7) reflection across y = -x

Answer

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)).
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.

Explanation:

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))