Let's first plot and connect the given vertices to draw △ PQR.
The given composition of isometries represents a glide reflection. Let's do these two things one at a time.
R_(y=-2)∘ T_(< 3, -1>)
Reflection across y=-2
Translation 3 units right and 1 unit down
To complete a glide reflection, we first perform the translation and then the reflection.
Translation
To translate △ PQR three units right and one unit down, we have to add 3 to each x-coordinate and subtract 1 from each y-coordinate.
(x,y) → (x+3,y-1)
Let's do this for the three vertices.
(x,y)
(x+ 3,y- 1)
(x',y')
P(0,5)
(0+ 3,5- 1)
P'(3,4)
Q(5,3)
(5+ 3,3- 1)
Q'(8,2)
R(3,1)
(3+ 3,1- 1)
R'(6,0)
With these points, we are able to draw the transformed image as △ P'Q'R'.
Reflection
To complete the reflection, we have to reflect all the vertices of △ P'Q'R' on the opposite side of the line y=-2. The distance from the vertices to the line y=-2 must remain the same. We will call the reflected image △ P''Q''R''.
Final Glide Reflection
The final glide reflection is the combination of the translation and the reflection.
The vertices of the image of △ PQR for the glide reflection are P''(3,-8), Q''(8,-6), and R''(6,-4).
