 In this video, I'm going to talk about transforming absolute value functions. It's a little bit different from the previous videos that I've been doing with absolute values. What we're going to be doing here is we're going to be transforming them. Now, that's a little bit different from translating. Translating just moves up, down, left, and right. Transforming is a little bit different. Transforming, we can either reflect it, we can stretch it, we can compress, we can do all sorts of different stuff. So that's what we're going to be doing, transforming absolute value functions. I got a couple of examples of this. The first one, we're going to reflect the graph of this function f of x is equal to the absolute value of x minus 2 plus 3. We're going to reflect it across the y-axis. Okay, so in a previous video, what we looked at is if we took a basic function and we moved it up, down, left, and right, our combination of both of those, we can figure out where the vertex was going to go. So actually, when I look at this function right here, I should be able to graph that function pretty easily, just based on these numbers I see inside. A normal absolute value function looks something like this. Looks something like this. Okay, it looks like a v. And then from that position, we move it up, down, left, and right based on these numbers here. So in this case, this 2 on the inside, this negative 2, this tells me to the right 2. And then this one right here, this plus 3 means up 3. So I'm actually going to take this vertex and go to the right 2 and then up 3. Okay, so now what I'm going to do is I'm going to get rid of those old lines, get rid of those old lines. And now I'm going to take my vertex and I'm going to, it was at 0, 0, but now I'm going to go to the right 2 and up 3. Right 2 and then up 1, 2, 3. So this is where the function is right now. This is where the function is right now. And then I just go up 1 over 1, up 1 over 1, up 1 over 1, up 1 over 1. So this is what our function looks like. Okay, call this 1f, call that 1f. Now what I want to do is I want to take that and I want to reflect it across the y-axis. So my y-axis is the one that goes up and down to my vertical axis. So I want to reflect it across the y-axis. So everything that's on the right side here is going to reflect over to the left. Okay, so now I'm going to do the notation first to show that and then we're going to do that actually on the graph. Then we're going to actually go to graph it. Okay, so the first thing I'm going to do is show the notation. So I'm going to show that f of x, my original function, I'm going to change it by reflecting it across the y-axis. Now how do I do that? To reflect across the y-axis, notice that all these x-coordinates, all these x-coordinates over here are going to be reflected across. So instead of a positive 2 for this x-coordinate here, if I reflect it across, it's going to be right over here with a negative 2 for my x-coordinate. So notice that the x-coordinates are the ones that change. We change the opposite number. So again, 2 for an x-coordinate changes to a negative 2. So what that does is I'm going to take my function and I'm going to take the opposite of the x-coordinates. So that tells you that you put a negative sign in there with the x. Negative sign in there with the x. Okay, so that's what I'm going to be doing. That's how I'm going to change my function. So this function here, that's how I'm going to change it. All right, so my new function is going to be g of x. That's a traditional new function that we use. My new g of x, I'm going to take the old one. I'm going to take the old one, f of x, and I'm going to take the opposite of the x. So wherever there's an x, I'm going to replace it with a negative x. Wherever there's an x, wherever there's an x, replace it with a negative x. Okay, so here's my original function. Oops, I made a mistake. That's a minus, not a plus. It's a minus, not a plus. So minus 2 plus 3. Now notice I wrote everything out first, but I left the x out. Okay, again, instead of writing an x, I'm going to write it with a negative x. So I'm going to put a negative x in here. It changes from an x to a negative x. So there we go. Now looking at that, there's no parentheses that I have to deal with. There's really no simplifying that I have to work with. So actually that's as good as it gets right there. So my new function, g of x is equal to absolute value negative x minus 2 plus 3. Okay, so that's what the new function is going to look like. That's the equation for this new function. Okay, so now let's actually look at what the graph is going to look like over here. Now again, when you look at a graph, it's actually, when you draw something out, it's much, much easier to see how it changes what the graph is going to look like when you graph the original and then you just save yourself. Okay, just reflect everything across the axis. It's much, much easier to see when you draw yourself a picture. So again, like earlier, take this point, reflect it across. Okay, I'm going to take this point, reflect it across. Now I'm going to be technical here. Basically what we're doing is that the corresponding points are going to be equidistant from the line of reflection. Okay, it's a very mathy way of saying that if this point is 2 away from the line of reflection, this point is, then the other point is going to be 2 away. This point was 1 away. This other point reflected is going to be 1 away. Okay, this point over here is 1, 2, 3 away. So it's going to be 1, 2, 3 away. There we go. This point is 0 away, so the new one is actually going to be 0 away. It's actually staying the same spot. This one over here is 1, 2, 3, 4 away, so 1, 2, 3, 4 away. There we go. Again, got a little bit, little bit mathy there, but the corresponding points, corresponding points are going to be equidistant from the line of reflection. Okay, kind of a very mathy way of reflecting the points across. There's my new G function, and that is reflecting across the Y axis. Okay, so there's my first example, showing the notation here, showing the notation, and showing what the graph is going to look like. Okay, so there's our first example. The second example that I'm going to do is a stretching example. So again, transforming absolute value functions. I'm going to stretch this graph. A little bit simpler equation here, a little bit simpler function I should say, and we're going to stretch this vertically by a factor of 2. Vertically by a factor of 2. So what I'm going to do is I'm going to draw, I'm going to draw this one. Now notice I'm not doing anything left and right inside. There's no numbers on the inside, so I'm not doing anything left and right. I'm just going down one. Okay, so take your original, your original V looking, your original V looking absolute value function, and you move everything down one. So my vertex is here, and then a couple of points here to create that V effect, to create that V. Okay, so there's that. Again, label it, that is my F function. Okay, you should have a pretty good idea of how to graph a very basic, again this is a very basic absolute value function. You should have a pretty good idea, I've had a graph that yet, either from homework or from previous videos or previous notes that you have. Okay, so now what we're going to do is we're going to take this function, we're going to stretch it vertically by a factor of 2. Stretch it vertically by a factor of 2. Okay, now when we stretch something vertically, when we stretch something vertically, these lines are actually going to get steeper. So it's going to be a, what you could call a thinner or taller V. It's basically what it's going to be. It's going to be a thinner or taller V. Okay, now when we stretch everything vertically, these arms are going to come in closer to the Y axis, and when everything gets stretched vertically, that's not only vertically up, that's vertically down also. It's also going to be stretched down. Okay, so now let's go to the notation to see what that's going to look like. Okay, so what I'm going to do is I'm going to take my old function, f of x, I'm going to change it, change it by vertically stretching by a factor of 2. So now I'm going to take my function, when I vertically stretch something, I multiply the entire function times the number. Now, okay, now you've got to do a little bit of thinking. Now, I'm either going to use 2 or I'm going to use 1 half. So I'm either going to multiply by 2 or I'm going to divide by 2 or multiply by 1 half, which is the same thing as divide by 2. Okay, now I've got to decide which one I'm going to use. Now, one way to do this is by looking at the slope. The slope of this line is just 1. Just 1 over 1, 1, 1, 1, 1. The slope is 1. And now what I'm going to do, if I vertically stretch this by a factor of 2, when I vertically stretch something, it's going to get taller and I'm going to go up. So just to kind of give a little bit of an example, my V-shape should actually do something like this. My V-shape should actually do something like that. That's what it's going to look like in the end. Something like that. Now, maybe not exactly like that, but something like that. So now, with that mental image of what it's supposed to look like, basically the slope of this has gotten bigger. The slope is steeper. So that tells me that do I multiply by 2 or do I multiply by 2? Well, if I want my slope to get bigger, I'm going to actually multiply times 2. So that's going to give me what this V-shape is over here. I'm going to rewrite that because I've got to undo this, get rid of my bad drawing there. I'll hold my 2 back. So what I'm doing is take the old function, change it by multiplying 2 times your entire function. And whenever you do something vertically, when you do a vertical stretch or compression, you always multiply that number times the entire function. Okay, so let's take g of x is the standby that we use for our second function. My new function is going to be the old one, but I'm going to multiply by 2, take the entire function times 2. So what does that look like? So take this entire function times 2. So two big parentheses here, absolute value of x minus 1. Okay, so 2 times the entire function. There's the entire function right there in these parentheses. Now, for this example, I'm going to have to simplify just a little bit. On the previous examples that I've done, there was no reason to simplify. I didn't have to add, subtract, multiply any numbers. But in this case, I do. This 2 is going to have to distribute to the absolute value of x, and this 2 is going to have to distribute to the negative 1 over there. So my new function, g of x is equal to 2 absolute value of x minus 2. Okay, so that's what it's going to look like. That's what it's going to look like. Okay. Now what we want to do is we want to graph this. Okay, we want to graph this. Now, every point, every x coordinate is going to have, or excuse me, not x coordinate, excuse me, if we're doing this vertically, every y coordinate, every y coordinate is going to be multiplied times 2. If we stretch something vertically, the y coordinates are going to be multiplied times 2. This is the factor of 2. So what I'm going to do is I'm going to look at all these points that I have here, and I'm going to stretch them by a factor of 2. I'm going to multiply the y coordinates times 2. So notice here, this point has a y coordinate of 2. I'm going to multiply that times 2 to go to 4. Same thing over here, y coordinate of 2 multiplied times 2 to go to 4. Okay, so this one here, y coordinate of 1 times 2 is 2. Same thing over here, y coordinate of 1 times 2 is 2. Now these ones right here, they have a y coordinate of 0, and 0 times 2 is still 0. So actually, these points are going to stay right where they're at. Those points are going to stay right where they're at. This one down here, our vertex has a y coordinate of negative 1. Negative 1 times 2 is negative 2. Notice here, negative 2 for your y intercept, right here, negative 2 for your y intercept. Slope intercept form, negative 2 right there. That's what happens on purpose. So my vertex is now down here, so now I'm going to connect all my red dots, create my v shape, create my v shape, try to get some straight lines in there, and that is my new function g. And everything has been stretched vertically by a factor of 2. So that's what the function is going to look like. Okay, now there's a couple of different ways that you can't graph this. If you have a graphing calculator or an application on an iPad that affects, there's a number of different ways that you can do this. But for now, this is kind of the easy way to do this without a calculator. There are easier ways to do it if you have such technology. But anyway, that is just a couple of examples of how to transform absolute value functions.