Welcome back, everyone. In the last video, we talked about transformations of functions. We discussed how there are three types of transformations you're going to see throughout this course. Now, for this video, we're specifically going to focus on the reflection transformation and with reflections, they can be kind of tricky because there are multiple different types of reflections that you're going to see and multiple different ways that your graph is going to fold. But in this video, we're going to be covering all those different situations. So hopefully this topic won't seem as tricky and will seem a lot more clear. So let's get right into this. A reflection transformation is a situation where the function appears to be folded. And we've discussed that a bit in the previous video, how it's like folding your graph in some way. But it's possible for the function to be folded over the X axis or the Y axis and how the function is folded is important for both the graph and the equation and what they're going to end up looking like. So let's say we have a fold or reflection over the X axis. When doing this, you can imagine taking your graph and increasing it at the x axis, like it's a piece of paper, you're then going to take it and fold it in this fashion and that's how your graph is going to form. So if we were to take this graph and fold it, it would end up looking like this. So this would be a reflection over the x axis. But now let's say we have a reflection over the Y axis. When reflecting over the Y axis, you're going to treat your graph at like a sheet of paper just like before. But now you're going to crease the graph at the Y axis. So the fold is going to be more like this kind of like opening or closing a book. So when doing this, your function is going to go from this position to that position, notice how it reflected over the Y because we folded it this way. So that's a reflection over the Y axis. Now, it's important to also notice how the function is going to change depending on how it reflects. So if we look at a reflection over the X axis, notice that we originally started at the point negative 11. And then after the reflection, we finished at the point negative one negative one. So notice how the X value stayed the same, but the Y values changed signs, we went from positive to negative one. So because of this, we could say that for our graph, when we reflect over the X axis, it's the Y values that change signs. So we basically go from positive Y to negative Y when reflecting over the X axis. So we could also say that our function goes from F of X to negative F of X because it's the outputs that become negative. But now let's take a look at a reflection over the Y axis. Notice how, when we reflected over the Y axis, we went from a point of negative 11 which we have on this left side and we reflected over to positive 11. So notice that our Y value stayed the same but our X values change signs. So because of this, we could say that we went from X to negative X. So it's actually going to be the inside of our function because we have X on the inside that's going to change signs. So this is how the function is going to change. But let's actually see if we can take this concept and apply it to an example. So in this example, we're given the function F of X is equal to X plus two. And we're told to let H of X be the function F of X with a reflection over the X axis. We're asked to sketch the graph of H of X and determine which of the functions below matches the new function H of X. So what I'm first going to do is see if I can sketch the reflection H of X on this graph over here. So this is the original function that we see F of X sketched for us. And I see in our example that it's given we're gonna have a reflection over the X axis. You can imagine taking this graph and increasing it at the X, you're then going to fold it and whatever gets folded down or up is going to be the new function. So if I take this portion of the graph and I fold it down, we're going to go from here all the way down through here and notice for this graph that we started at A Y value of positive two and finished at A Y value of negative two. That's because whenever you reflect over the X axis, it's the Y value that changes signs. Now I also need to fold this bottom portion of the graph up because we're doing a reflection over the X axis. So this bottom portion of the graph is going to extend up here like this after we reflect it. So this is going to be the reflected function H of X from our original function F of X. But we're also asked to figure out what the new equipment is going to look like based on the four options below. Well, our function H of X is simply F of X with a reflection over the X axis and we're told that when we reflect over the X axis, we need to make our entire function negative. So that means down here, our function is going to be negative F of X. So to draw this out, we're given the F of X is X plus two. So this whole thing is going to be negative X plus two. And if I go ahead and distribute this negative sign into the parentheses, we're going to end up with negative X minus two. So this is the equation for our reflected function H of X. And if I look at the four options below this matches with option C. So after reflecting our function over the X axis, our graph is going to look like this and our new equation is going to look like that. So this is the idea of reflection transformations when dealing with functions. Thanks for watching and let me know if you have any questions.