To determine the [tex]$y$[/tex]-coordinate of point [tex]$D$[/tex] after the given translation, follow these steps:
1. Identify the initial [tex]$y$[/tex]-coordinate for point [tex]$D$[/tex]. In this case, it is given as [tex]$3.5$[/tex].
2. Note the translation vector specified: [tex]$(x, y) \rightarrow (x + 6, y - 4)$[/tex]. This means we are translating the point by moving it 6 units to the right (affecting the [tex]$x$[/tex]-coordinate) and 4 units down (affecting the [tex]$y$[/tex]-coordinate).
3. Focus on the change in the [tex]$y$[/tex]-coordinate. The translation moves the [tex]$y$[/tex]-coordinate down by 4 units.
4. Subtract 4 from the initial [tex]$y$[/tex]-coordinate:
[tex]\[ y_{final} = y_{initial} - 4 \][/tex]
[tex]\[ y_{final} = 3.5 - 4 \][/tex]
5. Calculate the result:
[tex]\[ 3.5 - 4 = -0.5 \][/tex]
Thus, the [tex]$y$[/tex]-coordinate of point [tex]$D$[/tex] after the translation is [tex]$-0.5$[/tex].
