Question

what transformation transforms $(p,q)$ to $(q,p)$?

○ a reflection over $y = x$
○ a reflection over the $x$-axis
○ a reflection over the $y$-axis
○ a rotation of $90^{circ}$ about the origin

Explanation:

• Reflection over \(y = x\):
• The rule for reflection over the line \(y=x\) is \((x,y)\to(y,x)\). When we apply this to the point \((p,q)\), we get \((q,p)\).
• Reflection over the \(x -\)axis:
• The rule for reflection over the \(x -\)axis is \((x,y)\to(x, - y)\). For the point \((p,q)\), it would transform to \((p,-q)\), not \((q,p)\).
• Reflection over the \(y -\)axis:
• The rule for reflection over the \(y -\)axis is \((x,y)\to(-x,y)\). For the point \((p,q)\), it would transform to \((-p,q)\), not \((q,p)\).
• Rotation of \(90^{\circ}\) about the origin:
• The rule for a \(90^{\circ}\) rotation about the origin is \((x,y)\to(-y,x)\). For the point \((p,q)\), it would transform to \((-q,p)\), not \((q,p)\).

Answer:

a reflection over \(y = x\)