Question

graph quadrilateral qrst with vertices q(1, - 1), r(1, 2), s(4, - 1), and t(4, - 4) and its image after the glide reflection. translation: (x, y) → (x + 2, y) reflection: in the x - axis

Explanation:

Step1: Apply the translation

For a point \((x,y)\), the translation \((x,y)\to(x + 2,y)\) moves the point 2 units to the right.

• For \(Q(1,-1)\): \(Q'(1+2,-1)=(3,-1)\)
• For \(R(1,2)\): \(R'(1 + 2,2)=(3,2)\)
• For \(S(4,-1)\): \(S'(4+2,-1)=(6,-1)\)
• For \(T(4,-4)\): \(T'(4 + 2,-4)=(6,-4)\)

Step2: Apply the reflection

The rule for reflection over the \(x\) - axis is \((x,y)\to(x,-y)\)

• For \(Q'(3,-1)\): \(Q''(3,1)\)
• For \(R'(3,2)\): \(R''(3,-2)\)
• For \(S'(6,-1)\): \(S''(6,1)\)
• For \(T'(6,-4)\): \(T''(6,4)\)

Answer:

The vertices of the original quadrilateral \(QRST\) are \(Q(1,-1)\), \(R(1,2)\), \(S(4,-1)\), \(T(4,-4)\). The vertices of the image after glide - reflection (translation \((x,y)\to(x + 2,y)\) followed by reflection over the \(x\) - axis) are \(Q''(3,1)\), \(R''(3,-2)\), \(S''(6,1)\), \(T''(6,4)\)