Coordinate plane with plotted points and transformation instructions for geometric transformations.

I'll solve this problem step by step. The task is to apply a sequence of transformations to the given points and find their final coordinates.

Let me first understand what we're working with:

First, look closely:
- We have pre-image points: D(9,-3), E(6,-7), F(3,-3), G(5,-1)
- We need to apply three transformations in sequence: R_x=2° (reflection over the line x=2), then T_180° (rotation 180° about the origin), then T_<-10,-1> (translation by <-10, -1>)
- We need to fill in the table with coordinates after each transformation

I'll need to calculate the coordinates after each transformation step by step.

Let me start by applying the first transformation: reflection over the line x=2.

For reflection over x=2:
- The formula for reflecting a point (x,y) over the line x=a is (2a-x, y)
- So for x=2, the formula becomes (4-x, y)

Let me calculate the coordinates after the first transformation (D', E', F', G'):

For D(9,-3): D' = (4-9, -3) = (-5, -3)
For E(6,-7): E' = (4-6, -7) = (-2, -7)
For F(3,-3): F' = (4-3, -3) = (1, -3)
For G(5,-1): G' = (4-5, -1) = (-1, -1)

Now I'll apply the second transformation: rotation of 180° about the origin.
The formula for rotating a point (x,y) by 180° about the origin is (-x, -y).

Let me calculate the coordinates after the second transformation (D'', E'', F'', G''):

For D'(-5,-3): D'' = (5, 3)
For E'(-2,-7): E'' = (2, 7)
For F'(1,-3): F'' = (-1, 3)
For G'(-1,-1): G'' = (1, 1)

Finally, I'll apply the third transformation: translation by <-10, -1>.
This means we subtract 10 from the x-coordinate and subtract 1 from the y-coordinate.

Let me calculate the coordinates after the third transformation (D''', E''', F''', G'''):

For D''(5,3): D''' = (5-10, 3-1) = (-5, 2)
For E''(2,7): E''' = (2-10, 7-1) = (-8, 6)
For F''(-1,3): F''' = (-1-10, 3-1) = (-11, 2)
For G''(1,1): G''' = (1-10, 1-1) = (-9, 0)

Let me verify my calculations and organize them into the answer format.

Solution:

After applying the sequence of transformations R_x=2° → T_180° → T_<-10,-1> to the pre-image points, we get:

1. After first transformation (R_x=2°):
- D' = (-5, -3)
- E' = (-2, -7)
- F' = (1, -3)
- G' = (-1, -1)

2. After second transformation (T_180°):
- D'' = (5, 3)
- E'' = (2, 7)
- F'' = (-1, 3)
- G'' = (1, 1)

3. After third transformation (T_<-10,-1>):
- D''' = (-5, 2)
- E''' = (-8, 6)
- F''' = (-11, 2)
- G''' = (-9, 0)

These are the final coordinates that should be filled in the table.