 In this video, I want to talk about translating, I'm going to talk about translating absolute value functions, okay? So now before we start this video, you've got to have an idea of what an absolute value is and kind of what it does to numbers, what it does to a function, things like that. Now your basic understanding of an absolute value is, now from a very rudimentary point, it basically just makes everything inside the absolute value, whatever number is inside the absolute value, it makes it positive, it's a very rudimentary understanding of it. Basically, now what an absolute value actually is, is the distance on a number line from zero to that number, that's what an absolute value is. So that's kind of what we're dealing with here. Now absolute values do something a little bit different to graphs, we have to have an idea of what they look like, so we'll read through the direction and I'll graph what one looks like and then we'll make our transformations for that, because I have two different examples here, two different transformations. I should say translating, which moves everything either up, down, left, or right. All right, so directions, perform the following transformations of f of x equals the absolute value of x, then graph each transformed function. Okay, so what I'm going to do first is I'm actually going to take f of x, I'm going to take this function and I'm going to graph it on my grid here. That way we have an idea of what we're moving up, down, left, and right. Okay, so start at zero, zero for this function, start at zero, zero. Okay, now it's actually going to look very similar to a linear function. So the thing is when I plug in an x of one, so if I plug in an x of one, that's when I plug in one, the absolute value of one is one. So I plug in one and I get out of it one. If I plug in two, I get out of it two. I plug in three, I get out of it three. Okay, so as you can well imagine, right here, this is what your absolute value looks like. Okay, now let's go to the other side, a little bit different when I plug in these negative numbers. When I plug in negative one, it's actually going to take the absolute value of that number and it's going to get a positive one. So a negative one, when I plug it in, when I get out of it is a positive one. Negative two, when I plug it in, I get a positive two. Negative three, when I plug it in, I get a positive three. Okay, again, remember that rudimentary understanding of what an absolute value is. Whatever number I have in there, it just basically makes it positive because you're asking yourself the question, what is the distance from zero to that number on a number line? So what is the distance from zero to negative three on the number line? Well, it's just three. It's a y-coordinate of three, so that's what we get out of this. Anyway, so basically our absolute value function, we'll call this one f. Call this one f and block here. That is what an absolute value function looks like. Okay, so now we're going to take that function. We're going to move it up, down, left, right. We're going to do whatever we want to with it. Okay, so the first thing we're going to do is we're going to move everything five units down. Okay, change my colors here. We're going to move everything five units down. And the first thing I'm going to do is perform the following transformation, which kind of means I'm going to use notation to show this. And then I'm going to graph each one of them. And then I'm going to graph the new function. I'm going to code this so I can tell the difference between my different functions. Okay, so what am I going to do first? I'm going to move this five units down. So what I'm going to do is take my original function. I'm going to change it. Now, if I move it five units down, I take the function and I subtract five from it. Take the function and subtract five from it. Okay, that is moving the function five units down. Okay, now if you don't quite understand how to transform or translate units, I do have previous videos on how to transform and translate different functions. You can search for those. Okay, now, so this is basically what I'm going to do. Take your function, change it by taking the function and subtracting five from it. Okay, so my new function g of x. Okay, so the new one is g of x. The old one is f. The new one is g. Okay, it is going to be the old one, but I'm going to subtract five from it. Okay, so notice the different notation. Now, I'm actually introducing a new function here. Okay, I'm not using these arrows anymore. Okay, now I'm using equations. Okay, so g of x is then going to be, I take the old function, which is, now the old function is just the absolute value of x. That's all it is. Absolute value of x. That's this piece right here is the absolute value of x. This piece, absolute value of x, and subtract five from it. So my g of x function is equal to absolute value of x minus five. Okay, so that's what the equation looks like. That's what, excuse me, the function looks like. Now what I want to do is I want to graph that. I want to graph that. Now, actually these are pretty easy to graph. What I want to do is I want to take this absolute value of function and just move everything down five units. Move everything down five units. So take each one of these points that I have here and move them down five. So one, two, three, four, five. Now this is actually pretty easy to graph. We just go up one right one, up one right one, and up one left one, up one left one. So get a couple of points on here. Get some of my points on here. And get my arrows on there because that goes on forever. All right, there we are. Okay, so there is my new function. I called this one, I called it g. I'm going to label it g over here so I can tell the difference between all of them. Not only can I color code, but I can use letters to know the difference between the two of them. Okay, I'm going to change colors here and go down to this next one. And that was it. So we wrote the, we performed the transformation. So this is what the function is going to look like. And then we took the graph of the function, the old one, and then we moved everything five units down. So now we have what the new one looks like. Okay, so I'm going to do one more example. And this time I'm going to go one unit left. So I'm showing you an example of going up and down. And I'm going to show you an example of going left and right. Okay, it's going to be a little bit different, a little bit different. Okay, so I'm going to take my old function, I'm going to take my old function. I'm going to change it by taking the function and I'm going to add or subtract directly to the x. Whenever I move left or right, I add or subtract directly to the x. Now in this case, as I'm moving left, I'm going to add one directly to the x. Okay, so that's what the notation looks like. Notice the difference between moving left and right and then moving up and down. Left and right, up and down. Up and down is much, much easier. I will admit. Up and down is much, much easier. Left and right is a little bit harder because you have to add directly to the x. Alright, so that's kind of what the mapping notation looks like. So instead of using g, I'm going to use a different function notation. I'll use h. h of x is a common one we use. So h of x is going to be, this function is going to be our old function, x of x, excuse me. I actually don't want that parentheses there. There we go. So we take our old function and we're going to add one directly to the x. Okay, now our old function is this up here. It's a square root, or it's going to be the absolute value of x. So I want to take the absolute value of x, but instead of an x, I want an x plus one. Notice I'm adding one directly to this x. We're inside the absolute value here, inside the absolute value. A little bit different from doing it in the ups and downs. Ups and downs are outside, left and right are inside. So then my new function, h of x, is then going to be equal to the absolute value of x plus one. Nope, nope, nope. What am I doing there? Nope, there we go. Gotta put the lines in there, not parentheses. There we go. Okay, so that's what the notation looks like. That's what the function is going to look like. Now I'm simply going to graph it. Okay, now graphing is a little bit easier. Grapping is a little bit easier. We just want to move all the units, all the points once to the left. So I'm going to take my old function and move everything one to the left. So I'm going to take all these points, move them once to the left. So right here, right here, right here, right there. Just want to look to the left here, there, and over here. Y in between all the points. Y in between all the points. And this blue one here is my h function. And now you can see why we need to kind of label our functions. Notice how things are overlapping here. We need to have good labels on our functions so that we can read it. Okay, so that's just a couple of examples of translating absolute value functions, basically moving them up, down, left, and right, and showing some of the notation to go along with it. Okay, showing some of the notation that goes along with it. Now, just remember, with your absolute values, they look like these. They basically look like these. They're pretty simple functions to work with. One thing you just got to remember, if I'm moving up or down, I'm going to add or subtract from the whole function. Or if I'm moving left or right, I'm going to add or subtract directly to the x function. And the notation's a little bit harder to work with, but that's what we do when we want to move left and right. Alright, again, that's just a couple of examples of translating absolute value functions.