Question

parallelogram rstu is the image of parallelogram rstu under a translation left 2 units and down 6 units followed by a rotation about the origin.
write the rules for the translation and rotation.
translation: (x, y) ↦ (\boxed{ }, \boxed{ })
rotation: (x, y) ↦ (\boxed{ }, \boxed{ })

Explanation:

Step1: Determine the translation rule

A translation left \(2\) units changes the \(x -\) coordinate: \(x\to x - 2\). A translation down \(6\) units changes the \(y -\) coordinate: \(y\to y-6\). So the translation rule is \((x,y)\to(x - 2,y - 6)\).
Let's check with a point. For example, take point \(S(-3,8)\). After translation: \(x=-3-2=-5\), \(y = 8-6 = 2\). But we need to consider the rotation.

Step2: Analyze the rotation

Let's use the general rotation formula. If we assume the rotation is \(180^{\circ}\) about the origin, the formula for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\).
Let's check with point \(S(-3,8)\). After translation: \((-3 - 2,8 - 6)=(-5,2)\). After rotation \((-(-5),-2)=(5,-2)\). Looking at the image, if we assume the correspondence.
Another check: take point \(R(-5,5)\). After translation \((-5-2,5 - 6)=(-7,-1)\). After \(180^{\circ}\) rotation \((7,1)\). But if we consider the composition of translation \((x,y)\to(x - 2,y - 6)\) followed by \(180^{\circ}\) rotation \((x,y)\to(-(x - 2),-(y - 6))=(-x + 2,-y+6)\)

Let's use two - step approach.
First, translation: \((x,y)\to(x-2,y - 6)\)
Second, rotation. Let's assume the rotation is \(180^{\circ}\) about the origin. The rule for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\). If we have a point \((a,b)\) after translation, and then rotate, if \(a=x - 2\) and \(b=y - 6\), the final rule is \((x,y)\to(-(x - 2),-(y - 6))=(-x + 2,-y + 6)\)

Answer:

Translation: \((x,y)\to(x - 2,y - 6)\)
Rotation: \((x,y)\to(-x+2,-y + 6)\)