To determine the [tex]$y$[/tex]-coordinate of point D after a translation by the vector [tex]$(6, -4)$[/tex], we need to start by identifying the original coordinates of point D.
The coordinates of point D before translation are given as [tex]$(x_D, y_D)$[/tex], which specifies [tex]$x_D = 3.5$[/tex]. However, the value of [tex]$y_D$[/tex] is not provided in the problem. Because the initial [tex]$y$[/tex]-coordinate is unknown, we cannot precisely calculate the final [tex]$y$[/tex]-coordinate after translation.
In order to perform a translation, we would typically use the provided vector to adjust both the [tex]$x$[/tex] and [tex]$y$[/tex] coordinates of the point. The formula for translation of a point [tex]$(x, y)$[/tex] by a vector [tex]$(tx, ty)$[/tex] is given by:
- New [tex]$x$[/tex]-coordinate: [tex]$x + tx$[/tex]
- New [tex]$y$[/tex]-coordinate: [tex]$y + ty$[/tex]
Applying this translation vector [tex]$(6, -4)$[/tex] to a point [tex]$(x_D, y_D)$[/tex] involves the following:
- New [tex]$x$[/tex]-coordinate: [tex]$x_D + 6$[/tex]
- New [tex]$y$[/tex]-coordinate: [tex]$y_D - 4$[/tex]
Given that [tex]$x_D = 3.5$[/tex], the new [tex]$x$[/tex]-coordinate of [tex]$D'$[/tex] will be [tex]$3.5 + 6 = 9.5$[/tex].
However, the [tex]$y$[/tex]-coordinate [tex]$y_D$[/tex] is not provided. As a result, the new [tex]$y$[/tex]-coordinate after applying the translation is indeterminate since we cannot determine [tex]$y_D - 4$[/tex] without knowing the initial value of [tex]$y_D$[/tex].
To summarize, due to the lack of information regarding the initial [tex]$y$[/tex]-coordinate, it is insufficient to determine the [tex]$y$[/tex]-coordinate of point D after the translation.
