- [Instructor] So you see here, this is a screenshot of the Desmos online graphing calculator. You can use it at desmos.com, and I encourage you to use this after this video, or even while I'm doing this video, but the goal here is to think about reflection of functions. So let's just start with some examples. Let's saying that I had a function, f of x, and it is equal to the square root of x. So that's what it looks like. Fairly reasonable. Now, let's make another function, g of x, and I'll start off by also making that the square root of x. So no surprise there, g of x was graphed right on top of f of x. But what would happen if instead of it just being the square root of x, what would happen if we put a negative out front right over there? What do you think is going to happen when I do that? Well, let's just try it out. When I put the negative, it looks like it flipped it over the x-axis. It looks like it reflected it over the x-axis. Now instead of doing that way, what if we had another function, h of x, and I'll start off by making it identical to f of x. So once again, it's right over there. It traces out f of x. Instead of putting the negative out in front of the radical sign, what if we put it under the radical sign? What if we replaced x with a negative x? What do you think is going to happen there? Well, let's try it out. If we replace it, that shifted it over the y-axis. And then, pause this video, and think about how you shifted over both axes. Well, we could do a, well, I'm running out of letters, maybe I will do a, I don't know, k of x is equal to, so I'm gonna put the negative outside the radical sign, and then, I'm gonna take the square root, and I'm gonna put a negative inside the radical sign. And notice, it flipped it over both. It flipped it over both the x-axis and the y-axis to go over here. Now, why does this happen? Well, let's just start with the g of x. So when you get put the negative out in front, when you negate everything that's in the expression that defines a function, whatever value you would've gotten of the function before, you're now going to get the opposite of it. So when x is zero, we get zero. When x is one, instead of one now, you're taking the negative of it so you're gonna get negative one. When x is four, instead of getting positive two, you're now going to get negative two. When x is equal to nine, instead of getting positive three, you now get negative three. So hopefully, that makes sense why putting a negative out front of an entire expression is going to flip it over, flip its graph over the x-axis. Now what about replacing an x with a negative x? Well, one way to think about it, now is, whenever you inputted one before, that would now be a negative one that you're trying to evaluate the principle root of and we know that the principle root function is not defined for negative one. But when x is equal to negative one, our original function wasn't defined there when x is equal to negative one, but if you take the negative of that, well now you're taking this principle root of one. And so, that's why this is now defined. So, whatever value the function would've taken on at a given value of x, it now takes that value on the corresponding opposite value of x, and on the negative value of that x. And so that's why it flips it over the y-axis. And this is true with many types of functions. We don't have to do this just with a square root function. Let's try another function. Let's say, we tried this for e to the x power. So there you go. We have a very classic exponential there. Now let's say that g of x is equal to negative e to the x. And if what we expect to happen happens, this will flip it over the x-axis. So negative e to the x power and indeed that is what happens. And then, how would we flip it over the y-axis? Well, let's do an h of x. That's going to be equal to e to the, instead of putting an x there, we will put a negative x. Negative x. And there you have it. Notice, it flipped it over the y-axis. Now, both examples that I just did, these are very simple expressions. Let's imagine something that's a little bit more complex. Let's say that f of x, let's give it a nice, higher-degree polynomial, so let's say it's x to the third minus two x squared. That's a nice one and actually let's just add another term here. So plus two x. And I wanna make it, make it minus two x. I wanna see it accentuates some of those curves. All right, so that's a pretty interesting graph. Now, how would I flip it over the x-axis? Well the way that I would do that is I could define a g of x. I could do it two ways. I could say g of x is equal to the negative of f of x and we get that. So that's essentially just taking this entire expression and multiplying it by negative one. And notice, it's multiplying, it's flipping it over the x-axis. Another way we could've done it is instead of that, we could've said the negative of x to the third minus two x squared, and then minus two x, and then we close those parentheses, and we get the same effect. Now, what if we wanted to flip it over the y-axis? Well then instead of putting a negative on the entire expression, what we wanna do is replace our x's with a negative x. So you could do it like this. You could say that that's going to be f of negative x and that has the effect of everywhere you saw an x before you replaced it with a negative x. And notice, it did exactly what we expect. It flipped it over over the y-axis. Now, the other way we could've don't that just to make it clear, that's the same thing as negative x to the third power minus two times negative x squared minus two times negative x. And of course, we could simplify that expression, but notice, it has the exact same idea. And if we wanted to flip it over both the x and y-axis, well we've already flipped it over the y-axis, to flip it over the x-axis, oh whoops, I just deleted it, to flip it over the, I'm having issues here, to flip it over the x-axis as well, we would, oh and it gave me a parentheses already, I would just put a negative out front. So I put a negative out front and there you have it. This flipped it over both the x and y-axis. You can do them in either order and you will get to this green curve. Now, an easier way of writing that would've been just the negative of f of negative x and you would've gotten to that same place. So go to Desmos, play around with it, really good to build this intuition, and really understand why it's happening.