Question

graph the image of square uvwx after the following sequence of transformations: reflection across the line y = 3 rotation 180° counterclockwise around the origin

Explanation:

Step1: Find coordinates of square vertices

Assume \(U=( - 8,0)\), \(V=( - 8,4)\), \(W=( - 12,4)\), \(X=( - 12,0)\)

Step2: Reflection across \(y = 3\)

The formula for reflecting a point \((x,y)\) across the line \(y = k\) is \((x,2k - y)\). Here \(k = 3\).
For \(U=( - 8,0)\), the new - point \(U_1=(-8,2\times3 - 0)=(-8,6)\)
For \(V=( - 8,4)\), the new - point \(V_1=(-8,2\times3 - 4)=(-8,2)\)
For \(W=( - 12,4)\), the new - point \(W_1=(-12,2\times3 - 4)=(-12,2)\)
For \(X=( - 12,0)\), the new - point \(X_1=(-12,2\times3 - 0)=(-12,6)\)

Step3: Rotation 180° counter - clockwise around the origin

The formula for rotating a point \((x,y)\) 180° counter - clockwise around the origin is \((-x,-y)\)
For \(U_1=(-8,6)\), the new - point \(U_2=(8,-6)\)
For \(V_1=(-8,2)\), the new - point \(V_2=(8,-2)\)
For \(W_1=(-12,2)\), the new - point \(W_2=(12,-2)\)
For \(X_1=(-12,6)\), the new - point \(X_2=(12,-6)\)

Step4: Graph the new square

Plot the points \(U_2=(8,-6)\), \(V_2=(8,-2)\), \(W_2=(12,-2)\), \(X_2=(12,-6)\) and connect them to form the new square.

Answer:

Graph the square with vertices \((8,-6)\), \((8,-2)\), \((12,-2)\), \((12,-6)\) on the coordinate - plane.