Question

dentify a sequence of rigid transformations that would map ABCD → A" B" C'" D". Give your answer in the form of a composition transformation mapping.
& point)
• (х, у) → (-х, у - 4)
O (2.3) - (-2- 4,3)
0 （2-3）→（2-3+4）
0（2:3） → （2，-9-4）

Answer 1

To determine the sequence of rigid transformations that maps quadrilateral ABCD to A" B" C'" D", we need to analyze the provided transformations.

The transformation you've mentioned is:

- \((x, y) \to (-x, y - 4)\)

This transformation can be broken down into two separate transformations:

1. **Reflection** across the y-axis: This changes \((x, y)\) to \((-x, y)\).
2. **Translation** downward: This changes \((-x, y)\) to \((-x, y - 4)\).

Let's denote the points of quadrilateral ABCD as \(A(x_A, y_A)\), \(B(x_B, y_B)\), \(C(x_C, y_C)\), and \(D(x_D, y_D)\).

### Step 1: Reflection across the y-axis
The reflection across the y-axis transforms each point as follows:
- \(A'(-x_A, y_A)\)
- \(B'(-x_B, y_B)\)
- \(C'(-x_C, y_C)\)
- \(D'(-x_D, y_D)\)

### Step 2: Translation downward by 4 units
Next, we apply the translation downward by 4 units:
- \(A''(-x_A, y_A - 4)\)
- \(B''(-x_B, y_B - 4)\)
- \(C''(-x_C, y_C - 4)\)
- \(D''(-x_D, y_D - 4)\)

### Composition Transformation Mapping
To express the transformation from \(ABCD\) to \(A''B''C''D''\) in terms of composition, we can write it as:

\[
A''B''C''D'' = T_{(-4)} \circ R_{y} (ABCD)
\]

Where:
- \(R_{y}\) is the reflection across the y-axis, defined as \(R_{y}(x, y) = (-x, y)\),
- \(T_{(-4)}\) is the translation downward, defined as \(T_{(-4)}(x, y) = (x, y - 4)\).

Thus, the overall composition would look like this:

\[
(x, y) \xrightarrow{R_{y}} (-x, y) \xrightarrow{T_{(-4)}} (-x, y - 4)
\]

In terms of a single expression, you can also write this transformation as:

\[
(x, y) \mapsto (-x, y - 4)
\]

This sequence of transformations will effectively map quadrilateral \(ABCD\) to \(A''B''C''D''\).