Answer :
To solve this problem, we need to follow these steps:
Step 1: Draw the quadrilateral ABCD on the graph with the given vertices.
Step 2: Reflect the quadrilateral ABCD on the line y = 4 to obtain A'B'C'D'.
Step 3: Reflect the quadrilateral A'B'C'D' on the line y = 0 to obtain A"B"C"D".
Step 4: Find the coordinates of A", B", C", and D".
Step 5: Identify the single transformation that these two reflections represent.
Here's the step-by-step solution with the figure:
```
y
10 |
9 |
8 | A(4,8)
7 |
6 | B(6,6) D(3,6)
5 |
4 |-----------------------------
3 | C'(4,4) B'(6,2)
2 | A'(4,0) D'(3,2)
1 |
0 |-----------------------------
-1 | A"(4,0) D"(3,-2)
-2 | C"(4,-4) B"(6,-2)
-3 |
+-----------------------------
0 1 2 3 4 5 6 7 8
```
The coordinates of the vertices after the two reflections are:
A" = (4, 0)
B" = (6, -2)
C" = (4, -4)
D" = (3, -2)
These two reflections (reflection on y = 4 and then reflection on y = 0) represent a single transformation called a rotation of 180 degrees about the origin (0, 0).
Therefore, the single transformation that these two reflections represent is a rotation of 180 degrees about the origin.
Step 1: Draw the quadrilateral ABCD on the graph with the given vertices.
Step 2: Reflect the quadrilateral ABCD on the line y = 4 to obtain A'B'C'D'.
Step 3: Reflect the quadrilateral A'B'C'D' on the line y = 0 to obtain A"B"C"D".
Step 4: Find the coordinates of A", B", C", and D".
Step 5: Identify the single transformation that these two reflections represent.
Here's the step-by-step solution with the figure:
```
y
10 |
9 |
8 | A(4,8)
7 |
6 | B(6,6) D(3,6)
5 |
4 |-----------------------------
3 | C'(4,4) B'(6,2)
2 | A'(4,0) D'(3,2)
1 |
0 |-----------------------------
-1 | A"(4,0) D"(3,-2)
-2 | C"(4,-4) B"(6,-2)
-3 |
+-----------------------------
0 1 2 3 4 5 6 7 8
```
The coordinates of the vertices after the two reflections are:
A" = (4, 0)
B" = (6, -2)
C" = (4, -4)
D" = (3, -2)
These two reflections (reflection on y = 4 and then reflection on y = 0) represent a single transformation called a rotation of 180 degrees about the origin (0, 0).
Therefore, the single transformation that these two reflections represent is a rotation of 180 degrees about the origin.