About this video
Sal dilates points D, E and F about the origin with a scale factor of 1/2. Because the center is the origin, each image point is found by halving both coordinates: D(-8, -9) goes to (-4, -4.5), E(2, 7) goes to (1, 3.5) and F(6, -6) goes to (3, -3).
You'll be able to
- dilate a point about the origin by multiplying both coordinates by the scale factor.
- explain that a scale factor of 1/2 makes each point half as far from the center.
- write the coordinates of image points after a dilation.
Key words
- dilation
- A transformation that moves points toward or away from a center by the same scale factor.
- scale factor
- The number you multiply distances by; 1/2 makes everything half as far from the center.
- origin
- The point (0, 0), where the x-axis and y-axis cross.
Transcript
Plot the images of points D, E, and F after a dilation centered at the origin with a scale factor of 1/2. So we're going to center around the origin. We want to scale this thing down by 1/2. So one way to think about it is the points that will correspond to points D, E, and F are going to be half as far away from the origin, because our scale factor is 1/2 in either direction.
So for example, let's think about point D first. Point D is at negative 8. So if we have a scale factor of 1/2, what point D will map to is going to be at negative 4 on the x direction. And on the y direction, D is at negative 9, so this is going to be at negative 4.5.
Half of that. So that is going to be right over there. That's where point D is going to be, or the image of point D after the scaling. Now let's think about point E. E is 2 more than the origin in the x direction. So it's only going to be 1 more once we scale it by 1/2.
And it's 7 more in the y direction, so it's going to be at 3 and 1/2. 7 times 1/2 is 3 and 1/2. So we're going to stick it right over there. And then finally F, its x-coordinate is 6 more than the origin. Its y-coordinate is 6 less. So its image after scaling is going to be 3 more in the x direction and 3 less in the y direction.
So it's going to be right over there. So we've plotted the images of the points. So if you were to connect these points, you would essentially have dilated down DEF, and your center of dilation would be the origin. So let's just write these coordinates. Point D-- and point D, remember, was the point negative 8, negative 9.
That's going to map to-- take 1/2 of each of those. So negative 4 and negative 4.5. Point E maps to-- well, E was at 2, 7. So it maps to 1, 3.5. And then finally, point F was at 6, negative 6, so it maps to 3, negative 3. So the important thing to recognize is the center of our dilation was the origin.
So in each dimension, in the x direction or in the y direction, we just halved the distance from the origin, because the scale factor was 1/2. We got it right.
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