Question

how would you reflect the following points over the x–axis, the y–axis, and the line x = 3? points: a (2, 4) b (5, 2) c (3, −1) explain your process. justify your thinking with clear mathematical reasoning.

Explanation:

Step 1: Recall Reflection Rules

• Reflection over x - axis: For a point \((x,y)\), the reflected point is \((x, -y)\) (the x - coordinate remains the same, the y - coordinate is negated).
• Reflection over y - axis: For a point \((x,y)\), the reflected point is \((-x,y)\) (the y - coordinate remains the same, the x - coordinate is negated).
• Reflection over the line \(x = a\): For a point \((x,y)\), the reflected point is \((2a - x,y)\) (the distance between \(x\) and \(a\) is the same as the distance between the new x - coordinate and \(a\), so \(|x - a|=|x' - a|\), which gives \(x'=2a - x\) when solving for \(x'\)). Here \(a = 3\), so the formula is \((6 - x,y)\).

Step 2: Reflect Point A(2,4)

• Over x - axis: Using \((x,-y)\), substitute \(x = 2,y = 4\). The reflected point \(A_x=(2,-4)\).
• Over y - axis: Using \((-x,y)\), substitute \(x = 2,y = 4\). The reflected point \(A_y=(-2,4)\).
• Over \(x = 3\): Using \(x'=6 - x,y' = y\), substitute \(x = 2,y = 4\). \(x'=6 - 2=4\), so \(A_{x = 3}=(4,4)\).

Step 3: Reflect Point B(5,2)

• Over x - axis: Using \((x,-y)\), substitute \(x = 5,y = 2\). The reflected point \(B_x=(5,-2)\).
• Over y - axis: Using \((-x,y)\), substitute \(x = 5,y = 2\). The reflected point \(B_y=(-5,2)\).
• Over \(x = 3\): Using \(x'=6 - x,y' = y\), substitute \(x = 5,y = 2\). \(x'=6 - 5 = 1\), so \(B_{x = 3}=(1,2)\).

Step 4: Reflect Point C(3,-1)

• Over x - axis: Using \((x,-y)\), substitute \(x = 3,y=-1\). The reflected point \(C_x=(3,1)\) (since \(-(-1)=1\)).
• Over y - axis: Using \((-x,y)\), substitute \(x = 3,y=-1\). The reflected point \(C_y=(-3,-1)\).
• Over \(x = 3\): Using \(x'=6 - x,y' = y\), substitute \(x = 3,y=-1\). \(x'=6 - 3=3\), so \(C_{x = 3}=(3,-1)\) (since the point is on the line \(x = 3\), its reflection over \(x = 3\) is itself).

Answer:

• Reflection over x - axis:
• \(A(2,4)\to A_x(2,-4)\)
• \(B(5,2)\to B_x(5,-2)\)
• \(C(3,-1)\to C_x(3,1)\)
• Reflection over y - axis:
• \(A(2,4)\to A_y(-2,4)\)
• \(B(5,2)\to B_y(-5,2)\)
• \(C(3,-1)\to C_y(-3,-1)\)
• Reflection over \(x = 3\):
• \(A(2,4)\to A_{x = 3}(4,4)\)
• \(B(5,2)\to B_{x = 3}(1,2)\)
• \(C(3,-1)\to C_{x = 3}(3,-1)\)