Dilating points example

We're asked to plot the image of point A under a dilation about point P with a scale factor of three. So, what they're saying when they say under a dilation is they're saying stretching it or scaling it up or down around the point P.

That's what we're going to do. Just think about how far point A is, and then we want to dilate it with a scale factor of 3. However far A is from point P, it's going to be three times further under the dilation—three times further in the same direction.

So, how do we think about that? Well, one way to think about it is to go from P to A. You have to go one down and two to the left—so minus one and minus two. If you dilate it with a factor of three, then you're gonna want to go three times as far down, so minus three, minus three, and three times as far to the left, so you'll go minus six.

So, one—let me do this—so negative one, negative two, negative three, negative four, negative five, negative six. You will end up right over there, and you can even see that this is indeed three times as far from P in the same direction.

We could call the image of point A maybe A prime. So, there you have it, it has been dilated with a scale factor of three. You might be saying, "Wait, I'm used to dilation being stretching or scaling. How have I stretched or scaled something?"

Well, imagine a bunch of points here that represent some type of picture. If you push them all three times further from point P—which you could use as your center of dilation—then you would expand the size of your picture by a scale factor of 3.

Let's do another example with a point. Here, we're told to plot the image of point A under a dilation about the origin with a scale factor of one-third.

So, first of all, we don't even see point A here, so it's actually below the fold. Let's see, there we go—that's our point A. We want it to be about the origin, so about the point (0, 0). This is what we want—the dilation about the origin with a scale factor of one-third.

How do we do this? Well, here, however far A is from the origin, we now want to be in the same direction but one-third as far. One way to think about it is to go from the origin to A. You have to go six down and three to the right.

So, one-third of that would be two down and one to the right. Two is one-third of six, and one is one-third of three. You will end up right over here—that would be our A prime.

Notice you are one-third as far away from the origin as we were before, because once again, this is point A under a dilation about the origin with a scale factor of one-third.