To find the new [tex]$y$[/tex]-coordinate of point [tex]\(D\)[/tex] after the translation, let's follow these steps:
1. Identify the original coordinates of point [tex]\(D\)[/tex]: The given coordinates of point [tex]\(D\)[/tex] are [tex]\( (3.5, y_D) \)[/tex].
2. Determine the translation vector: The translation vector given is [tex]\((x + 6, y - 4)\)[/tex]. This means we are adding 6 to the [tex]$x$[/tex]-coordinate and subtracting 4 from the [tex]$y$[/tex]-coordinate of point [tex]\(D\)[/tex].
3. Apply the translation to the [tex]$y$[/tex]-coordinate:
- The original [tex]$y$[/tex]-coordinate of point [tex]\(D\)[/tex] is [tex]\(y_D = 0\)[/tex].
- According to the translation vector, we need to subtract 4 from the [tex]$y$[/tex]-coordinate: [tex]\( y_D - 4 \)[/tex].
4. Calculate the new [tex]$y$[/tex]-coordinate:
- [tex]\( y_D - 4 = 0 - 4 = -4 \)[/tex].
Therefore, after applying the translation to point [tex]\(D\)[/tex], the new [tex]$y$[/tex]-coordinate of [tex]\(D'\)[/tex] is [tex]\(-4\)[/tex].
So, the translated point [tex]\(D'\)[/tex] will have coordinates [tex]\((3.5, -4)\)[/tex].
