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? Select the three correct answers.
Max is correct.
Josiah is correct.
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 y = x should give.
If one reflects a figure first across the x-axis from quadrant II then reflects 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 across the line y = x will have the coordinates of (-y, x).
Max is correct.
Josiah is correct.
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 y = x should give.
If one reflects a figure first across the x-axis from quadrant II then reflects 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 across the line y = x will have the coordinates of (-y, x).
Answers
GPT-4o mini
To analyze the situation, let's break down the reflections and see which student's claim holds true.
1. **Reflection across the x-axis:** This reflection changes a point \((x, y)\) to \((x, -y)\).
2. **Reflection across the y-axis:** This reflection changes a point \((x, y)\) to \((-x, y)\).
If we reflect a point in the second quadrant, say \((-a, b)\) where \(a > 0\) and \(b > 0\), first across the x-axis, we get:
\[
(-a, b) \rightarrow (-a, -b) \text{ (now in quadrant III)}
\]
Next, reflecting this point across the y-axis gives:
\[
(-a, -b) \rightarrow (a, -b) \text{ (now in quadrant IV)}
\]
The result after both reflections \((a, -b)\) does not correspond to a point obtained by reflecting across the line \(y = x\), which would require that the point would become \((-b, a)\) for a point originally located at \((-a, b)\).
Now let's evaluate which statements support Max's claim and refute Josiah's assertion:
1. **Max is correct.** - This is true.
2. **Josiah is correct.** - This is false.
3. **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 y = x should give.** - This statement supports Max's claim. True.
4. **If one reflects a figure first across the x-axis from quadrant II then reflects across the y-axis from quadrant III, the image will end up in quadrant IV.** - This statement also supports Max's claim regarding the final location of the image after two reflections. True.
5. **A figure that is reflected from quadrant II to quadrant IV across the line y = x will have the coordinates of (-y, x).** - This statement does not directly support or refute either student's argument about the specific reflections conducted and their results. False or irrelevant.
Based on the analysis above, the three correct statements to support Max's claim are:
1. Max is correct.
2. 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 y = x should give.
3. If one reflects a figure first across the x-axis from quadrant II then reflects across the y-axis from quadrant III, the image will end up in quadrant IV.
1. **Reflection across the x-axis:** This reflection changes a point \((x, y)\) to \((x, -y)\).
2. **Reflection across the y-axis:** This reflection changes a point \((x, y)\) to \((-x, y)\).
If we reflect a point in the second quadrant, say \((-a, b)\) where \(a > 0\) and \(b > 0\), first across the x-axis, we get:
\[
(-a, b) \rightarrow (-a, -b) \text{ (now in quadrant III)}
\]
Next, reflecting this point across the y-axis gives:
\[
(-a, -b) \rightarrow (a, -b) \text{ (now in quadrant IV)}
\]
The result after both reflections \((a, -b)\) does not correspond to a point obtained by reflecting across the line \(y = x\), which would require that the point would become \((-b, a)\) for a point originally located at \((-a, b)\).
Now let's evaluate which statements support Max's claim and refute Josiah's assertion:
1. **Max is correct.** - This is true.
2. **Josiah is correct.** - This is false.
3. **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 y = x should give.** - This statement supports Max's claim. True.
4. **If one reflects a figure first across the x-axis from quadrant II then reflects across the y-axis from quadrant III, the image will end up in quadrant IV.** - This statement also supports Max's claim regarding the final location of the image after two reflections. True.
5. **A figure that is reflected from quadrant II to quadrant IV across the line y = x will have the coordinates of (-y, x).** - This statement does not directly support or refute either student's argument about the specific reflections conducted and their results. False or irrelevant.
Based on the analysis above, the three correct statements to support Max's claim are:
1. Max is correct.
2. 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 y = x should give.
3. If one reflects a figure first across the x-axis from quadrant II then reflects across the y-axis from quadrant III, the image will end up in quadrant IV.