Video Transcript
In the figure below, determine the
coordinates of the points 𝐴 prime, 𝐵 prime, 𝐶 prime, and 𝐷 prime, where 𝐴 prime
𝐵 prime 𝐶 prime 𝐷 prime is a translation of 𝐴𝐵𝐶𝐷 by negative two, negative
five.
A translation is a transformation
which moves a shape up or down and from side to side. A translation of negative two,
negative five means the shape moves two places to the left and five places down. Seen as we’re translating a shape
with four points, we have four coordinates to translate. There are a couple of ways we can
find the images of each point after the translation. But let’s start by doing this
graphically.
We’ll start with point 𝐴. If we move the point 𝐴 two units
to the left and five units down, we find that the image of 𝐴, 𝐴 prime, is at the
point six, one. Let’s do the same with point
𝐵. We move point 𝐵 two units to the
left and five units down. And we find that the image of 𝐵,
𝐵 prime, is at the point six, negative four. Now let’s translate the point
𝐶. We move the point 𝐶 two units to
the left and five units down. And we find that the image of 𝐶,
𝐶 prime, is at the point negative one, negative four. Finally, let’s translate the point
𝐷. We move the point 𝐷 two units to
the left and five units down. And we find that the image of 𝐷,
𝐷 prime, is at the point one, negative one. Then, joining all our points, we
see the translated shape 𝐴 prime 𝐵 prime 𝐶 prime 𝐷 prime.
Now we can read the coordinates of
the translated points directly from the graph to get our answer. But let’s also see a different
method we could have used. For a translation with horizontal
displacement of 𝑎 units and a vertical displacement of 𝑏 units, the point 𝑥, 𝑦
maps to 𝑥 add 𝑎, 𝑦 add 𝑏. Here are the coordinates of the
four points we were given in the question: 𝐴, 𝐵, 𝐶, and 𝐷. We can use the formula above to map
each point to its image. The horizontal displacement is
negative two, so 𝑎 is negative two. The vertical displacement is
negative five, so 𝑏 is negative five.
So we calculate the image of each
of these coordinates by adding negative two to the 𝑥-coordinate and negative five
to the 𝑦-coordinate, which then allows us to calculate the images of each of these
points. That is, 𝐴 prime is six, one; 𝐵
prime is six, negative four; 𝐶 prime is negative one, negative four; and 𝐷 prime
is one, negative one. Then, we can check these points by
looking at the translation we carried out on the graph. And we see that the points match
up.
