Name:___________________
Composition of Transformations
Date:_____ Period:____
Ms. Anderle
Composition of Transformations
When two or more transformations are combined to form a new transformation,
the result is called a composition of transformations. In a composition, the first
transformation produces an image upon which the second transformation is then
performed.
The symbol for composition is an open circle.
The notation rx-axis ◦ T3,4 is read as a reflection in the x-axis FOLLOWING a
translation of (x+3,y+4). Basically, this process is done in REVERSE. So whenever
you see a composition of a transformation, you work BACKWARDS!!!
Examples:
1. What is RO ◦ RO(2,3)?
2. D2 ◦ D1/2(4,6) =
3. What is ry=2 ◦ rx-axis(-2,6)?
4. What is R180˚ ◦ T-2,3(5,14)?
5. What is R-270˚ ◦ D-1(8,3)?
6. What is D4 ◦ D1/4(x,y)?
7. What is T2,1 ◦ ry=-x ◦ D3(2,1) ?
8. What is T7,-2 ◦ D1/2 ◦ RO (8,10)?
9. The coordinates of triangle FUN are F(-5,1), U(-1,1), and N(-1,7). On a
coordinate plane, draw and label ΔFUN. Perform the composition
transformation ry-axis ◦ rx-axis.
10. Graph ΔABC with vertices A(1,4), B(3,7), and C(5,1). Graph and label the
composition T-5,-2 ◦ rx-axis.
11. Graph ΔTOM with vertices T(-2,3), O(4,5), and M(0,5). Graph and label the
composition R90˚ ◦ ry-axis.
12. Graph ΔEMY with vertices E(2,5), M(8,7), and Y(6,4). Graph and label the
composition R-90˚ ◦ T-5,1.
13. Graph ΔBUG with vertices B(1,1), U(1,5), and G(6,1). Graph and label the
composition ry=x ◦ ry-axis. Is there a single transformation that can
accomplish these transformations?
Glide Reflection
When a translation (a slide) and a reflection are performed one after the other, a
transformation called a glide reflection is produced. In a glide reflection, the line
of reflection is parallel to the direction of the translation. It does not matter
whether you glide first and then reflect, or reflect first and then glide. This
transformation is commutative.
Properties preserved under a glide reflection:
1. distance is preserved (lengths of segments are the same)
2. angle measures (remain the same)
3. parallelism (parallel lines remain parallel)
4. colinearity (points stay on the same lines)
5. midpoint (midpoints remain the same in each figure)
***ORIENTATION IS NOT PRESERVED! LETTERING IS NOT THE SAME***
Examples:
1. ΔJKL has vertices J(6,-1), K(10,-2), and L(5,-3). Graph ΔJKL and its image
after a T0,4 and a reflection in the y-axis.
2. ΔPQR has vertices P(1,1), Q(2,5), and R(4,2). Graph ΔPQR and its image
after the glide reflection rx-axis ◦ T-3,-3.
3. Quadrilateral LISA has vertices L(-3,4), I(-1,3), S(-1,1), and A(-4,2). Graph
LISA and its image after rx-axis ◦ T5,0
4. ΔTUV has vertices T(2,-1), U(5,-2), and V(3,-4). Graph ΔTUV and its image
after a transformation (x,y)(x – 1, y + 5) and a rotation of 180˚. Is this a
glide reflection?
5. ΔDFG has vertices D(2,8), F(1,2), and G(4,6). Graph ΔDFG and its image
after ry=x ◦ T3,3. Is this a glide reflection?
6. Examine the graph below. Is this an example of a glide reflection?