Question

PLEASE HELP!!!!! Sometimes two transformations, one performed after the other, have a nice description as a single transformation. For example, instead of translating 2 units up followed by translating 3 units up, we could simply translate 5 units up. Instead of rotating 20 degrees counterclockwise around the origin followed by rotating 80 degrees clockwise around the origin, we could simply rotate 60 degrees clockwise around the origin. Can you find a simple description of reflecting across the x-axis followed by reflecting across the y-axis?

Answer 1

Answer: A rotation of 180° around the origin.

Explanation:

When we have a point (x, y) and we want to reflect it over a given line, the distance between the point and the line remains invariant under the transformation.

This is equivalent as a rotation around a point, the "radius" of the circle that we are creating when we do a rotation is constant.

Now, let's analyze our case:

First a reflection over x-axis, and then a reflection over the y-axis.

This means that if our point starts in quadrant 1 then:

-The reflection over the x-axis will leave our point in quadrant 4.

-The reflection over the y-axis will leave our point in quadrant 3.

Now, if our point stated on quadrant 2.

-The reflection over the x-axis will leave our point in quadrant 3.

-The reflection over the y-axis will leave our point in quadrant 4.

So essentially, these transformations "move" our point to the opposite quadrant, such that the rules of the reflection are applied.

This means that:

The distance between our point and the x-axis does not change.

The distance between our point and the y-axis does not change.

But if the distances to both axes do not change, then the distance between our point and the origin also does not change.

This means that we can find a rotation that is equivalent to this transformation.

So we must look at a rotation that moves our point 2 quadrants.

a rotation that moves our point 2 quadrants will be:

A rotation of 180° around the origin.