 In this video, we're going to practice some more of our graph transformations that we've been learning about in this lecture here. And all of these pictures are going to be based upon the graph f of x equals the square root of x. If you're not familiar with what that picture looks like, let's switch over to Desmos for a second. With this function, we can switch the f of x to whatever we want. We could graph the function as x to the one half power, but a little bit more natural. We're actually going to put in the square root here, which if you ever get typed something in Desmos, you can actually use this keypad at the bottom to try to insert some mathematical symbols like the square root symbol. But there's a lot of shortcuts you can use. Like for the square root, we use type in sqrt. It'll automatically switch to the square root symbol. This isn't just for Desmos, but many mathematical softwares will use sqrt as the shortcut for the square root. Insert the x right there. I get into this keypad. You then see our graph right here of the square root. It's an upward-increasing function towards infinity, although it does grow very slowly. Switching back to our slides over here, we want to graph the function f of x equals the square root of x. But in this situation, we're going to do the same three transformations, but in a different order. For the first graph, what we're going to do is we're going to first shift the graph left by two. You take the square root graph and you shift it to the left by two. You're going to get this graph here in yellow marked by step one. What this does to the formula, we start off with y equals the square root of x. Shifted left by two, you're going to replace the x with x plus two and you get this picture right here. This is the square root that's been shifted left by two. The next transformation, we're going to shift things up by three. Graphically, to go from step one to step two, you're going to shift things up by three. We're going to go from this point, we're going to go one, two, three points up, so shift up by three. What that does graphically is we're going to add three to the function. We take the square root of x plus two, we add a three to it, and that has the effect of shifting everything up by three. The last transformation we're going to do is we're going to reflect across the y-axis here. Graphically, we're going to take our picture, which it's in stage two right here. We're going to reflect it across the y-axis and then we get the final picture, graph three right here. This is the final graph we're looking for. How do you reflect across the y-axis? We're going to, like we saw on our table before, to reflect across the y-axis. You take the x and you replace it with a negative x. The function, the final form, after all three of these transformations, you're going to get the square root of negative x plus two plus three. Now the plus two is inside the square root, the plus three is outside the square root, thus they affect the graph differently. Then this solid yellow graph right here is the graph of that function. If we do those three transformations. On the other hand, what if we switch the order of these operations? We reflect first, then shift up, then shift left. How does that affect things? Well, if you want to begin, we're going to start off with the square root of x. That part's the exact same. If we then reflect across the y-axis, what that means, looking at our table, we're going to take x and replace it with a negative x. That reflects across the y-axis. Taking the standard square root, if we reflect it across the y-axis, you get this graph right here. Like we said, that replaces the x with a negative x. Then we're going to shift things up by three. You're just going to add three to the function you had before, which gives us this picture right here. Then we're going to go from step one to step two, so you just move everything up by three. Starting at this point, you're going to go one, two, three above. That gives us picture two. Then finally, the third transformation is if you shift everything left by two. What this means is you're going to take the x in the formula. Whenever you do a horizontal transformation, because you're in the horizontal zone, you're going to replace each and every x with the expression x plus two to shift it to the left. You're going to replace the x with an x plus two. The fact they have a negative sign actually distributes across here. The final formula is going to look like the square root of negative x minus two plus three. Then step three right here, if you take this picture and you shift it two to the left, you're going to get the picture here at stage three. You see that these pictures are not the same thing. If you compare this graph with this one, you'll see that both of them have been reflected so that the square root is pointing to the left and not pointing to the right. Both of them have been shifted up by three, but one of them looks like it got shifted to the right by two and the other one looks like it got shifted to the left by two. The reason why this happened here is that if you shift things to the left and then you reflect, look at yourself in a mirror sometime. If you take two steps to the left, for the perspective of your mirror image, they just took two steps to the right. Changing the order of operations does affect the outcome on the graph things in this way. Now, why is it a shift up in both situations? Well, it turns out that a horizontal reflection reflecting across the y-axis is horizontal reflection because you're going to replace the x with a negative x inside of the horizontal zone. Thus, horizontal transformations don't affect vertical transformations and vertical transformations don't affect horizontal transformations. They are independent of each other. As such, shifting up wasn't affected by the reflection whatsoever. If we reflected across the x-axis, which is a vertical reflection, shifting up then reflecting across the x-axis would have the effect of shifting down. These do affect one another. Horizontal transformations can affect other horizontal transformations and vertical transformations can affect other vertical transformations. And so, attention to the order we go in. This one can be a very dangerous thing right here because when you look at the second graph right here, you may be very tempted to say something like, oh yeah, this got reflected and then it got shifted to the right and got shifted up by three. But that's not what this formula does. That's what this one does right here. And so, to kind of help us explain and keep track of these things, I'm going to tell you to follow the following convention. To avoid these complications, always following order. First, you deal with reflections. Then, you deal with compressions and stretches. And finally, you then deal with shifts of some kind. And honestly, a student gave me the following mnemonic device to try to help me out and help other students remember this process. When I teach this class sometimes in the spring semester, we start in January, so people just have New Year's resolutions on mind. So think of this as a New Year's resolution. You look at yourself on the scale, it's like, I could probably lose 20 or 30 pounds for Christmas and Thanksgiving, right? So you often think that you reflect upon your health, right? I could probably eat better. I should exercise more. You have this type of reflection, so you reflect upon your life and your health, right? And then after that, you're like, okay, I decided to exercise. So before you start exercising, what do you have to do? You have to stretch. Stretch your muscles so you don't get a cramp or something like that. And then after you've reflected on the need to exercise, then you've stretched, only after you've done those two things, now are we welcome to start exercising, we can move around. And so if you think about that pneumonic device there, reflect, stretch, then move, you'll always get these in the correct order. Now coming...