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 labelled $ABCD$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 labelled $A'B'C'D'$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 Cartesian 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
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 Cartesian 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.
Translation of $a$a horizontal units and $b$b vertical units
$(x,y)\to(x+a,x+b)$(x,y)→(x+a,x+b)
- If $a$a is positive, then the $x$x-value of the image point is to the right $a$a units from the original point.
- If $a$a is negative, then the $x$x-value of the image point is to the left $a$a units from the original point.
- If $b$b is positive, then the $y$y-value of the image point is up ‘$b$b’ units from the original point.
- If $b$b is negative, then the $y$y-value of the image point is down $b$b units from the original point.
- For example, $(x,y)\to(x−5,y+2)$(x,y)→(x−5,y+2) means that each of the image points are translated left $5$5 units and up $2$2 units from the original point.
Reflections about the $x$x or $y$y-axes
- A shape reflected in the $x$x-axis has a mapping rule $(x,y)\to(x,−y)$(x,y)→(x,−y). For example, the vertex of the shape originally positioned at $(2,3)$(2,3) is reflected about the $x$x-axis to $(2,−3)$(2,−3).
- A shape reflected in the $y$y-axis has a mapping rule $(x,y)\to(−x,y)$(x,y)→(−x,y). For example, the vertex of the shape originally positioned at $(2,3)$(2,3) is reflected about the $y$y-axis to $(−2,3)$(−2,3).
Rotations about the origin
- A shape rotated $90°$90° counterclockwise has a mapping rule $(x,y)\to(−y,x)$(x,y)→(−y,x).
- A shape rotated $180°$180° counterclockwise has a mapping rule $(x,y)\to(−x,−y)$(x,y)→(−x,−y).
- A shape rotated $270°$270° counterclockwise has a mapping rule $(x,y)\to(y,−x)$(x,y)→(y,−x).
Practice questions
Question 1
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 2
The point $A\left(6,-1\right)$A(6,−1) is first rotated $180^\circ$180° about the origin, and then it is reflected across the $x$x-axis. This produces the point $A'$A′.
Enter the coordinates of $A'$A′.
$A'\left(\editable{},\editable{}\right)$A′(,)
Which of the following transformations also takes $A$A to $A'$A′?
A reflection across the $y$y-axis.
AA $90^\circ$90° clockwise rotation about the origin.
BA vertical translation.
CThere is no other way to describe the transformation.
D
Question 3
Points $A\left(-5,-7\right)$A(−5,−7), $B\left(4,4\right)$B(4,4) and $C\left(9,1\right)$C(9,1) are the vertices of a triangle. What are the coordinates $A'$A′, $B'$B′ and $C'$C′ that result from reflecting the triangle across the $y$y-axis, and translating it $3$3 units right and $5$5 units up?
Ensure that you have performed both transformations before submitting your answer.
$A'\left(\editable{},\editable{}\right)$A′(,)
$B'\left(\editable{},\editable{}\right)$B′(,)
$C'\left(\editable{},\editable{}\right)$C′(,)