Question

Max is trying to prove to his friend that two reflections, one across the x-axis and another across the y-axis, will not result in a reflection across the line y = x for a pre-image in quadrant II. His friend Josiah is trying to prove that a reflection across the x-axis followed by a reflection across the y-axis will result in a reflection across the line y = x for a pre-image in quadrant II. Which student is correct, and which statements below will help him prove his conjecture? Check all that apply

Max is correct.
Josiah is correct.
If one reflects a figure across the x-axis from quadrant II, the image will end up in quadrant III.
If one reflects a figure across the y-axis 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 x-axis, the points of the image can be found using the pattern (x, y) (x, –y).
If one reflects a figure across the y-axis, 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.

Answer 1

Answer:

  Max is correct

  If one reflects a figure across the x-axis, the points of the image can be found using the pattern (x, y) ⇒ (x, –y).

  If one reflects a figure across the y-axis, 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.

Step-by-step 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.

Answer 2

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 x-axis, the points of the image can be found using the pattern (x, y) ⇒ (x, –y).
• If one reflects a figure across the y-axis, 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:

$(x,y) \to (x,-y)$ ---- reflection across the x-axis

$(x,-y) \to (-x,-y)$  ---- followed by a reflection across the x-axis

So, the transformation rule is:

$(x,y) \to (-x,-y)$

The transformation rule for a reflection across the line y = x is:

$(x,y) \to (y,x)$

This means that:

Josiah's claim is incorrect, and Max is correct

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