Max is trying to prove to his friend that two reflections, one across the xaxis and another across the yaxis, will not result in a reflection across the line
Question
Max is correct.
Josiah is correct.
If one reflects a figure across the xaxis from quadrant II, the image will end up in quadrant III.
If one reflects a figure across the yaxis from quadrant III, the image will end up in quadrant IV.
A figure that is reflected from quadrant II to quadrant IV will be reflected across the line y = x.
If one reflects a figure across the xaxis, the points of the image can be found using the pattern (x, y) (x, –y).
If one reflects a figure across the yaxis, the points of the image can be found using the pattern (x, y) (–x, y).
Taking the result from the first reflection (x, –y) and applying the second mapping rule will result in (–x, –y), not (y, x), which reflecting across the line should give.
2 Answer

1. User Answers sqdancefan
Answer:
Max is correct
If one reflects a figure across the xaxis, the points of the image can be found using the pattern (x, y) ⇒ (x, –y).
If one reflects a figure across the yaxis, the points of the image can be found using the pattern (x, y) ⇒ (–x, y).
Taking the result from the first reflection (x, –y) and applying the second mapping rule will result in (–x, –y), not (y, x), which reflecting across the line should give.
Stepbystep explanation:
The answer above pretty well explains it.
The net result of the two reflections will be that any figure will retain its orientation (CW or CCW order of vertices). It is equivalent to a rotation by 180°. The single reflection across the line y=x will reverse the orientation (CW ⇔ CCW). They cannot be equivalent.

2. User Answers MrRoyal
When a point is reflected, it must be reflected across a line.
The true statements are:
 Max is correct
 If one reflects a figure across the xaxis, the points of the image can be found using the pattern (x, y) ⇒ (x, –y).
 If one reflects a figure across the yaxis, the points of the image can be found using the pattern (x, y) ⇒ (–x, y).
 Taking the result from the first reflection (x, –y) and applying the second mapping rule will result in (–x, –y), not (y, x), which reflecting across the line should give.
For Josiah's claim, we have:
[tex](x,y) \to (x,y)[/tex]  reflection across the xaxis
[tex](x,y) \to (x,y)[/tex]  followed by a reflection across the xaxis
So, the transformation rule is:
[tex](x,y) \to (x,y)[/tex]
The transformation rule for a reflection across the line y = x is:
[tex](x,y) \to (y,x)[/tex]
This means that:
Josiah's claim is incorrect, and Max is correct
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