Translating points

- [Instructor] What we're going to do in this video is look at all of the ways of describing how to translate a point and then to actually translate that point on our coordinate plane. So, for example, they say plot the image of point P under a translation by five units to the left and three units up. So let's just do that at first, and then we're gonna think about other ways of describing this. So we want to go five units to the left. So we start right over here. We're gonna go one, two, three, four, five units to the left, and then we're gonna go three units up. So that's going to be one, two, three. And so the image of point P, I guess, would show up right over here, after this translation described this way. Now, there are other ways that you could describe this translation. Here, we described it just in plain English, by five units to the left and three units up. But you could, and this will look fancy, but, as we'll see, it's hopefully a pretty intuitive way to describe a translation. You could say, look, I'm gonna take some point with the coordinates x comma y. And the x coordinate tells me what's my coordinate in the horizontal direction to the left or the right. And so I want that to be five less. So I would say x minus five comma y. And what do we do to the y coordinate? Well, we're going to increase it by three. We're going to translate three units up, so y plus three. So all this is saying is whatever x and y coordinates you have, this translation will make you take five from the x. That's what, meaning this is, this right over here, is five units to the left. And then this right over here, is saying three units up. Increase your y coordinate by three. Decrease your x coordinate by five. And so let's just test this out with this particular coordinate, with this particular point. So at this point right over here, P has the coordinates, its x coordinate is three, and its y coordinate is negative four. So let's see how that works. If I have three comma negative four, and I want to apply this translation, what happens? Well, let me just do my coordinates. And so I started off with three and negative four, and I'm going to subtract five from the three. So subtract five here, we see that right over there, and we're going to add three to the y. So notice, well, instead of an x, now I have a three. Instead of an x, now I have a three. Instead of a y, now I have a negative four. Instead of a y, now I have a negative four. And so another way of writing this, we're going from three comma negative four to three minus five is negative two, and negative four plus three is negative one. So what are the coordinates right over here? Well, the coordinate of this point is indeed negative two comma negative one. So notice how this, I guess you could say this formula, the algebraic formula that shows how we map our coordinates, how it's able to draw the connection between the coordinates. And so you'll see questions where they'll tell you, hey, plot the image, and they'll describe it like this. Translate x units to the left or the right or three units up or down. You'll sometimes see it like this, but just recognize this is just saying just take your x and subtract five from it, which means move five to the left. And this just means take your y coordinate and add three to it, which means move three up. And sometimes they'll ask you, hey, what's the new coordinate? Or sometimes they'll ask you to plot something like that, but just realize that it's all the same underlying idea.