Question

draw and write functions for all the reflectional symmetries (if any) in the regular octagon. reflection functions r_y(hwlgomiu) r_x(hwlgomiu)

Explanation:

Step1: Recall reflection - function rules

For a reflection of a point \((x,y)\) over the \(x\) - axis, the transformation is \((x,y)\to(x, - y)\). For a reflection over the \(y\) - axis, the transformation is \((x,y)\to(-x,y)\). In a regular octagon centered at the origin with vertices labeled as shown, assume the center of the octagon is at the origin \((0,0)\).
Let's assume a vertex \(P(x,y)\) of the octagon.

Step2: Reflection over the \(x\) - axis (\(R_x\))

If we have a vertex \(P(x,y)\) of the octagon, the reflection of \(P\) over the \(x\) - axis, denoted as \(R_x(P)\) will have coordinates \((x,-y)\). For example, if a vertex \(A\) has coordinates \((a,b)\), \(R_x(A)=(a, - b)\).

Step3: Reflection over the \(y\) - axis (\(R_y\))

If we have a vertex \(P(x,y)\) of the octagon, the reflection of \(P\) over the \(y\) - axis, denoted as \(R_y(P)\) will have coordinates \((-x,y)\). For example, if a vertex \(B\) has coordinates \((c,d)\), \(R_y(B)=(-c,d)\).
In general, for any vertex \((x,y)\) of the regular octagon:
The reflection function \(R_x(x,y)=(x, - y)\) and \(R_y(x,y)=(-x,y)\)

Answer:

The reflection function over the \(x\) - axis is \(R_x(x,y)=(x, - y)\) and the reflection function over the \(y\) - axis is \(R_y(x,y)=(-x,y)\)