Problem 28 - Algebra 2

Question

For Exercises \(28-31,\) use rectangle \(A B C D\) with vertices \(A(-4,4), B(4,4),\) \(C(4,-4),\) and \(D(-4,-4) .\) Find the coordinates of the image in matrix form after a reflection over the \(x\) -axis followed by a reflection over the \(y\) -axis.

Step-by-Step Solution

Answer

The image's coordinates are \((4, -4), (-4, -4), (-4, 4), (4, 4)\).

Step 1: Identify Initial Vertices

The given rectangle has vertices at specific coordinates. These are: \( A(-4, 4) \), \( B(4, 4) \), \( C(4, -4) \), and \( D(-4, -4) \).

Step 2: Reflection Over the x-axis

Reflecting a point \((x, y)\) over the \(x\)-axis results in the point \((x, -y)\). Applying this transformation, the new vertices become: \( A'(-4, -4) \), \( B'(4, -4) \), \( C'(4, 4) \), and \( D'(-4, 4) \).

Step 3: Reflection Over the y-axis

Reflecting a point \((x, y)\) over the \(y\)-axis results in the point \((-x, y)\). Applying this transformation to the previously reflected points, the new vertices are: \( A''(4, -4) \), \( B''(-4, -4) \), \( C''(-4, 4) \), and \( D''(4, 4) \).

Step 4: Matrix Form

To express the coordinates in matrix form, arrange them as rows in a matrix: \[ \begin{bmatrix} 4 & -4 \ -4 & -4 \ -4 & 4 \ 4 & 4 \end{bmatrix} \]. This matrix represents the coordinates of the image after both reflections.

Key Concepts

Reflection over x-axis

When dealing with **reflection over the x-axis**, we essentially mirror the figure across this horizontal line. This means every point shifts vertically while their horizontal positions remain unchanged.
To achieve this mathematically, for each point \( (x, y) \), the transformation becomes \( (x, -y) \).
In other words:

• The x-coordinates of the points do not change.
• The y-coordinates of the points become their negatives.

For example, consider a point A at \( (3, 5) \) on the plane. Upon reflecting over the x-axis, A becomes \( (3, -5) \).
This operation effectively flips the point to the opposite side of the x-axis.

Reflection over y-axis

For a **reflection over the y-axis**, the transformation mirrors the shape across the vertical line, changing our perspective to the opposite side.
In mathematical terms, the coordinates of each point \( (x, y) \) transform to \( (-x, y) \).

• The y-coordinates remain the same.
• The x-coordinates switch to their opposites.

Let's take a point B located at \( (6, -2) \). When you reflect it over the y-axis, B shifts to \( (-6, -2) \).
This reflection operation effectively swaps the point's location from one side of the y-axis to the other while maintaining its vertical position.

Matrix representation

**Matrix representation** provides a concise and efficient way to handle multiple points and transformations in geometry. In this context, a matrix lists points as rows or columns.
This is particularly useful for reflection transformations as it allows for simplicity and uniformity.
For example, for our given rectangle, reflection transformations can be applied by arranging the coordinates \( (x, y) \) into a matrix:
\[ \begin{bmatrix} x_1 & y_1 \ x_2 & y_2 \ \vdots & \vdots \ x_n & y_n \end{bmatrix} \]

• Applying the transformation uniformly across all rows reflects each point.
• Resultant matrices show transformed coordinates.

This approach neatly organizes the coordinates for further analyses or visualization, streamlining the processing of geometric transformations.

Geometric transformations

**Geometric transformations** involve altering the position or size of a shape. They are fundamental in geometry for studying figures' properties and relationships.
Reflections, like those over the x-axis and y-axis, are a type of geometric transformation specifically aimed at flipping a shape over a designated axis.

• Other transformations include translations, rotations, and scaling.
• Maintaining the figure's shape and size is a key aspect while changing its orientation or position.

By understanding these transformations, one can visualize the dynamic and interactive nature of geometric figures across different conditions.
They are not merely mathematical exercises; they also permit practical applications in fields like computer graphics, engineering, and art.