 In this video, I'm going to talk about translating functions. Specifically, I'm going to talk about translating linear functions. Notice this line that I have here. This is a linear function. Call this one F. And what we're going to do is we're going to translate them. We're going to move them up, or down, or left, or right. In this case, for this example, I'm going to do a horizontal shift, which is moving left and right. So you can kind of see, after we read this, we're going to move this either left or right. And then on the next slide, I'm going to show what happens when you move them up and down. So how am I going to do this? Well, first, I have this line. I have this example here. Translate f of x equals 2x plus 1, two units to the right. So we're going to take this line with a y intercept of 1, and then a slope of 2, a positive slope of 2. And we're going to translate it by moving it two units to the right. And then what we're going to do is we're going to rewrite the equation for this new line. And then we're going to see kind of what happens when you move functions left and right. So the first thing I need to do is I need to find a couple of points on this line. So there's a y intercept right there of 0, 1. Here's a point right here. What is that? 1, 3. 1, 3 is another point. Another point right here. Negative 1, negative 1, negative 1. OK, I'm going to use these three points. It is as simple as you might think. Let's just take these three points and move them two units to the right. That's all we really need to do. So take this point, move it two units to the right, right there, two units to the right, right there, and two units to the right, right there. Now I want to be exact with this. So what I'm going to do is I'm going to change this up just a little bit. And I'm going to draw myself a really nice straight line with this. I need to make this as straight as possible. Now if you're doing this with paper, if you're doing this with paper, what you want to do is you want to try and do this maybe with a ruler or maybe do this with maybe a straight edge of some sort. That way your lines are nice and straight just like this one that I created here. OK, now that that's done, what I need to do is I need to write the equation of this new line. Now F was the name of our first function. I'm going to call this next one G. It's usually what the new ones are going to be called, kind of alphabetical order, F, G, that kind of thing. But now the new function, what we need to do is we need to write it in slope-intercept form. So we have y equals mx plus b, that's our slope-intercept form. I need to know what the slope and the y-intercept of a line is. So what I need to do is take a look at my new line and figure out what the y-intercept and what the slope is. OK, so first of all, the y-intercepts all the way down here. Now this is why you need a straight line because you need to know exactly what this point is right here. As you can see, it's at 0, negative 3. So I have minus 3 for my y-intercept. And then my slope, actually, if you look at these points, rise to run 1, rise to run 1. My slope is still going to stay 2. My slope still stays 2. OK, now as you can kind of see, what we can do now is we can compare. We can compare these two to see how everything changed when we moved this function two units to the right. Now notice that the slope actually stays the same. You can tell that these are parallel lines. The slope stays the same. But the y-intercept here seems to be the only thing that changed. So instead of plus 1, now it's negative 3. So basically what it looks like by moving everything right 2, we've basically moved this equation down four slots. The y-intercept is right down here. It used to be a positive 1, and now it's a negative 3. Now basically what this is, in general, is that when you want to move something left or you want to move something right, what you need to do is you need to add or subtract it directly to the x. So this is kind of what this notation reads. If I have a function, a normal function, and I want to change it, and if I want to change it by subtracting some number directly to the x, this is going to be a horizontal shift to the right. So this is going to be a right. We'll just call this a right shift. If I want to move anything to the right, I'm going to actually subtract this number. Now you can look at it from this example. If I move two units to the right, this number didn't go up. This number went down. This y-intercept right here, it went down. So we're actually subtracting from our function. We're actually subtracting from the x, moving things to the right. And then opposite, you can well imagine, if I add something to the x directly to the x, this is going to be a left does not have an i in it. This is going to be a left shift. This is going to be a left shift. Now I will do an example here later that will show this a little bit more detail of if we move something left or right, how we add or subtract directly to the x. But this is just kind of a very basic idea of what this is right now. So that is our horizontal shift. Now let's go to the vertical shifts. So this is a new one, translating functions. We're going to do a vertical shift. This actually is a little bit easier to understand than the horizontal shift. So we did the harder one first. Now we'll do one of the easier ones. So just like last time, what I'm going to do is I'm going to, here's my function. Kind of have the same one, 2x plus 1. And I want to move this three units up. I want to use it, move it three units up. So just like the last time, I'm going to take some of these points that I have. So I'm going to use the same points that I used last time. Negative 1, negative 1, 0, 1, and then 1, 3. I'm going to take these three points, and I'm simply just going to move them three units up. That's really it. That's all I'm doing. So I'm going to take this point. I'm going to move it three units up, 1, 2, 3. I'm going to take this point, move it up 3, 1, 2, 3. I can already start to see that this point here, 1, 2, 3, is going to be kind of off of my grid. I don't know if that's exactly where that's supposed to go. So actually, I'm going to come down here. I'm going to find one additional point to use just to make sure I do this right, 1, 2, 3. There we go. It doesn't hurt to always put an extra point in there to make sure you're doing this correctly. All right, now what I want to do is I'm going to draw my straight line. And again, I'm going to use my computer here to kind of help me with that. OK, draw a straight line here, draw a straight line here. Now, again, you do need a really straight line, because now when I write the equation for the new one, equation for the new line, so f is the blue line, g is this black line here, it's the new one. Now what I want to do is I want to write the equation for that line, again, I'm going to write it in slope-intercept form, y equals mx plus b. And so I need a slope. So my slope is down 1, right 2, but this is a positive slope. So that's going to be 2x for my slope. And then my y-intercept, I can plainly see that right there, is 1, 2, 3, 4. So plus 4 for my y-intercept. OK, now compare these two. Look at the blue here and look at the black here. How did those two functions change? Well, notice here that we used to have a 1 for a y-intercept. Now we have a positive 4 for y-intercept. The y-intercept went up by 3, it looks like. Well, coincidentally, we were moving everything up by 3. So that's that right there. That's actually no coincidence at all. That's by design. So basically what happens when you vertically shift something, you move it up or down by some number. You're only really changing this y-coordinate. Now, as you saw from the other example, it wasn't. Actually, let me flip back to that other example. This example here, if plus 1 to minus 3, didn't really have much to do with the 2. If you compare 1 and negative 3, looks like I went down by 4. I went down by 4. So if I go 2 units to the right, that's the same as down by 4. So it's not quite clear yet of why this happens. We'll explore that a little bit later. Oh, I went too far on that. OK, now here, going back to my, sorry, there we go. Flipping around a bunch of different slides here. Now I'm back to my vertical shift. Now here, it's actually blatantly obvious that if I go 3 units up, all I have to do is just add 3 to my function. So notice down here in general, if I have my function and I want to change it by vertically shifting it, what I want to do is I just want to add directly to the function. And this is going to be shifting up. If I add to the function, it's going to be shifting up, just as we saw here with this example. And you can well imagine you could expand on that. And if I subtract some number to the function, this is going to be a shift down. This is going to be a shift down. All right, that's just a couple of examples of how to do a horizontal shift and a vertical shift. Sorry about sliding around all those different slides and all. But anyway, yes, that is translating functions, translating linear functions by either using a vertical shift or by using a horizontal shift.