Certainly! Let's go through the problem step-by-step so that you can understand how to find the translation vector [tex]\( T \)[/tex] and then apply it to another point.
### Step 1: Find the translation vector [tex]\( T \)[/tex]
You are given the point [tex]\( (3, -4) \)[/tex] and its image [tex]\( (1, 5) \)[/tex] after translation by vector [tex]\( T \)[/tex].
1. The original point is [tex]\( (3, -4) \)[/tex].
2. The translated point is [tex]\( (1, 5) \)[/tex].
We need to find vector [tex]\( T = \binom{a}{b} \)[/tex].
To find the components [tex]\( a \)[/tex] and [tex]\( b \)[/tex] of the translation vector [tex]\( T \)[/tex], we calculate:
- [tex]\( a \)[/tex] is the horizontal change: [tex]\( 1 - 3 = -2 \)[/tex]
- [tex]\( b \)[/tex] is the vertical change: [tex]\( 5 - (-4) = 5 + 4 = 9 \)[/tex]
Thus, the translation vector is:
[tex]\[ T = \binom{-2}{9} \][/tex]
### Step 2: Apply the translation vector [tex]\( T \)[/tex] to the new point [tex]\( (-4, -2) \)[/tex]
We are now asked to find the image of the point [tex]\( (-4, -2) \)[/tex] when translated by the vector [tex]\( T = \binom{-2}{9} \)[/tex].
1. The new point we are translating is [tex]\( (-4, -2) \)[/tex].
2. The translation vector is [tex]\( T = \binom{-2}{9} \)[/tex].
To get the translated point, we add the translation vector to the new point:
- The new [tex]\( x \)[/tex]-coordinate is: [tex]\(-4 + (-2) = -4 - 2 = -6\)[/tex]
- The new [tex]\( y \)[/tex]-coordinate is: [tex]\(-2 + 9 = 7\)[/tex]
Therefore, the image of the point [tex]\( (-4, -2) \)[/tex] under the same translation is:
[tex]\[ (-6, 7) \][/tex]
### Summary
- The translation vector [tex]\( T \)[/tex] is [tex]\( \binom{-2}{9} \)[/tex].
- The image of the point [tex]\( (-4, -2) \)[/tex] after translation is [tex]\( (-6, 7) \)[/tex].
