Video Transcript
The following translation 𝐴𝐵 is
equivalent to a horizontal displacement from one to five and a vertical displacement
from four to two. Find the image of point 𝐶 by
performing translation 𝐴𝐵 in the direction of the ray 𝐴𝐵.
We’re told that this translation
𝐴𝐵, which is the mapping needed to map point 𝐴 onto point 𝐵, is equivalent to a
horizontal displacement from one to five and a vertical displacement from four to
two. So we can calculate both the
horizontal and vertical displacements. The horizontal position increases
by four units, and the vertical position decreases by two units. We can express this translation in
mapping notation as the point 𝑥, 𝑦 is mapped to the point 𝑥 plus four, 𝑦 minus
two.
From the figure, we can determine
that the point 𝐶 has coordinates one, two. So, applying this mapping, point 𝐶
will be mapped to the point with coordinates one plus four, two minus two, which is
the point five, zero. We can check that this answer is
correct by applying the translation graphically. Since the translation maps 𝐴 to
𝐵, point 𝐶 must be mapped the same distance and direction as point 𝐴. So we can draw a ray starting at
point 𝐶, which is the same length as the line segment 𝐴𝐵 and in the direction of
the ray 𝐴𝐵, to find the point 𝐶 prime. This confirms that the coordinates
of 𝐶 prime, which is the image of point 𝐶 following translation 𝐴𝐵 in the
direction of the ray 𝐴𝐵, are five, zero.
