To determine the new [tex]\( y \)[/tex]-coordinate of point [tex]\( D \)[/tex] after a translation, we need to follow these steps:
1. Identify the initial coordinates of point [tex]\( D \)[/tex]: Let's denote the initial coordinates as [tex]\( (x, y) \)[/tex].
2. Translation amounts: The problem specifies a translation of [tex]\( (x, y) \rightarrow (x+6, y-4) \)[/tex]. This means the [tex]\( x \)[/tex]-coordinate increases by 6 units and the [tex]\( y \)[/tex]-coordinate decreases by 4 units.
3. Initial [tex]\( y \)[/tex]-coordinate: For the point [tex]\( D \)[/tex], suppose the initial coordinates are [tex]\( (3.5, y_{initial}) \)[/tex]. The initial [tex]\( y \)[/tex]-coordinate is referred to as [tex]\( y_{initial} \)[/tex].
4. Calculate the new [tex]\( y \)[/tex]-coordinate:
- According to the translation rule, the new [tex]\( y \)[/tex]-coordinate will be [tex]\( y_{initial} - 4 \)[/tex].
Without the initial [tex]\( y \)[/tex]-coordinate given, we can't calculate an exact numerical value. However, we can express the new [tex]\( y \)[/tex]-coordinate in terms of the initial [tex]\( y \)[/tex]-coordinate.
So, if the initial coordinates of [tex]\( D \)[/tex] are [tex]\( (3.5, y_{initial}) \)[/tex], after the translation, the new coordinates of [tex]\( D \)[/tex] (denoted as [tex]\( D' \)[/tex]) will be [tex]\( (3.5 + 6, y_{initial} - 4) \)[/tex].
This simplifies to:
[tex]\[ D' (9.5, y_{initial} - 4) \][/tex]
To summarize:
- The [tex]\( y \)[/tex]-coordinate of point [tex]\( D' \)[/tex] after the translation will be [tex]\( y_{initial} - 4 \)[/tex].
