Question

graph △rst with vertices ( r(4,1) ), ( s(7,3) ), and ( t(6,4) ) and its image after the glide reflection. translation: ( (x,y)\to(x - 3,y) ) reflection: in the line ( y=-1 )

Explanation:

Step1: Apply translation

For point \(R(4,1)\):
\((x,y)\to(x - 3,y)\) gives \(R'(4-3,1)=(1,1)\)
For point \(S(7,3)\):
\((x,y)\to(x - 3,y)\) gives \(S'(7 - 3,3)=(4,3)\)
For point \(T(6,4)\):
\((x,y)\to(x - 3,y)\) gives \(T'(6-3,4)=(3,4)\)

Step2: Apply reflection

The formula for reflection over the line \(y = k\) is \((x,y)\to(x,2k - y)\). Here \(k=-1\)
For \(R'(1,1)\):
\((x,y)\to(x,2(-1)-y)=(1,-2 - 1)=(1,-3)\)
For \(S'(4,3)\):
\((x,y)\to(x,2(-1)-y)=(4,-2 - 3)=(4,-5)\)
For \(T'(3,4)\):
\((x,y)\to(x,2(-1)-y)=(3,-2 - 4)=(3,-6)\)

Answer:

The vertices of \(\triangle RST\) are \(R(4,1)\), \(S(7,3)\), \(T(6,4)\). After translation \((x,y)\to(x - 3,y)\) the vertices are \(R'(1,1)\), \(S'(4,3)\), \(T'(3,4)\). After reflection over \(y=-1\) the vertices of the image are \(R''(1,-3)\), \(S''(4,-5)\), \(T''(3,-6)\)