Question

In Exercises 11-14, 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 horizontal reflection across the \(y\) axis, followed by a vertical reflection across the \(x\) axis, compared to (b) a counterclockise rotation of \(180^{\circ}\)

Short answer

Both transformation sets produce the same outcome: a mapping of points to (-x, -y), so they are equivalent.

Step-by-step solution

Understand the initial transformation

We start with two operations for transformation set (a). First, there's a horizontal reflection across the y-axis. This means that any point (x, y) on the unit square will be reflected to (-x, y). Next, there is a vertical reflection across the x-axis, turning the point (-x, y) into (-x, -y). This completes all transformations for option (a).

Analyze second transformation

For transformation set (b), there is a single operation: a counterclockwise rotation of 180°. When this rotation is applied to any point (x, y), it becomes (-x, -y). It is equivalent to placing the point directly across from its starting location on both axes.

Compare transformations

Now, we compare the outcomes of both transformations. In transformation set (a), through two reflections, any point (x, y) turns into (-x, -y) after both reflections. Similarly, a 180° rotation also results in transforming any point (x, y) into (-x, -y) in transformation set (b).

Conclusion

Since both transformation sets (a and b) result in each point (x, y) being mapped to (-x, -y), the transformations result in the same transformed unit square and are equivalent.

Key concepts

Reflections

Reflections are fundamental geometric transformations that flip a shape over a specific line called the line of reflection. When reflecting a point across the y-axis, the x-coordinate switches signs, while the y-coordinate remains unchanged. Thus, the point (x, y) becomes (-x, y). Another common reflection is across the x-axis, where the x-coordinate stays the same, and the y-coordinate changes sign, changing (x, y) to (x, -y). Reflections can change the orientation of shapes but preserve distances and angles between points.

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

Combining these reflections in the exercise results in each point (x, y) becoming (-x, -y), the same outcome as a 180° rotation. This indicates that sequential reflections across perpendicular axes effectively flip the shape twice, giving it a half-turn. Contemplating how reflections work helps us understand why they might sometimes offer the same result as other transformations.

Rotations

Rotations are a type of geometric transformation that involves turning a shape around a fixed point or origin. For a rotation by 180 degrees, every point (x, y) becomes its opposite (-x, -y). This is as though the shape has been turned halfway around a circle.
If we consider a rotation of 180 degrees, it is effectively moving the shape to its diametrically opposite position. The operation doesn't change the shape or size but alters its position and orientation. Just like reflections, rotations preserve the distance and angles between points.
More generally, rotations can be about any angle, and for 180-degree rotations, the rule is straightforward:

• Rotation by 180°: (x, y) → (-x, -y)

Understanding these transformations helps to see why a 180-degree rotation achieves the same final positioning as two sequential perpendicular reflections, as explored in this exercise.

Geometric Transformations

Geometric transformations include various operations that alter the position, orientation, or size of shapes on a coordinate plane. Some of the most common transformations are reflections, rotations, translations, and scalings. These transformations maintain certain properties, such as:

• Shape: The figure's basic form remains constant.
• Distance: Points in the figure remain the same distance from each other.
• Angles: Angles within the shapes are preserved.

In the case of this exercise, both reflections and rotations have been used to transform a unit square. Examples like these reveal how different types of transformations can sometimes produce identical outcomes, as with the equivalent results of two reflections and one rotation. Understanding various transformations provides a fundamental grasp of spatial reasoning and its mathematical properties.

Unit Square

A unit square is a square with all sides of length 1, usually positioned with its bottom-left corner at the origin (0,0) on a coordinate plane. The vertices of a unit square in standard position are at the coordinates (0, 0), (1, 0), (1, 1), and (0, 1).
The unit square is often used in geometric problems because of its simplicity and ease of manipulation. It's a basic building block that helps visualize the effects of transformations. Through transformations like reflections and rotations, the shape of the unit square is invariant, meaning the relative geometry stays the same, though its position may shift.
By applying vertical and horizontal reflections or a 180-degree rotation, as in this exercise, one can track how each point on the unit square is moved. This demonstrates how transformations interact with the square's geometry and familiarizes learners with concepts via practical examples.