Dilating algebraically about a point

0:00- [Tutor] We're told to, "Graph the image of triangle PQR after a dilation by a scale factor of 2/3 about the point (-1, 2)." So pause this video and have a go at it before we do this together. So in other videos, we've done pure dilations by a scale factor about the origin, but what's different here is we're about a point that's not the origin.

0:21So what I'd like to do is first, do a translational transformation, so that the point (-1,2) gets shifted to the origin. Then we do a scale factor about that origin, about 0, 0, and then we can do the translation back from the origin to the point (-1, 2), and the whole time, I'm gonna describe each of these compounding dilations in coordinate notation, so we can get the final dilation.

0:50So the first thing I wanna do, is I wanna shift the point (-1, 2), so that's (-1, 2), this point right over here. I want to shift that to the origin. In order to shift that to the origin, I have to shift to the right by 1, and down by 2. So the first transformation I'm going to do, is I'm gonna take (x, y), and whatever my x is, I'm gonna add 1, that shifts me to the right by 1, and whatever my y is, I'm going to subtract 2.

1:21Now, that will definitely shift (-1, 2) to the origin, but what does it do to our triangle? Well, everything shifts to the right by 1 and down 2, to the right by 1 and down 2, to the right by 1 and down 2. And so after this transformation, my triangle looks like this, my triangle looks like this, and we can see what these coordinate points are, which is going to be important as we are about to do the actual dilation.

1:51This point right over here is the point (6.3), (6.3), this is the point (9,0), and this is the point -3, oh, sorry, it's (0,-3), I get confused sometimes, (0,-3) right over there, did I do that right? Yep, (6,3), (9,0), (0,-3). Now, the next thing we wanna do is dilation, by a scale factor of 2/3.

2:19To do that, you essentially just multiply every x and y by 2/3. So if we wanna write it in coordinate notation, whatever our coordinates are now, we're gonna multiply by 2/3. So now, we just multiplied 2/3 times each of these, so (2/3)(x+1), and then 2/3 (y-2), and then if we look at this, if we see what each of these points become, 0 times 2/3 is still 0, -3 times 2/3, lemme do this in a new color.

2:49- 3 times 2/3 is negative -2, 9 times 2/3 is 6, 0 times 2/3 is 0, so it maps over here, and then 6 times 2/3 is 4, and then 3 times 2/3 is 2, so this becomes 4, 2, 4, 2. And so now, our triangle has been dilated about the origin to look like this. Now, the last thing we wanna do is shift back, so that the origin goes back to (-1,2), so we just wanna undo this transformation right over here.

3:25And so that last step, what I'm going to do, I start with this point, (2/3 times x +1), and now I wanna subtract that 1 to be able to shift back to the left. And then likewise, here, this coordinate, 2/3(y - 2), this y coordinate, I want to shift it back up by 2.

3:49So each of these points, I'm going to shift to the left by 1 and up by 2, actually, lemme do this in another new color. So to the left by 1 and up by 2, to the left by 1 and up by 2, to the left by 1 and up by 2. And so my triangle, finally looks like this. And then this is my final dilation in coordinate notation, but I wanna simplify it a little bit.

4:21Well, actually, let's see, is there...? Yeah I guess I might as well simplify it, do a little bit of algebra. So this is going to be the x one, the x coordinate is (2/3 x + 2/3 - 1), and then this is 2, the y coordinate is (2/3 y - 4/3 + 2) Now 2/3 - 1, is the same thing as -1/3, so this is (2/3 x - 1/3), and then this over here, if I have (-4/3 + 2), 2 is 6/3, so this is gonna be the same thing as 2/3, so it's going to be (2/3 y + 2/3), and we're done, I could put this in here, this is 2/3 x - 1/3, and then this is 2/3 y + 1/3, and we are done.