Let's solve the problem step-by-step to find the [tex]$y$[/tex]-coordinate of point [tex]$D$[/tex] after the translation.
1. Understand the Translation:
The translation given is [tex]$(x, y) \rightarrow (x+6, y-4)$[/tex]. This means that to find the new coordinates of any point after the translation:
- Add 6 to the original [tex]$x$[/tex]-coordinate.
- Subtract 4 from the original [tex]$y$[/tex]-coordinate.
2. Initial Coordinates of Point D:
The given coordinates of point D are [tex]$D(3.5, \square)$[/tex]. This means:
- The [tex]$x$[/tex]-coordinate is 3.5.
- The [tex]$y$[/tex]-coordinate needs to be determined.
3. Applying the Translation to the [tex]$y$[/tex]-coordinate:
Suppose the original [tex]$y$[/tex]-coordinate of point D is [tex]$y$[/tex]. After applying the translation [tex]$(x, y) \rightarrow (x+6, y-4)$[/tex], the new [tex]$y$[/tex]-coordinate becomes [tex]$y - 4$[/tex].
4. Resultant Coordinates:
Given that we need the new coordinates after translation and knowing the given transformation affects both [tex]$x$[/tex] and [tex]$y$[/tex] in specified ways.
- The [tex]$x$[/tex]-coordinate after translation is [tex]$3.5 + 6 = 9.5$[/tex].
- Similarly, determining the [tex]$y$[/tex]-coordinate directly from the results given, [tex]$y - 4 = -4$[/tex].
Therefore, since the translation reduces the original [tex]$y$[/tex] value by 4 and results in [tex]$-4$[/tex],
- Solving [tex]$y - 4 = -4$[/tex],
we get [tex]$y = 0$[/tex].
So, the original [tex]$y$[/tex]-coordinate was [tex]$0$[/tex], and after applying the translation [tex]$(x, y) \rightarrow(x+6, y-4)$[/tex], we get the new coordinate.
Thus, after the translation, point [tex]$D$[/tex] has new coordinates: [tex]$(3.5, 0) \rightarrow (9.5, -4)$[/tex].
Hence, the [tex]$y$[/tex]-coordinate of point [tex]$D$[/tex] after the translation is [tex]\(-4\)[/tex].
