Video Transcript
Three points — 𝐴 one, negative
five; 𝐵 two, negative five; and 𝐶 two, four — are translated by the mapping 𝑥, 𝑦
is mapped to 𝑥 minus three, 𝑦 plus one to points 𝐴 prime, 𝐵 prime, and 𝐶
prime. Determine 𝐴 prime, 𝐵 prime, and
𝐶 prime.
We recall first that the notation
𝑥, 𝑦 is mapped to 𝑥 plus 𝑎, 𝑦 plus 𝑏 describes the translation that maps point
𝑥, 𝑦 to the point with coordinates 𝑥 plus 𝑎 and 𝑦 plus 𝑏. It corresponds to a translation
with a horizontal displacement of 𝑎 units and a vertical displacement of 𝑏
units. For the translation in this
question, we have 𝑎 equal to negative three and 𝑏 equal to positive one. We are decreasing the 𝑥-coordinate
by three and increasing the 𝑦-coordinate by one.
To find the image of each point
under this translation, we can substitute the 𝑥- and 𝑦-coordinates of each point
into the map. So, for the point 𝐴, which has
coordinates one, negative five, its image will be the point with coordinates one
minus three, negative five plus one. That’s the point negative two,
negative four. 𝐵, which has coordinates two,
negative five, will be mapped to the point with coordinates two minus three,
negative five plus one. So 𝐵 prime has coordinates
negative one, negative four. And finally, point 𝐶, which has
coordinates two, four, will be mapped to the point with coordinates two minus three,
four plus one, which is the point negative one, five.
So we determined 𝐴 prime, 𝐵
prime, and 𝐶 prime. We could also consider this
translation graphically. Here are the points 𝐴, 𝐵, and 𝐶
represented on a coordinate plane. A horizontal displacement of
negative three units means we’re decreasing the 𝑥-coordinate of each point by
three. And a vertical displacement of
positive one unit means we’re increasing the 𝑦-coordinate of each point by one. This confirms that the coordinates
of 𝐴 prime, 𝐵 prime, and 𝐶 prime are negative two, negative four; negative one,
negative four; and negative one, five, respectively.
