To complete this translation, we need to apply the given translation rule [tex]\((x, y) \rightarrow (x+2, y-2)\)[/tex] to each of the original coordinates. This means we will add 2 to the [tex]\( x \)[/tex]-coordinate and subtract 2 from the [tex]\( y \)[/tex]-coordinate for each point. Let's go through this step by step:
1. First point [tex]\( C(-5, -1) \)[/tex]:
[tex]\[
\begin{aligned}
x & = -5 + 2 = -3 \\
y & = -1 - 2 = -3
\end{aligned}
\][/tex]
So, [tex]\( C' \)[/tex] becomes [tex]\( (-3, -3) \)[/tex].
2. Second point [tex]\( D(-4, 3) \)[/tex]:
[tex]\[
\begin{aligned}
x & = -4 + 2 = -2 \\
y & = 3 - 2 = 1
\end{aligned}
\][/tex]
So, [tex]\( D' \)[/tex] becomes [tex]\( (-2, 1) \)[/tex].
3. Third point [tex]\( E(-3, 2) \)[/tex]:
[tex]\[
\begin{aligned}
x & = -3 + 2 = -1 \\
y & = 2 - 2 = 0
\end{aligned}
\][/tex]
So, [tex]\( E' \)[/tex] becomes [tex]\( (-1, 0) \)[/tex].
4. Fourth point [tex]\( F(-2, 0) \)[/tex]:
[tex]\[
\begin{aligned}
x & = -2 + 2 = 0 \\
y & = 0 - 2 = -2
\end{aligned}
\][/tex]
So, [tex]\( F' \)[/tex] becomes [tex]\( (0, -2) \)[/tex].
Putting this all together, we get the translated coordinates:
[tex]\[
\begin{aligned}
C' & (-3, -3) \\
D' & (-2, 1) \\
E' & (-1, 0) \\
F' & (0, -2)
\end{aligned}
\][/tex]
