 We've recently learned about graph transformations, that is ways we can change the formula of a function so it changes its graph and vice versa. And sometimes students ask me the question, why don't things act so weird in the horizontal zone? Do, do, do, do, do, do, do. Well, honestly, I'll have to be honest with you all. I'm an imposter. I haven't been one who watches the Twilight Zone. That was sort of before my time. But actually, as I grew up, there was a horror sci-fi show that I did watch a lot, they called the Outer Limits. And one thing I learned about the Outer Limits is that no matter what you think happened in the episode, it looks like the problem was resolved by the end. It turns out it always ends the same way that the problem didn't go away, that you are still stuck in the same sci-fi horror situation that you were in the entire episode. The reason I bring this up is that students are asking me, why is the horizontal zone so weird? I have a revelation for you. We live in the horizontal zone. That is, it acts weird everywhere. We just don't notice it. What I mean is the following. When we learned about graph transformations, we learned a couple of different things. How do you shift the graph left, right, up, and down? How do you stretch the graph vertically, horizontally? How do you reflect it horizontally, vertically? We learned all the addition techniques, right? And it basically comes down to two categories. You have your scaling category and your shifting. We did kind of differentiate between reflecting, reflecting just means you're scaling by a negative. So there's really just the two categories. And we saw that when you wanted to shift things to the left or right, you would replace the x with an x minus h or an x plus h. But then, like when you have your function f of x, like y equals f of x, to do a horizontal shift, you would replace y with f of x minus h. That's how you shifted things to the right. But if you wanna do a vertical shift, what you did is you would take y equals f of x plus k, right? So why is it adding k and subtracting h in this different situation? Well, let's now enter the horizontal zone here, right? If I actually were to take this equation right here and replace the y with y minus k, notice you're gonna get y minus k equals f of x. But since it's a function, we're often insistent on solving for y. If you were to solve for y on both sides, you'd add k to both sides, and you get up where you get right here, right? So the only reason that y looks like it's acting different from x is actually the insistence that we have on setting y equal to whatever else there is. If we don't require that y equals in our function formula, then it turns out that we can actually get away with the same type of stuff. So backing this up a little bit, if we don't require that we have to get y equals whatever, then a horizontal shift would look like replace x with x minus h, and a vertical shift would look like y minus k equals f of x. That is you replace the y with a y minus k, and that makes the graph go up. Replacing y with y plus k makes the y go down. Same thing with vertical shifting and scaling and such, I mean, if you were to do a horizontal stretch, that means take the x corner and replace it with x over a. If you wanna do a vertical stretch, that means replace the y corner with y over b, that equals f of x, but if you insist on solving for y, what's gonna happen is you're gonna get y equals b times f of x, right? Something like that. And so the only reason they look like they're acting differently is because we insist on moving the transformation from the left side of the equation to the right side of the equation. And as you switch from the left side to the right side, you're gonna switch signs on your operations, multiplication comes to division, subtraction comes addition, all of that type of stuff. So what if we were to, one reason I wanna bring this up, right, is that we can do graph transformations to any graph, not just a function graph, the fact that the graph passes the vertical line test has really no bearing on whether we can do these transformations or not. And so this right here provides us a generalized notion of graph transformations. How do we transform a graph defined by an equation involving x and y's? Well, to shift it, you replace the variable with that variable minus a number. If you wanna stretch it, you replace a variable with, that same variable divided by a number. And so let's use an example of a function that's not gonna necessarily be something we're familiar with. We're gonna take the function x to the y equals two. That is our variable x is raised to the y power and that equals two. So we're gonna switch over to Desmos right here and show you what this graph looks like. It turns out this is a function. It is, it does pass the vertical line test, but solving for y is not a simple task here. So we're gonna leave it as it is. If we want to start transforming this graph, we can do the following thing. Notice that the point two comma one is on this graph. Two to the first is equal to two. So if we wanna do some type of transformations, let's say we wanna do a shift. Let's do x minus h and add the little slider right here and to compensate shifting by h, we have to add h to the coordinate there. If we start making h get bigger, this makes the graph go to the right. If we make h get smaller, this makes the graph go to the left. And if we bring it back to zero, it's as if no shift happened whatsoever. If I wanna do a vertical shift, I replace y with my minus k. I don't add k, I replace y with y minus k. Make sure I add k right here as well. And so now as I make k get bigger, it makes the whole graph go up. If I make k get smaller, it makes the whole graph go down. And if I set k equal to zero, it's as if no vertical shift happens whatsoever. What about scaling some kind if I wanna stretch it? If I wanna stretch it, what I'm gonna do is I'm gonna divide everything by a. So we add this factor of a. In the formula, we divide by a, but in the point, it has the effect of timesy by a. So as a gets bigger, your horizontally stretching the graph knows how it's stretching away from the y-axis. If I make the a value get smaller, and we're now compressing it, squishing it, squishing it, squishing it. If I take a negative a value, it reflects it on the other side. And as the absolute value increases, it stretches, stretches, stretches, stretches. If I go back to one, it's as if no horizontal stretch happened whatsoever. I can do the same thing for the y-coordinate. If I divide the y-coordinate by some value b, which we'll add b into this game here, we're gonna just, that has the effect that you're timesing the coordinates by b. So as b gets bigger, you're vertically stretching the graph. It's getting taller, taller, taller, taller, taller, taller. If you make b get smaller, it's getting compressed vertically, vertically, vertically, oh, it's squish, squish, squish, squish. And then when you switch to a negative, it's gonna cause a vertical reflection, reflection across the x-axis. And as the absolute value increases, you see that it gets stretched, stretched, stretched, stretched, sitting it back to one. We see that we have no transformation whatsoever. And so as we do these different transformations, they have these effects on the graph. If I start playing a movie right here, we stretch it. We're gonna stretch it horizontally now. You get some shifting going on. You get some shifting going on, right? You see all these different transformations of the same function. Oh, my point seems to be going crazy. We'll kick that one out for a moment. Oh, turn off the label here. So we start doing all these stretches all at once. We see all these different possibilities. And all of these transformations are occurred on a function that doesn't have the form y equals, whatever, right? We can also do this for any function we want. We can do this for any graph we want. It doesn't have to be this graph x to the y equals two. We could do these same things for the graph x squared plus y squared equals one, which gives us the unit circle. If we start doing these transformations to a unit circle, what's gonna happen if we start stretching it and shifting it? Well, that's for you to answer, isn't it?