Question

reflection 3: reflect the points across the line y = x
fill out table:
what seems to be the rule?
describe in coordinate mapping notation a translation that will reflect a figure across the line y = x:
(x,y)→(,)
reflection 4: reflect the points across the line y = -x:
fill out table:
what seems to be the rule?
describe in coordinate mapping notation a translation that will reflect a figure across the line y = -x:
(x,y)→(,)
lets summarize our results:
reflection across the x - axis:
reflection across the y - axis:
reflection across the line y - x:
reflection across the line y = -x:

Explanation:

Reflection 3: Reflect the points across the line \(y = x\)

Step1: Apply the reflection rule

When reflecting a point \((x,y)\) across the line \(y=x\), the rule is \((x,y)\to(y,x)\).
For the point \((3,2)\), \(x = 3,y = 2\). So \(x'=2,y'=3\).
For the point \((- 1,3)\), \(x=-1,y = 3\). So \(x'=3,y'=-1\).
For the point \((1,0)\), \(x = 1,y=0\). So \(x'=0,y'=1\).

Step2: Fill the table

\(x\)	\(y\)	\(x'\)	\(y'\)
\(-1\)	\(3\)	\(3\)	\(-1\)
\(1\)	\(0\)	\(0\)	\(1\)

Step3: Write the coordinate - mapping notation

The coordinate - mapping notation for reflection across the line \(y = x\) is \((x,y)\to(y,x)\)

Reflection 4: Reflect the points across the line \(y=-x\)

Step1: Apply the reflection rule

When reflecting a point \((x,y)\) across the line \(y =-x\), the rule is \((x,y)\to(-y,-x)\)
For the point \((3,2)\), \(x = 3,y = 2\). So \(x'=-2,y'=-3\)
For the point \((-1,3)\), \(x=-1,y = 3\). So \(x'=-3,y' = 1\)
For the point \((1,0)\), \(x = 1,y=0\). So \(x'=0,y'=-1\)

Step2: Fill the table

\(x\)	\(y\)	\(x'\)	\(y'\)
\(-1\)	\(3\)	\(-3\)	\(1\)
\(1\)	\(0\)	\(0\)	\(-1\)

Step3: Write the coordinate - mapping notation

The coordinate - mapping notation for reflection across the line \(y=-x\) is \((x,y)\to(-y,-x)\)

Summarize the results

• Reflection across the \(x\) - axis: \((x,y)\to(x,-y)\)
• Reflection across the \(y\) - axis: \((x,y)\to(-x,y)\)
• Reflection across the line \(y = x\): \((x,y)\to(y,x)\)
• Reflection across the line \(y=-x\): \((x,y)\to(-y,-x)\)

Answer:

• Reflection 3 - Table:

\(x\)	\(y\)	\(x'\)	\(y'\)
\(-1\)	\(3\)	\(3\)	\(-1\)
\(1\)	\(0\)	\(0\)	\(1\)

• Reflection 3 - Rule: \((x,y)\to(y,x)\)
• Reflection 4 - Table:

\(x\)	\(y\)	\(x'\)	\(y'\)
\(-1\)	\(3\)	\(-3\)	\(1\)
\(1\)	\(0\)	\(0\)	\(-1\)

• Reflection 4 - Rule: \((x,y)\to(-y,-x)\)
• Summary:
• Reflection across \(x\) - axis: \((x,y)\to(x,-y)\)
• Reflection across \(y\) - axis: \((x,y)\to(-x,y)\)
• Reflection across \(y = x\): \((x,y)\to(y,x)\)
• Reflection across \(y=-x\): \((x,y)\to(-y,-x)\)