 In this video, I'm going to talk about translating and reflecting functions, okay, translating and reflecting functions. I'm just going to go over a couple of examples of how to translate and reflect a function. So this little graph that I have down here, this M-looking shape here, this is a function. If you remember your vertical line test, if I put a vertical line at any point on this function, I only hit it once, okay, so that makes this a function. So what I'm going to do is I'm going to use a table to perform each transformation. So I'm going to have two of them, actually. This first one, I'm going to translate two units up, and another example, I'm actually going to do a reflection. So that's what we're going to do. We're going to translate this two units up, and we're going to use a table to do this. Now this is a simple enough function, you just might be able to translate two units up without any problem, but I'm going to show you how to use a table. So if you get to something a little bit more complicated, a function that's a little bit more complicated, you'll know what a table is and know how to use it. Okay, but first, before I can actually have a table, I need some points to use. So I'm going to come over here to my function and I'll find a couple of points. Here's one point that's here on the arm of my M, which is 1, 2, 3, 4, 1, 2. So this is negative 4, 2, there's one point. Here's a peak, which is going to be negative 2, 0, negative 2, 0. And then right here, I have another valley, okay, that's kind of an important point. So that's going to be just down 2, so that's 0, negative 2 is that point there. This one over here, another peak, is going to be 2, 0. And then over here, I'm going to choose another point, but I'm going to choose something a little bit different from this one over here. I'm going to go down a little bit farther. And this one over here is going to be 5, negative 3, I believe is what it is. 1, 2, 3, 4, 5, 1, 2, 3. Yes it is. Oh, I think I messed up right here. My mistake is going to be a negative 2, negative 4, negative 2. And the third quadrant, both the X and Y coordinates, it's got to be negative. Over here, 5, 3, just checking my points, yep. Okay, now you might ask yourself, why would you need 5 points? Well, you need enough points to make an accurate table. I just chose 5 because the peaks, the valley, and then one arm over here, one arm over there. It kind of depends on what the function looks like. Sometimes you'll need 2 points, sometimes you'll need 5, sometimes you'll need 8. It just kind of depends on what the function looks like. Okay, so I'm going to make my table, I'm going to make my table. This is going to be an X, Y coordinate table, but I'm going to have an extra column over here. This extra column, we're going to figure out what we're going to do because when I translate a function, the X or the Y coordinates are going to change. Now we're going to translate this 2 units up, so if I'm taking everything and moving it up, that's going to affect the Y axis, the Y axis or the Y coordinates. So what we're going to do is we're going to take the Y coordinates and we're going to add 2 to them, and we're going to create a whole new set of points to use for graphing. So what I'm going to do is I'm going to take all my points that I have over here and I'm going to put them in my column. So this is starting on the left side, so negative 4, negative 4, negative 2, and negative 2, 0, and 0, negative 2, and 2, 0, and then also 5, negative 3. So what I'm going to do is I'm going to add 2 to the Y coordinates, so I'm going to take negative 2 plus 2, and now you can see with this table, this is going to be very fast, very, very quick. So I'm going to take negative 2 and add 2 to it to get 0. I'm going to add 0 plus 2 to get 2. I'm going to take negative 2 plus 2 to get 0. I'm going to take 0 plus 2 to get 2, and I'm going to take negative 3 plus 2 to get negative 1. So what that does is this gives me a whole new set of Y coordinates to use. Now my whole new set of Y coordinates, the X's are going to be the same, the X's don't change, but it's the Y coordinates that are now going to change. So my new points, I actually use these X coordinates and then use these new Y coordinates. So my new point is going to be 5, negative 1. So that's one of my new points, so 5 and then negative 1, right here. So I'm going to take the rest of these points, so I have 2, 2, that's another point, and then I have 0, 0, which is right here at the origin, then I have negative 2, 2, so negative 2, 2, right here, and then I also have, last but not least, negative 4, 0. So negative 4, 0, right there, right there. Okay, now what this does, now amongst all the other points that I have here, what this is is what my new function is going to look like after it's been translated. So now what I can do is basically connect the dots. So I'm going to go here for one of my arms, I'm going to go down here for, there's a peak, there's a valley. There's my last peak, and then I'm going to connect the dots down here, and there's my second arm there on the right side. So there, right here, is my new function. Right here is my new function, kind of missed all of my points there. Anyway, that's how you use a table, this is how you use a table to translate a function two units up. First of all, you've got to find some of the points to use, and then you've got to translate them, then you've got to add two to the Y coordinate. Now, it depends on if you're translating up, down, left, or right, sometimes you'll change the Y coordinate, sometimes you'll change the X coordinate, just kind of depends on the problem. Okay, but what you'll do is you'll first find some points, you'll change those points, and then you'll redraw your graph, and this is my function that has been translated two units up. My function that's been translated two units up. Okay, so now let's go into a different example where I'm going to actually translate, and I'm actually going to reflect across the X axis. I'm going to take this exact same function, and I'm going to reflect it across the X axis. I'm going to reflect it across the X axis. Now, of course, the first thing that I need to do, just like last time, is I need to find some points to use. I'm going to find a couple of points to use. I'm going to use the peaks and valleys, just like last time, and then I'm also going to use a point on each one of these arms. I'm going to use the same points I used last time. So I'm going to label out these points again. So we have this is negative 4, negative 2. This point is negative 2, 0. This point is 0, negative 2. This point is 2, 0. And this point over here is going to be 5, negative 3, 5, negative 3. So I'm going to use these same points, same function. I'm just doing something a little bit different. I'm going to reflect across the X axis now. Now, remember, if I do any reflections across the X axis, what that does is that's actually going to take these points and reflect them across. OK, so here's the X axis right here. It's going to reflect them up. It's going to reflect them up in this case. Now, what that does is that doesn't change the X coordinate at all. That changes the Y coordinate. Now, if you remember from last time, if you remember from one of my previous videos, what we need to do is we need to actually multiply the Y coordinates of all these different points by negative 1. So let's see what that looks like if we do a table for that. So I'm going to have my X, Y coordinates, just like last time. I'm going to have a third column over here for any changes that I make. So I'm going to write down my points again. So negative 4, negative 2, negative 2 and 0, 0 and then also 5, negative 3. So what we're going to do is we're going to take the Y coordinate and we're going to multiply by negative 1. Take the Y coordinate and multiply by negative 1. You could also think of it as just simply changing the sign on the Y coordinate. So we're going to take negative 1 times, OK, I'm going to use a little parenthesis here, times negative 2. To get a positive 2, we're going to take negative 1 times 0 to get, well, 0. We're going to take negative 2 times 0, or excuse me, not negative 2 times 0. Let me go back again. Got a little off track with my numbers there. One more time. So I'm going to take negative 1 times negative 2, which is going to get me 2. And I'm going to take negative 1 times 0 again to get 0. And I'm going to take negative 1 times negative 3 to get a positive 3. OK, so now, just like last time, these are the X coordinates that I'm using, and these are going to be my new Y coordinates that I use for my new function. Now what I need to do is take all these new points, take all the X's and Y's, and graph them. So 5, 3, 5, 3 is my first point. So 1, 2, 3, 4, 5, 1, 2, 3, there's my first point. And then I have 2, 0, 2, 0. It looks like we're in that same spot right there. And then we are 0, 2. So 0, 1, 2, right here. And then we are at negative 2, 0. So negative 2, 0. Looks like we're at that point again. See what this looks like in a minute. And then we also have negative 4, 2. So negative 1, 8, 2, negative 3, negative 4, and then 1, 2, right there. OK, so now I'm going to connect the dots, a little arm going out here, arm going out there. Peaks have now become valleys. Valleys have now become peaks. And this is my function that has been reflected across the X axis. So notice the points that are actually sitting on the X axis, they just stayed where they're at. But other points got flipped. Notice that we got distance of 2, distance of 2. This point got flipped, distance of 2, distance of 2. And then this point here got flipped, distance of 3, and distance of 3. So that is an example of reflecting a function. And this is also an example of kind of using a table to do that, using a table to do that. All right, that was translating functions and reflecting functions.