Question

if pqrs is translated along $(x,y)\to(x + 3,y - 2)$, reflected in the x - axis, rotated $90^{circ}$ counterclockwise about the origin, and reflected in the y - axis, what are the coordinates of $pqrs$

Explanation:

Step1: Find the original coordinates

From the graph, \(P(-4,2)\), \(Q(-2,3)\), \(R(0,1)\), \(S(-2,-1)\)

Step2: Translation \((x,y)\to(x + 3,y-2)\)

For \(P\): \(x=-4+3=-1\), \(y = 2-2=0\), so \(P'(-1,0)\)
For \(Q\): \(x=-2 + 3=1\), \(y=3-2 = 1\), so \(Q'(1,1)\)
For \(R\): \(x=0+3=3\), \(y=1-2=-1\), so \(R'(3,-1)\)
For \(S\): \(x=-2+3 = 1\), \(y=-1-2=-3\), so \(S'(1,-3)\)

Step3: Reflection in the \(x -\)axis \((x,y)\to(x,-y)\)

For \(P'\): \(P''(-1,0)\) (since \(y = 0\))
For \(Q'\): \(Q''(1,-1)\)
For \(R'\): \(R''(3,1)\)
For \(S'\): \(S''(1,3)\)

Step4: Rotation \(90^{\circ}\) counter - clockwise about the origin \((x,y)\to(-y,x)\)

For \(P''\): \(P'''(0,-1)\)
For \(Q''\): \(Q'''(1,1)\)
For \(R''\): \(R'''(-1,3)\)
For \(S''\): \(S'''(-3,1)\)

Step5: Reflection in the \(y -\)axis \((x,y)\to(-x,y)\)

For \(P'''\): \(P''''(0,-1)\)
For \(Q'''\): \(Q''''(-1,1)\)
For \(R'''\): \(R''''(1,3)\)
For \(S'''\): \(S''''(3,1)\)

Answer:

\(P''''(0,-1)\)
\(Q''''(-1,1)\)
\(R''''(1,3)\)
\(S''''(3,1)\)