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 will help him prove his conjecture? choose three correct answers.
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.
max is correct.
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).

Explanation:

Step1: Understand reflection rules

• Reflection across the \(x -\)axis: \((x,y)\to(x, - y)\)
• Reflection across the \(y -\)axis: \((x,y)\to(-x,y)\)
• Reflection across the line \(y = x\): \((x,y)\to(y,x)\)

Step2: Analyze each statement

• Statement 1: Josiah is correct

Let's assume a general point \((x,y)\). First reflection across the \(x -\)axis gives \((x,-y)\), then reflection across the \(y -\)axis gives \((-x,-y)\). The mapping rule is \((x,y)\to(-x,-y)\), not \((y,x)\). So this statement is incorrect.

• Statement 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

As above, two - step reflection ( \(x -\)axis then \(y -\)axis) gives \((-x,-y)\), reflection across \(y = x\) gives \((y,x)\). So this statement is correct as it points out the wrong mapping rule for the two - step reflection compared to reflection across \(y=x\).

• Statement 3: Max is correct

A figure reflected from quadrant II to quadrant IV across the line \(y = x\) has coordinates \((-y,x)\). Let \((x,y)\) be a point in quadrant II (\(x\lt0,y>0\)). After reflection across \(y = x\), the point is \((y,x)\) which is in quadrant IV (\(y>0,x\lt0\) when considering the coordinate transformation). So this statement is correct.

• Statement 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

Let \((x,y)\) be in quadrant II (\(x\lt0,y > 0\)). Reflection across \(x -\)axis: \((x,-y)\) (quadrant III as \(x\lt0,-y\lt0\)). Reflection across \(y -\)axis: \((-x,-y)\) (quadrant IV as \(-x>0,-y\lt0\)). So this statement is correct.

Answer:

• 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
• Max is correct.
• 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.