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11 Transformations and the number plane
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11.05 Composing transformations

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Composition 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.

Rectangle A B C D is in the third quadrant. Ask your teacher for more information.

This rectangle has vertices labelled ABCD. Let's perform a composition of transformations involving a translation followed by a reflection.

Rectangle ABCD starts in the 3rd quadrant.

Rectangle A B C D is in the third quadrant and two transformed rectangles. Ask your teacher for more information.

First, let's translate the rectangle 5 units to the left and 11 units up. This translated rectangle has vertices labelled A'B'C'D'.

Next we'll reflect the rectangle A'B'C'D' across the y-axis to produce the rectangle A''B''C''D''. Both transformations are shown on the number plane.

Rectangle A B C D is in the third quadrant and two transformed rectangles. Ask your teacher for more information.

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.

Examples

Example 1

The given triangle is to undergo two transformations.

-8
-7
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-1
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-8
-7
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y
a

First, plot the triangle that results from reflecting the given triangle across the x-axis.

Worked Solution
Create a strategy

If we reflect across the x-axis, the x-values of the coordinates will remain the same and the y-values will change sign.

Apply the idea
-8
-7
-6
-5
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-2
-1
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x
-8
-7
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y

The vertices (0, -1), (-8, 4), and (-4, -8) have reflection points of (0, 1), (-8, -4), and (-4, 8) across the x-axis.

b

Now translate the reflected triangle 4 units to the right.

Worked Solution
Create a strategy

Move every vertex of the reflected triangle from part (a) the required units to the right.

Apply the idea
-8
-7
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x
-8
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y

To find the new coordinates of each point, we add 4 to the x-coordinate.

The translated vertex of (0, 1) is (0+4,1)=(4,1).

The translated vertex of (-8, -4) is (-8+4,-4)=(-4,-4).

The translated vertex of (-4, 8) is (-4+4,8)=(0,8).

By plotting and joining the translated points we get the triangle shown.

Example 2

The point A\left(6,\,-1\right) is first rotated 180\degree about the origin, and then it is reflected across the x-axis. This produces the point A'.

a

What are the coordinates of A'?

Worked Solution
Create a strategy

When a point is rotated 180\degree about the origin, the coordinates change sign.

Then, if we reflect across the x-axis, the x-values of the coordinates will remain the same and the y-values will change sign.

Apply the idea

Rotating (6,\,-1) by 180\degree will be (-6,\,1).

Reflecting (-6,\,1) across the x-axis will be (-6,\,-1).

A' (-6,\,-1)

b

Which of the following transformations also takes A to A'?

A
A reflection across the y-axis.
B
A 90\degree clockwise rotation about the origin.
C
A vertical translation.
D
There is no other way to describe the transformation.
Worked Solution
Create a strategy

Use the following characteristics of transformations:

  • If reflected across the y-axis, the x-values change sign and the y-values of the coordinates remain the same.

  • If rotated 90\degree clockwise about the origin, the coordinates switch then the y-values change sign.

  • If translated vertically, the x-coordinates remain the same and the value of y-coordinates change.

Apply the idea

From A (6, -1) to A' (-6, -1), the x-value changed sign and the y-value remained the same.

The transformation is also a reflection across the y-axis, so the answer is option A.

Example 3

Points A\left(-5,-7\right), B\left(4,4\right), and C\left(9,1\right) are the vertices of a triangle. What are the coordinates of A', B', and C' that result from reflecting the triangle across the y-axis, and translating it 3 units right and 5 units up?

Worked Solution
Create a strategy

For reflection across the y-axis, change the sign of the x-coordinates and keep the y-coordinates the same.

For the translation, add the required units right to the x-coordinates and add the required units up to the y-coordinates of the reflected points.

Apply the idea

To find the reflected points, change the sign of the x-coordinates: \text{Reflected points}=(5,\,-7), \,(-4,\,4), \,(-9,\,1)

To find the translated points add 3 to the x-coordinates and 5 to the y-coordinates:

\displaystyle \text{Translated points}\displaystyle =\displaystyle (5+3,\,-7+5),\, (-4+3,\, 4+5), (-9+3,\, 1+5)
\displaystyle =\displaystyle (8, -2),\, (-1, 9),\, (-6, 6)

The coordinates are A' (8,\,-2), B' (-1,\,9), and C' (-6,\,6).

Idea summary

A composition of transformations is a list of transformations that are performed one after the other. The order that you perform the transformations can effect the result.

Outcomes

VCMMG261

Describe translations, reflections in an axis, and rotations of multiples of 90° on the Cartesian plane using coordinates. Identify line and rotational symmetries

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