Example translating points

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 5 units to the left. So we start right over here. We're going to go 1, 2, 3, 4, 5 units to the left, and then we're going to go 3 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 5 units to the left and 3 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 going to take some point with the coordinates x, comma, y, and the x-coordinate tells me how what's my coordinate in the horizontal direction to the left or the right.

And so I want that to be 5 less, so I would say x minus 5, comma, y. And what do we do with the y-coordinate? Well, we're going to increase it by three. We're gonna 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 5 from the x. That's what meaning this is.

This right over here is 5 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 in this point, right over P has the coordinates; its x-coordinate is 3, and its y-coordinate is negative 4. So let's see how that works. If I have 3, comma, negative 4, and I want to apply this translation, what happens?

Well, let me just do my coordinates. And so I started off with 3 and negative 4, and I'm going to subtract 5 from the 3. So subtract 5 here, we see that right over there, and we're going to add 3 to the y. So notice we're instead of an x now I have a 3; instead of an x now I have a 3; instead of a y now I have a negative 4; instead of y now I have a negative 4.

And so another way of writing this, we're going from 3, comma, negative 4 to 3 minus 5 is negative 2 and negative 4 plus 3 is negative 1. So what are the coordinates right over here? Well, the coordinate of this point is indeed negative 2, comma, negative 1.

So notice how this, because 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 take your x and subtract 5 from it, which means move 5 to the left, and this just means take your y-coordinate and add 3 to it, which means move 3 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.