College Algebra: Reflections

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Okay, now sometimes we want to take a graph of a function and look at various reflections. We were talking about we have this notion of being symmetric with respect to the x-axis or the y-axis or the origin, but what about something that actually has the feature that we have occur, and we want to take a look at what it would look like if we reflected it? For example, take a look at this curve right here. Suppose that's some function. I don't know what it is--let's call it f(x). But suppose I want to write down the equation of the function whose graph looks just like this? The only difference is I basically just reflect the entire curve right along the x-axis. So imagine this thing being able to pivot, and I want to take the whole thing and reflect it so it would look like this. I'm just going to reflect it right over the x-axis and it'll look like that. It's the exact same function; the only thing is I just reflected it over the x-axis. I'll do it again. Now, how did I do that? Well, actually, we could think about that and see how to actually reflect a function over the x-axis. So think about it, what do I want to do? I want to take that value, which is a y value, but I want to make it negative that value, right? I want to make that value negative. I want to make this value now positive. I want to take all these positive values and make them all negative values. I want to take all these negative values and make them all positive values. The exact same numerical value, but I want to switch the sign. In fact, I want to switch the sign everywhere. So, in fact, what I would do if I want to actually reflect a function, f(x), over to the other side, flip it over with respect to the x-axis, all I would do is consider the new function -f(x). Because think about it, all that -f(x) does is it takes the function f, but it puts a minus sign in front of all the values. So all these positive values will now be negative values. All these negative values will all be positive values. So now I'll be positive, and then I'll be negative, and then I'll be positive, and then I'll be negative. It flips it.
So just by creating a new function, the new function being called -f(x), that new function is exactly the same as the old function except it's been flipped over the x-axis. So that's how you flip a function. If someone says here's a function like x^2, and I say to you I want you to flip it over the x-axis, all I do is say, okay, put a minus sign in front of it and say -x^2, and that's it. Now it's a sad face parabola. But if I look at the negative of that, that's now being - -x^2, that's +x^2 and I'm back to here. So to go back and forth, just stick a negative sign in front of the entire function--not just the first term--in front of the entire function, -f(x).
Okay, what if you wanted to now flip with respect to the y-axis? Let's go back to the original function I was looking at. Suppose I want to flip this picture with respect to the y-axis. So take all these points here and move them to the other side, and take this point here and move it to this side. What would it look like visually? I'll do it for you right now. I'm just going to pivot right along here--pivot, pivot, pivot--here we go--pivot. It looks like that. The exact same function, but now I've just flipped it over the y-axis. Let me do that again. If I pivot, pivot, pivot, pivot, it looks like that.
So how do you flip the points from one side to the other? What do I want to do? What I want to do is I want to graph on this side all the values that I used to graph over here. Those are the negative x values. And then the things I graphed here, I now want to graph here, so I want to switch the role of x's and negative x's. So to actually flip over the y-axis, what I consider is the new function which looks like f(-x), because I take every value of x, like this value right here--look at this one--and instead of graphing that, I'm going to graph f(-x). That means I go over -x. See that value? There it is. So over here I plot that point. Well, that would require me to flip this over. In fact, if I were to disappear for a second you'd see this really clearly. So this point right here... Here's x. I go to -x, that's here, and that's the point I'm going to plot, so I'm going to plot that point. Do you see how that now flipped over to here? You see that value of x now is way up here, and it came from the -x. So just by looking at the function, f(-x), that's actually going to allow me to flip right over the y-axis.
So if someone gives you a function... In fact, let's do a simple little example here. Let's do the parabola again. With the parabola, if I take a look at this and say to you, "I want you to flip this with respect to the y-axis," would that look any different? Absolutely not. It should be the same. And what happens if in the function f(x) = x^2, you replace x by -x? Well, -x^2 is still just x^2, so in fact, there there's no change. Flipping doesn't do anything. But if the parabola were way over here, flipping would have a dramatic impact. So if someone gives you a function and they say, "Gee, I wish I knew the related function, the different function that would be a flip over the x-axis," just think to yourself what you want to do. You want to take all the y values that used to be positive and make them all negative. So I look at -f(x), it's the function that flips over the x-axis. If I want to take a look at a related function which is a flip over the y-axis, all I do is flip the roles of x and -x and I look at the new function, f(-x).
Okay, I hope you get a sense of these reflection things, but it really does require sometimes to reflect on them.
Relations and Functions
Manipulating Graphs- Symmetry and Reflections
Reflections Page [1 of 1]