Question

the coordinates of the vertices of parallelogram abcd are a(-2,2), b(3,5), c(4,2), and d(-1,-1). state the coordinates of the vertices of parallelogram abcd that result from the transformation ( r_{y - axis}circ t_{2,-3} ). the use of the set of axes below is optional.

Explanation:

Step1: Translation formula

The translation \(T_{2,-3}\) has the rule \((x,y)\to(x + 2,y-3)\).
For point \(A(-2,2)\): \(x=-2,y = 2\), after translation \(A'(x_1,y_1)=(-2 + 2,2-3)=(0,-1)\)
For point \(B(3,5)\): \(x = 3,y=5\), after translation \(B'(x_2,y_2)=(3 + 2,5-3)=(5,2)\)
For point \(C(4,2)\): \(x = 4,y=2\), after translation \(C'(x_3,y_3)=(4 + 2,2-3)=(6,-1)\)
For point \(D(-1,-1)\): \(x=-1,y=-1\), after translation \(D'(x_4,y_4)=(-1 + 2,-1-3)=(1,-4)\)

Step2: Reflection formula

The reflection \(r_{y - axis}\) has the rule \((x,y)\to(-x,y)\)
For \(A'(0,-1)\): \(x = 0,y=-1\), after reflection \(A''(x_1',y_1')=(0,-1)\)
For \(B'(5,2)\): \(x = 5,y=2\), after reflection \(B''(x_2',y_2')=(-5,2)\)
For \(C'(6,-1)\): \(x = 6,y=-1\), after reflection \(C''(x_3',y_3')=(-6,-1)\)
For \(D'(1,-4)\): \(x = 1,y=-4\), after reflection \(D''(x_4',y_4')=(-1,-4)\)

Answer:

\(A''(0,-1)\), \(B''(-5,2)\), \(C''(-6,-1)\), \(D''(-1,-4)\)