 In this video, I'm going to talk about reflecting functions. This is going to be a video where I just go over a couple of examples of how to reflect a function either across the x-axis or the y-axis. OK, so let's go ahead and get started. Let g of x be the indicated transformation of f of x. OK, and then write the rule for g of x. Now remember when it says write the rule for g of x, basically we're just writing the equation for g of x. So I did try not to get that too confused. OK, so we're taking this f of x function, negative 2 thirds x minus 1, and we are going to reflect it across the x-axis. OK, now the first thing I'm going to do is I'm going to show you using the notation, and then I'm going to show you using the graph. So there's kind of two ways that you can do this. But since they ask us to write the rule for g of x, what we need to do is write the equation, write the function itself. So that's the end goal, is to write the function. But I'm going to show you two different ways to do this. OK, so to take this function and reflect it across the x-axis, so here we go. We have to take f of x, and I want to change it. So now what I'm doing is I'm writing the mapping notation, the arrow notation I guess is another way to say it. So we're going to take f of x, and we're going to change it. If we want to reflect across the x-axis, basically everything is going to change. Our slope is going to change. Our y-intercept is going to change. Everything is going to change. So what I'm going to do is I'm going to take a negative times my entire function, a negative times the entire function. That's what happens when you reflect something across the x-axis. OK, so that's what the notation looks like, so let's look at the actual functions. So my g of x function, the new one, is simply just going to be the old one. It's going to be the old function, except I'm going to multiply everything times a negative. Now notice what we do here. This g of x, this right here, you can see from the previous videos that I've done about reflection, is that it always matches up with what we have right here. Basically this right part after the arrow, that's basically what we're doing. We're going to take the function itself and multiply by a negative. Now you've got to be careful here. You've got to use parentheses. Negative 2-thirds x minus 1, the entire function is going to be multiplied by a negative. Notice we took the f of x, replaced it with the entire function. Remember what f of x is up here. And then after we take this negative and distribute it, the new g of x function, the new rule, the new rule is going to be a positive 2-thirds x plus 1, positive 2-thirds x plus 1. So there's my answer. I can actually stop right there. If you understand how to do the notation, you can pretty much stop right there. That's the new rule for g of x after flipping f of x across the x-axis. Now I'm going to actually do this a little bit differently. I'm going to do this one more time. But I'm going to show you using a graph. I'm going to show you using this graph over here because there's basically two types of people. There's the people that understand the notation, which is right here. And then there's the people who need a picture. I'm kind of one of those people who need a picture. So I'm actually going to graph everything to see what it looks like. The first thing I'm going to do is I'm going to graph f of x. So I got negative 1 for a y-intercept. And I have a slope of negative 2-thirds. So 1, 2, 1, 2, 3. 1, 2, 1, 2, 3. There we go. And then down here, 1, 2, 1, 2, 3. I think that's a good enough point right there. Draw my nice straight line. And this is my f function. There we go. All right, now when I reflect this across the x-axis, the fancy way of saying this is that all the points are going to be equidistant from the line of reflection. What that basically means is all the points that I have on here. So this 1, 2, 3, 4 points, they're going to be reflected across the x-axis. Remember, your x-axis is here. Your y-axis is there. It's going to be reflected across the x-axis. And all these points are going to be equidistant from that line of reflection. So I'm going to change my color here. This point is 1, 2, 3 away. It's 3 away. And so on this other side, it's going to be 1, 2, 3, 1, 2, 3. It's going to be about right there. And then this point is 1 away. So it's going to be 1 away on the other side. This one is 1 away. So it's going to be 1 away on the other side. And this one is 1, 2, 3 on the other side, or on this side, so it's going to be 1, 2, 3 on this other side. And here is what my g function looks like. Now you notice now, if you're doing this just with the graphing, you wouldn't have all this notation over here. But notice, we have a y-intercept of positive 1, positive 1 for the y-intercept. And then our slope from point to point, rise 2, run 1, 2, 3. Rise 2, run 1, 2, 3. Because the slope is a positive 2 thirds. Slope is a positive 2 thirds. So there's two ways you can look at it. You can either do the notation and figure out what the new rule is going to be, or you can graph everything and figure out what's going to be. I'm one of those people who likes to graph. I like to visually see what's going on. So this is over here is what I prefer. But again, there's those of us out here who understand the notation, and so we can do that. All right, one more example, and then I'll be done with this, one more example. Same direction, so let g of x be the indicated transformation of f of x, and then write the rule for g of x. Now notice it's the exact same function. But notice what I'm going to do here. I'm going to reflect across the y-axis. So basically what I'm doing is I'm showing you reflecting this equation across the y-axis and then across the x-axis. And I'm actually going to flip-flop this. I'm actually going to graph this first and show you what it looks like, and then I'm going to use the notation afterwards. It's a little bit different order. So I'm going to graph the first one. I'm going to graph f of x first. Do this kind of quickly, so negative 1, then 1, 2, 1, 2, 3. So I went the wrong way. It's got to be a negative slope, not a positive slope. So 1, 2, 1, 2, 3, 1, 2, 1, 2, 3, 1, 2, 1, 2, 3. That little mistake that I made with the slope, that's one thing you always got to check. Do I have a positive slope or a negative slope? Because I did my slope correctly, I just went in the wrong direction. Make sure that you always graph. In this case, I was graphing a negative slope, so I got to go downhill. OK, and then now the other one is my g of x. I use a different color for this. It's going to be reflected across the y-axis. So the y-axis is here, x-axis is here, so everything's going to be reflected across this line. Now again, same as last time, points are going to be equidistant from your line of reflection. So basically, like this point here, is 1, 2, 3 away, so it's going to be 3 away on the other side, 1, 2, 3. So it's going to be about right there. This point is going to stay where it's at. It's on the line, so it doesn't really get reflected. It just stays right there. All right, so this one is 1, 2, 3 away, 1, 2, 3 away. And 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, and 6 again. All right, connect all the dots to my nice straight line here, and this is my g function. OK, all right, for now, from that, what we can do is we can figure out what the rule, again, we're finding the rule for g, what the equation for g is. So g of x, the rule for it is going to be, OK, so now it has a y-intercept of negative 1. I'll put that right there on the end. Put that my x right there, and I need the slope. What's a positive slope? I'm going to go up 2 over 1, 2, 3, so 1, 2, 1, 2, 3. So I've got 2 thirds for a slope, 2 thirds for a slope. Now compare that to what our original equation was. It looks like the only thing that changed was our slope. The negative 1 for the y-intercept stayed the same. As you can see right here, the y-intercept stayed the same. But then it looks like the only thing that changed was our slope. Now that actually gives us a little bit of a hint of what we're going to do for the notation. So that's one way to solve it. That's one way to figure out what the rule is. Now the second way is to use your notation to figure it out. So take your old function, and we're going to change it by, now, when you're reflecting across the y-axis, we only take the negative just to the x portion of our equation, just to the x portion of our equation. Now this is a little bit different. Last one, I'm going to flip back to my last slide. My last slide here, a little bit different. If I'm reflecting across the x-axis, it's a negative times the entire function. Going forward again, if I'm reflecting across the y-axis, I take my negative, and I only multiply times the x portion. So my new g of x is going to be rewrite this. My old function, but I'm going to replace the x, replace the x with a negative x. Replace the x with a negative x. So that's what I'm going to do. So I have a negative 2 thirds, and then replace the x with a negative x. Replace the x with a negative x. Now what happens there is that a negative 2 thirds times a negative x, the negatives are going to cancel, and so that leaves me with a positive 2 thirds x. So that's how we can see with the notation how to get the rule for g of x. All right, those are my two examples of going over reflecting functions. Now notice I use the same function, and I just reflected it differently each time, but that will kind of give you a way to compare and contrast the different ways of reflecting across either the x-axis or the y-axis.