Question

Let α be the transformation on R² given by α(x,y)=(−x,y+2). (a) Using the definition of a glide reflection, explain why α is a glide reflection. (b) Find three reflections μ₁, μ₂, μ₃ such that α = μ₃∘μ₂∘μ₁.

Answer 1

The transformation α is a glide reflection explained by a reflection and vertical translation. To find three reflections μ₁, μ₂, and μ₃ that compose α, we reflect across the y-axis, the x-axis, and the y-axis again respectively. The resulting transformation is α = (x, -y-2).

Step-by-step explanation:

(a) A glide reflection is a transformation that combines a reflection and a translation. It is defined by the composition of these two transformations. In this case, the transformation α consists of a reflection across the y-axis followed by a vertical translation of 2 units in the positive direction. The reflection across the y-axis flips the x-coordinate of a point, while the translation shifts the point vertically.

(b) To find three reflections μ₁, μ₂, μ₃ such that α = μ₃∘μ₂∘μ₁, we need to manipulate the given transformation α.

Let's start with μ₁ as the reflection across the y-axis. This reflection will change the sign of the x-coordinate, so μ₁(x,y) = (-x, y+2).

Next, let μ₂ be the reflection across the x-axis. This reflection will change the sign of the y-coordinate, so μ₂(-x, y+2) = (-x, -y-2).

Finally, let μ₃ be the reflection across the y-axis again. This reflection will change the sign of the x-coordinate, so μ₃(-x, -y-2) = (x, -y-2).

Putting it all together, α = μ₃∘μ₂∘μ₁ = (x, -y-2).