9.04 Compositions of transformations
A composition of transformations is a list of transformations that are performed one after the other. For example, we might first translate a shape in some direction, then rotate that shape about the origin. The first transformation is the translation, the second transformation is the rotation, and the composition is the combination of the two.
Exploration
The rectangle below has vertices labeled $ABCD$ABCD. Let's perform a composition of transformations involving a translation followed by a reflection.
Rectangle $ABCD$ABCD starts in the 3rd quadrant.
First, let's translate the rectangle $5$5 units to the left and $11$11 units up. This translated rectangle will have vertices labeled $A'B'C'D'$A′B′C′D′.
Next we'll reflect the rectangle $A'B'C'D'$A′B′C′D′ across the $y$y-axis to produce the rectangle $A"B"C"D"$A"B"C"D". Both transformations are shown on the number plane below.
Translating $ABCD$ABCD to get $A'B'C'D'$A′B′C′D′, then reflecting to get $A"B"C"D"$A"B"C"D".
The number of dashes on each vertex of the shape allows us to keep track of the number and order of transformations. Notice that if we reverse the order of the composition we get a different result after both transformations.
The order of transformations is important.
This is the case for compositions in general, although there are some special compositions for which the order does not matter.
Worked example
Question 1
The vertices of triangle $ABC$ABC have the coordinates $A\left(-2,4\right)$A(−2,4), $B\left(-1,3\right)$B(−1,3), and $C\left(-3,2\right)$C(−3,2). The following composition of transformations is performed:
- Rotation by $180^\circ$180° clockwise about the origin, then
- Reflection across the $x$x-axis.
What equivalent single transformation will take triangle $ABC$ABC to triangle $A"B"C"$A"B"C"?
Think: We can perform the rotation to get triangle $A'B'C'$A′B′C′, then the reflection to get triangle $A"B"C"$A"B"C". Then we can compare the location and orientation of the triangles.
Do: Both transformations are shown on the number plane below.
Rotating $ABC$ABC to get $A'B'C'$A′B′C′, then reflecting to get $A"B"C"$A"B"C".
We can see that the vertices of triangle $A"B"C"$A"B"C" have coordinates $A"\left(2,4\right)$A"(2,4), $B"\left(1,3\right)$B"(1,3), and $C"\left(3,2\right)$C"(3,2). Comparing to the vertices of $ABC$ABC, only the sign of the $x$x-coordinates have changed, so the single transformation from $ABC$ABC to $A"B"C"$A"B"C" is a reflection across the $y$y-axis.
Practice questions
Question 2
For the point on the plane, plot the point that will result from a translation of $5$5 units left and $5$5 units down, followed by a rotation of $90^\circ$90° counterclockwise about the origin. Ensure that you have performed both transformations before submitting your answer.
- Loading Graph...
Question 3
The given triangle is to undergo two transformations.
First, plot the triangle that results from reflecting the given triangle across the $x$x-axis.
Loading Graph...The original triangle and the reflected triangle from the previous part are given. Now plot the triangle that results when the original triangle is reflected across the $x$x-axis and translated $4$4 units right.
Loading Graph...
Question 4
Triangle $ABC$ABC is to undergo two separate reflections.
Plot triangle $A'B'C'$A′B′C′, the result of reflecting triangle $ABC$ABC across the line $y=x$y=x.
Loading Graph...Now suppose triangle $A'B'C'$A′B′C′ is reflected across the $y$y-axis to form triangle $A''B''C''$A′′B′′C′′. What single transformation would overlap triangle $ABC$ABC onto triangle $A''B''C''$A′′B′′C′′?
A reflection across the $x$x-axis.
AA reflection across the $y$y-axis.
BA $90^\circ$90° rotation clockwise about the origin.
CA $90^\circ$90° rotation counterclockwise about the origin.
D