Question

What type of transformation has the same effect as each composition of transformations?

translation (x,y) ---> (x+4,y) followed by a reflection across the line y=-4

Translation (x,y)---> (x+4,y+8) followed by (x,y)---> (x-2,y+9)

Reflection across the line y=7, and then across the line y=3

Reflection across the line y=x, and then across the line y=2x+5

These are all separate problems, not one.

Answer 1

The transformations can be simplified to a single type: the first is a glide reflection; the second is a translation; the third is a translation; and the fourth is a rotation.

Step-by-step explanation:

Each of the four transformation sequences mentioned can be evaluated to determine the overall effect and can be simplified to a single transformation:

1. A translation followed by a reflection across a line parallel to the translation vector results in a glide reflection, where the object is reflected and translated along the direction of the reflection axis.
2. Two translations in sequence simply result in a single translation that combines the effects of both individual translations.
3. Reflecting an object across two parallel lines results in a translation perpendicular to those lines, with the distance being twice the distance between the lines.
4. Reflecting an object across two intersecting lines results in a rotation around the point of intersection, with the angle of rotation being twice the angle between the two lines.

For applying these principles:

• The first transformation sequence is a glide reflection.
• The second results in a single translation by adding the effects of both translations.
• The third sequence leads to a translation since the lines y=7 and y=3 are parallel.
• The fourth sequence amounts to a rotation around the point of intersection of the lines y=x and y=2x+5.

Answer 2

identify the type of transformation is reflection