Question

Two sets of transformations are given. Sketch the transformed unit square under each set of transformations. Are the transformations the same? Explain why/why not. (a) a reflection across the line \(y=\) \(x\) followed by a reflection across the \(x\) axis, compared to (b) a reflection across the the \(y\) axis, followed by a reflection across the line \(y=x\)

Short answer

The transformations result in the same final unit square.

Step-by-step solution

Understanding the original unit square

The original unit square has its vertices at (0,0), (1,0), (1,1), and (0,1). This means it is positioned in the first quadrant of the coordinate plane.

Transformation for set (a)

First, reflect the unit square across the line \(y = x\). The reflection across \(y = x\) swaps the x and y coordinates, so the vertices become (0,0), (0,1), (1,1), and (1,0). Next, reflect the resulting figure across the \(x\) axis, which negates the y-coordinates. The resulting vertices are (0,0), (0,-1), (1,-1), and (1,0).

Transformation for set (b)

First, reflect the unit square across the \(y\) axis, which negates the x-coordinates. The vertices become (0,0), (-1,0), (-1,1), and (0,1). Next, reflect the resulting figure across the line \(y = x\), swapping the x and y coordinates. The resulting vertices become (0,0), (0,-1), (1,-1), and (1,0).

Comparing both transformations

After performing both transformations, both sets result in the vertices being (0,0), (0,-1), (1,-1), and (1,0). This means that the final shape and position of the unit square after each sequence of transformations is the same.

Key concepts

Reflection

A reflection is a type of geometric transformation that flips an object over a specific line, known as the line of reflection. This transformation creates a mirror image of the original shape. Reflections can be performed across different lines, such as the x-axis, y-axis, or the line y = x.
For example, when reflecting a point across the x-axis, the x-coordinate remains the same, but the y-coordinate is negated. Similarly, reflecting across the line y = x swaps the x and y coordinates.

• Reflection across x-axis: (x, y) → (x, -y)
• Reflection across y-axis: (x, y) → (-x, y)
• Reflection across y = x: (x, y) → (y, x)

In our exercise, reflection plays a crucial role in transforming the unit square from its original position to the final outcome.

Coordinate Plane

The coordinate plane, also known as the Cartesian plane, is a two-dimensional surface on which positions are defined using pairs of numbers. These are called coordinates and are typically written as (x, y), where x is the horizontal distance from the origin and y is the vertical distance.

• Origin: The point (0,0) which is the intersection of the x-axis and y-axis.
• Quadrants: The plane is divided into four quadrants, each defined by the positive and negative values of x and y.

The original unit square, situated at (0,0), (1,0), (1,1), and (0,1), lies in the first quadrant. Understanding the coordinate plane is essential for visualizing the transformations and knowing how the unit square's position changes.

Transformation Sequence

A transformation sequence is a series of operations applied to a figure to change its position, orientation, or size. In geometry, these operations include translations, rotations, reflections, and dilations. Sequencing matters because applying transformations in different orders can result in different outcomes.
In the exercise, we observe two sequences of reflections applied to a unit square:

• Sequence (a): Reflection across y = x, then across the x-axis.
• Sequence (b): Reflection across the y-axis, then across y = x.

Despite starting differently, both sequences converge to the same final coordinates, illustrating that order can sometimes be rearranged without affecting the final result, depending on the operations involved.

Unit Square

A unit square is a simple geometric figure with sides of length one unit. It serves as a basic building block in geometry and is helpful for demonstrating transformations because of its regular shape and symmetry.
In the coordinate plane, the unit square can be represented by its vertices: (0,0), (1,0), (1,1), and (0,1). These coordinates provide a clear reference for visualizing transformations. After transformation, checking these points helps determine if the operation was performed correctly.
In our exercise, the unit square undergoes reflections. By tracking the vertices' changes through the transformations, we verify the sequence's effect on the figure's orientation and position.

Matrices in Geometry

Matrices in geometry are powerful tools used to perform and represent transformations. A matrix is a rectangular array of numbers that can encode translation, rotation, scaling, and reflection. Using matrices, transformations can be conducted in a systematic and consistent manner.
For reflections, specific matrices can simplify the process:

• Reflection across x-axis: \[\begin{bmatrix}1 & 0 \0 & -1\end{bmatrix}\]
• Reflection across y-axis: \[\begin{bmatrix}-1 & 0 \0 & 1\end{bmatrix}\]
• Reflection across y = x: \[\begin{bmatrix}0 & 1 \1 & 0\end{bmatrix}\]

By applying the appropriate matrices to the coordinates of the unit square, students can computationally achieve the same transformations discussed in the exercise and gain a deeper understanding of how reflections work within geometrical frameworks.