Hey, everyone and welcome back. So up to this point, we've been talking about transformations of functions. Now, in the previous couple of videos, we took a look at reflections and shifts. Now, in this video, we're going to see how we can graph shifted and reflected functions. Now, it's very common throughout this course that you're going to see a combination of transformations to a single function. And this process can oftentimes be a bit tricky because we're not really used to seeing multiple transformations at once. But in this video, we're going to take a look at some scenarios and examples and I think that you'll find this process is actually pretty straightforward. So without further ado let's get right into this. Now, we've already taken a look at the reflection transformation where you can imagine folding your graph in some kind of way. We've also taken a look at the shift where you imagine taking your graph and literally just moving it to another location. So we started here at 00 and moved to 32. This would be an example of a shift. Now, what you can do is take both the reflection and the shifts and you can combine them into a single transformation or I should say really a combination of transformations to a single function. And what this would look like is literally just the two transformations combined. So notice how we have our graph here, which started by pointing up and then it got reflected over the X axis. So it was pointing down, then this graph got shifted to a new location specifically the location that we had over here. So this would be an example of a combination of multiple transformations to a single function. And the way that the notation is going to change is actually pretty straightforward as well because notice for the reflection, our function negative when we reflected over the X axis and our shift to 32 was shown in the function where we had our horizontal shift of three and our vertical shift of two. And notice for the combined reflection and shift, we had this negative sign which showed up for the reflection. But then we had this three and this two which showed up for the horizontal and vertical transformations respectively as well when we did the shift. So notice how the function notation and the graph is actually pretty straightforward when you combined multiple transformations. But let's actually see if we can do an of this ourselves. So this example says if G of X is a transformation of the function F of X is the absolute value of X write the equation for G of X. Now notice on this graph, we have the original function F of X and we also have the transformation G of X. So based on this graph, we're trying to figure out how G of X is going to change F of X or basically what the final equation for G of X is going to look like and to solve this. Well, to figure out X, we need to figure out how F of X has changed. So we know that G of X is a transformation of our original function F of X. And let's see what we notice about this graph. Well, notice that this graph was initially pointing up and now it's pointing down. This is an example where we reflect over the X axis. So because of this, whenever we reflect over the X axis, our function becomes negative. So what that means is that in the example, we have down here, our F of X is going to become negative F of X when we do this transformation. But this is not the only thing that happened to this graph because notice this graph has also been shifted. So we were originally going to be at this position after the reflection, but now this graph got shifted two units to the left. So we started here at 00, the origin and we finished centered over here at negative 20. Now whenever we have a shift transformation, you're going to have that your function becomes F of X minus the horizontal shift plus the vertical shift. But we don't have a vertical shift of any kind. So we only have to deal with the horizontal shift. And in this example, the horizontal shift which I'll call H is negative two. So that means we're going to have our function negative F of X minus negative two because that's our shift transformation. But notice that these negative signs will just cancel. So all we're going to end up having is negative F of X plus two. And by the way, recall that when you see this plus sign, that means that you shift to the left. And whenever you see this minus sign, that means you shift to the right. So just something to recall from our video on shifts. So this is what the transformation is going to ultimately look like. So if I were to put this all into our transform function G of XG of X is going to be the negative absolute value of X plus two because notice how we had X plus two as the inside function and we made our entire function negative to match all the transformations that happened on the graph over here. So our transform function is going to look like this. And this right here is the solution to the problem. So that is how you can deal with multiple transformations on a single function. Hope you found this video helpful. Thanks for watching.