Question

j(-4, 4), k(-3, 4), l(-1, 1), m(-4, 1)
reflect in the x-axis, and then rotate 180° about the origin.

Explanation:

Step1: Reflect over x - axis

The rule for reflecting a point \((x,y)\) over the \(x\) - axis is \((x,y)\to(x, - y)\).

• For point \(I(-4,4)\): Applying the rule, we get \(I'(-4,-4)\).
• For point \(K(-3,4)\): Applying the rule, we get \(K'(-3,-4)\).
• For point \(L(-1,1)\): Applying the rule, we get \(L'(-1,-1)\).
• For point \(M(-4,1)\): Applying the rule, we get \(M'(-4,-1)\).

Step2: Rotate \(180^{\circ}\) about the origin

The rule for rotating a point \((x,y)\) \(180^{\circ}\) about the origin is \((x,y)\to(-x,-y)\).

• For point \(I'(-4,-4)\): Applying the rule, we have \(-x = -(-4)=4\) and \(-y=-(-4) = 4\), so \(I''(4,4)\).
• For point \(K'(-3,-4)\): Applying the rule, we have \(-x=-(-3) = 3\) and \(-y=-(-4)=4\), so \(K''(3,4)\).
• For point \(L'(-1,-1)\): Applying the rule, we have \(-x=-(-1)=1\) and \(-y=-(-1) = 1\), so \(L''(1,1)\).
• For point \(M'(-4,-1)\): Applying the rule, we have \(-x=-(-4)=4\) and \(-y=-(-1)=1\), so \(M''(4,1)\).

Answer:

After reflection over the \(x\) - axis and \(180^{\circ}\) rotation about the origin, the new points are \(I''(4,4)\), \(K''(3,4)\), \(L''(1,1)\), \(M''(4,1)\)