Question

1. reflection across y = -x

Explanation:

1. Recall the transformation rule for reflection across the line \(y =-x\):

• The transformation rule for reflecting a point \((x,y)\) across the line \(y=-x\) is \((x,y)\to(-y,-x)\).

1. Assume the coordinates of the points:

• Let's assume \(F=( - 4,-2)\), \(G=( - 4,-1)\), \(H=( - 2,-1)\), \(I=( - 2,-3)\) (by counting the grid - squares).
• For point \(F=( - 4,-2)\):
• Using the rule \((x,y)\to(-y,-x)\), we substitute \(x = - 4\) and \(y=-2\). Then \(F'=(2,4)\).
• For point \(G=( - 4,-1)\):
• Substitute \(x=-4\) and \(y = - 1\) into the rule \((x,y)\to(-y,-x)\). We get \(G'=(1,4)\).
• For point \(H=( - 2,-1)\):
• Substitute \(x=-2\) and \(y=-1\) into the rule \((x,y)\to(-y,-x)\). We get \(H'=(1,2)\).
• For point \(I=( - 2,-3)\):
• Substitute \(x=-2\) and \(y=-3\) into the rule \((x,y)\to(-y,-x)\). We get \(I'=(3,2)\).

The new points after reflection across \(y =-x\) are \(F'=(2,4)\), \(G'=(1,4)\), \(H'=(1,2)\), \(I'=(3,2)\). You can then plot these points on the coordinate - plane to get the reflected figure.

Step1: Recall transformation rule

The rule for reflecting \((x,y)\) across \(y=-x\) is \((x,y)\to(-y,-x)\).

Step2: Find image of \(F\)

Given \(F=( - 4,-2)\), substitute \(x=-4,y = - 2\) into the rule: \((-(-2),-(-4))=(2,4)\).

Step3: Find image of \(G\)

Given \(G=( - 4,-1)\), substitute \(x=-4,y=-1\) into the rule: \((-(-1),-(-4))=(1,4)\).

Step4: Find image of \(H\)

Given \(H=( - 2,-1)\), substitute \(x=-2,y=-1\) into the rule: \((-(-1),-(-2))=(1,2)\).

Step5: Find image of \(I\)

Given \(I=( - 2,-3)\), substitute \(x=-2,y=-3\) into the rule: \((-(-3),-(-2))=(3,2)\).

Answer:

The new points are \(F'=(2,4)\), \(G'=(1,4)\), \(H'=(1,2)\), \(I'=(3,2)\)