. Let's see, does this final point fit on here? I think it just does. Okay, so this is a rough sketch of the original cubic function. Now what we're going to do again is going to look at our different transformations. So I'll start inside. Anything inside by the X is going to be our horizontal transformations. So when we look inside we just have the plus one which is just going to be one change. And in this case that is going to be a shift to the left one. So that's the first thing that we're going to do. And that's the only horizontal change that we have here. So that's all that we have to worry about. Now if you look on the outside of your function we're looking at any vertical changes. So like I said before we always start with any multiplication here. First, whether it just be a negative sign or a number. We have the negative 0.5 and what that is going to do is two things. It's going to compress by one half our function and then it's also because it's negative that is going to reflect over the X axis. Because remember it's a vertical change so reflecting over the X axis is what changes our Y values. Now we still have one more change to look at here. We have this minus two at the way end of our function and subtracting two is a shift. In this case it's going to shift our graph down. So we have four different transformations to do and we can just apply them in this order that we're looking at here. So I will first do the horizontal shift. We're going to shift everything to the left one. So I'll just try to make a few little dots here in order to make that happen. I can't quite see down here at the bottom but I think that that would be our first change so everything shifted to the left one and that was our only horizontal change. Now our vertical change is going to compress everything by one half is our next thing. So compressing by one half means all of our Y values are going to be divided by two. So I'm going to divide all of these values by two just the Y value. Notice they're just staying in line with the X value that they were with. This one's a little hard to see because I made it kind of messy before but see how that's a little squished together vertically? So that's a nice vertical compression in orange. And then we'll go to green. Green we're just going to reflect everything so all of our Y values we're just going to make the opposite. We'll make that a positive four. That's a positive one half. This one's at a negative one half and here is down at a positive four. And we get our green function and then finally we always finish with our shifting up and down. So here we're going to shift the entire graph down to. So each point that we had before we'll shift it down to. This one's a little tricky because it's at a half but we can figure it out and there we go. In blue we have our final graph and always make sure you clearly label what the final graph is on each of yours.