To find the [tex]$y$[/tex]-coordinate of point [tex]$D$[/tex] after the given translation, we need to understand what translation means. A translation moves every point of a figure or a space by the same distance in a given direction.
Given translation rule: [tex]\((x, y) \rightarrow (x+6, y-4)\)[/tex].
Let's apply this rule to point [tex]\(D\)[/tex]. The starting coordinates of point [tex]\(D\)[/tex] are given as [tex]\(D'(3.5, y)\)[/tex].
1. The [tex]$x$[/tex]-coordinate of point [tex]\(D\)[/tex] is initially 3.5.
2. The [tex]$y$[/tex]-coordinate of point [tex]\(D\)[/tex] is initially unknown, let's denote it as [tex]\(y\)[/tex].
According to the translation rule [tex]\((x, y) \rightarrow (x+6, y-4)\)[/tex]:
- The new [tex]$x$[/tex]-coordinate will be [tex]\(3.5 + 6\)[/tex].
- The new [tex]$y$[/tex]-coordinate will be [tex]\(y - 4\)[/tex].
From the information given, we know that the final coordinates after the translation are:
[tex]\[
\left( 3.5 + 6, y - 4 \right) = (9.5, -4)
\][/tex]
To find the new [tex]$y$[/tex]-coordinate:
[tex]\[
y - 4 = -4
\][/tex]
Solving for [tex]\(y\)[/tex]:
[tex]\[
y - 4 = -4 \\
y = -4 + 4 \\
y = 0
\][/tex]
Thus, the [tex]$y$[/tex]-coordinate of point [tex]\(D\)[/tex] before the translation was [tex]\(0\)[/tex].
Therefore, the [tex]$y$[/tex]-coordinate of point [tex]$D$[/tex] after the translation is:
[tex]\[
D^{\prime}(3.5, -4)
\][/tex]
So, the [tex]$y$[/tex]-coordinate is [tex]\(-4\)[/tex].
