Question

1 find the following reflections of the point (2, -8) a. $r_{x-axis}$ b. $r_{y-axis}$ c. $r_{y=x}$ d. $r_{y=-x}$

Explanation:

Step1: Recall reflection rules

• Reflection over \( x \)-axis: \((x, y) \to (x, -y)\)
• Reflection over \( y \)-axis: \((x, y) \to (-x, y)\)
• Reflection over \( y = x \): \((x, y) \to (y, x)\)
• Reflection over \( y = -x \): \((x, y) \to (-y, -x)\)

Step2: Apply to point \((2, -8)\)

Part A: \( R_{x\text{-axis}} \)

Using rule \((x, y) \to (x, -y)\), substitute \( x = 2 \), \( y = -8 \):
\((2, -(-8)) = (2, 8)\)

Part B: \( R_{y\text{-axis}} \)

Using rule \((x, y) \to (-x, y)\), substitute \( x = 2 \), \( y = -8 \):
\((-2, -8)\)

Part C: \( R_{y = x} \)

Using rule \((x, y) \to (y, x)\), substitute \( x = 2 \), \( y = -8 \):
\((-8, 2)\)

Part D: \( R_{y = -x} \)

Using rule \((x, y) \to (-y, -x)\), substitute \( x = 2 \), \( y = -8 \):
\(-y = -(-8) = 8\), \(-x = -2\), so \((8, -2)\)

Answer:

A. \( R_{x\text{-axis}} \): \(\boldsymbol{(2, 8)}\)
B. \( R_{y\text{-axis}} \): \(\boldsymbol{(-2, -8)}\)
C. \( R_{y = x} \): \(\boldsymbol{(-8, 2)}\)
D. \( R_{y = -x} \): \(\boldsymbol{(8, -2)}\)