Let's analyze the reasoning of each student and determine who correctly translates the point [tex]\( N(-2, -4) \)[/tex] to [tex]\( N^{\prime}(2, 4) \)[/tex].
1. Raheem: Raheem suggests the rule [tex]\( (x \cdot (-1), y \cdot (-1)) \)[/tex].
- Applying this rule to the point [tex]\( N(-2, -4) \)[/tex]:
- [tex]\( x = -2 \)[/tex], so [tex]\( x \cdot (-1) = -2 \cdot (-1) = 2 \)[/tex]
- [tex]\( y = -4 \)[/tex], so [tex]\( y \cdot (-1) = -4 \cdot (-1) = 4 \)[/tex]
- The result is the point [tex]\( (2, 4) \)[/tex], which matches [tex]\( N^{\prime}(2, 4) \)[/tex].
2. Andrew: Andrew suggests the rule [tex]\( (x + 2, y + 4) \)[/tex].
- Applying this rule to the point [tex]\( N(-2, -4) \)[/tex]:
- [tex]\( x = -2 \)[/tex], so [tex]\( x + 2 = -2 + 2 = 0 \)[/tex]
- [tex]\( y = -4 \)[/tex], so [tex]\( y + 4 = -4 + 4 = 0 \)[/tex]
- The result is the point [tex]\( (0, 0) \)[/tex], which does not match [tex]\( N^{\prime}(2, 4) \)[/tex].
3. Lo: Lo suggests the rule [tex]\( (x + 4, y + 0) \)[/tex].
- Applying this rule to the point [tex]\( N(-2, -4) \)[/tex]:
- [tex]\( x = -2 \)[/tex], so [tex]\( x + 4 = -2 + 4 = 2 \)[/tex]
- [tex]\( y = -4 \)[/tex], so [tex]\( y + 0 = -4 + 0 = -4 \)[/tex]
- The result is the point [tex]\( (2, -4) \)[/tex], which does not match [tex]\( N^{\prime}(2, 4) \)[/tex].
4. Correct Rule: The suggested rule [tex]\( (x + 4, y + 8) \)[/tex].
- Applying this rule to the point [tex]\( N(-2, -4) \)[/tex]:
- [tex]\( x = -2 \)[/tex], so [tex]\( x + 4 = -2 + 4 = 2 \)[/tex]
- [tex]\( y = -4 \)[/tex], so [tex]\( y + 8 = -4 + 8 = 4 \)[/tex]
- The result is the point [tex]\( (2, 4) \)[/tex], which matches [tex]\( N^{\prime}(2, 4) \)[/tex].
Based on the analysis, the correct student who has found the rule that translates [tex]\( N(-2, -4) \)[/tex] to [tex]\( N^{\prime}(2, 4) \)[/tex] is Raheem.
Therefore, the answer is: Raheem.
