Unit 7 Section 6 : Combining Transformations

In this section we combine transformations. We see that sometimes 2 transformations are equivalent to a single transformation.
Here we use transformations from the following types:

Translations Enlargements Reflections Rotations

Example 1

The shape shown in the diagram is reflected first in the y-axis and its image is then reflected in the x-axis.
What single transformation would have the same result as these two transformations?

The diagram shows how the original shape A is first reflected to B, and B is then reflected to C.

A rotation of 180° about the origin would take A straight to C.

Example 2

A triangle is to be enlarged with scale factor 2, using the origin as the centre of enlargement. Its image is then to be translated along the vector .
The coordinates of the corners of the triangle are (2, 1), (2, 4) and (4, 1).
What single transformation would have the same result?

The diagram shows the original triangle, A; the enlargement takes it to B, which is then translated to C.

The triangle A could be enlarged with scale factor 2 to give C.

This diagram shows that the centre of enlargement would be the point (8, –1).

The single transformation that will move triangle A to triangle C is an enlargement, scale factor 2, centre (8, –1).

Exercises

Question 1

State the order of rotational symmetry of each of the following shapes:

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(a)

Reflect the shape shown in the x-axis and then reflect its image in the y-axis.

(b)

What single transformation would have the same result as these two transformations?

Question 2

A rectangle has corners at the points with coordinates (1, 2), (3, 1), (5, 5) and (3, 6). It is first reflected in the x-axis and then its image is rotated through 180° about the origin.
Describe how to move the rectangle from its original position to its final position, using only one transformation.

Question 3

A shape is rotated through 180° about the origin and then its image is reflected in the x-axis.
What single transformation would have the same result as the two transformations described above?

Question 4

A triangle has corners at the points with coordinates (2, 2), (3, 6) and (8, 6).

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(a)

Draw the triangle and enlarge it with scale factor 2, using the origin as the centre of enlargement.

(b)

Translate the enlarged shape along the vector .

(c)

Describe fully the enlargement that would produce the final triangle from the original triangle.

Enlargement, centre ( ) scale factor .

Question 5

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(a)

Draw the triangle with corners at the points (2, 1), (4, 1) and (4, 2).

(b)

Reflect this shape in the line y = x.

(c)

Reflect the new triangle in the y-axis.

(d)

What single transformation would have the same result as the two transformations described above?

Question 6

A shape is reflected in the line y = x and then the image is reflected in the line y = –x.
Describe a single transformation that would have the same result.

Question 7

The shape shown in the following diagram is to be reflected in the line x = 4 and then its image is to be reflected in the line y = 5.

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(a)

Draw a diagram to show how the shape moves.

(b)

What single transformation would have the same result?

Question 8

The shape shown in the diagram is translated along the vector .

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(a)

Draw the final position of the shape.

The shape could be moved to this position using two reflection.
E.g. Reflect the original chevron in the line x = 4 and then reflect its image in the line x = 9, as in the diagram.
Question 9

The shape shown in the diagram is to be enlarged with scale factor 3 using the point (0, 4) as the centre of enlargement.

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(a)

Draw the enlarged shape.

(b)

The enlarged shape is translated along the vector . Draw the new position of the shape.

(c)

Describe the single enlargement that would have the same result as the two transformations used above.

Enlargement, centre ( ), scale factor .
Question 10

A shape is reflected in the line y = x, then in the line y = –x, and finally in the x-axis.
What single transformation would have the same result?

E.g. The transformations of shape (3, 1), (5, 1), (5, 3), (3, 2) is shown in the diagram.
Question 11

The following design is based on a Celtic pattern.

Part of the pattern is shown below:

The pattern is made of two rectangular blocks, A and B.
Complete the instructions to map block A onto block B.

E.g. rotate ° around (7,) and then reflect in the .