Lesson 13.3 · 25 min

Combining Transformations

Carrying out one transformation after another, and finding the single transformation that does the same job.

In this lesson you will learn to…

Carry out two transformations in sequence
Find the single transformation equivalent to two combined
Know that two reflections in parallel lines make a translation
Decide whether the final image is congruent to the object

Why this matters

Animators and game designers build complicated movements from simple ones — a character might be flipped, turned and slid in one frame. Knowing how transformations combine lets them replace several steps with one.

Try it on a point

The quickest way to find a combined effect is to follow a single point, such as $(2, 1)$ :

Reflect in $x = 4$ : $(2, 1) \to (6, 1)$ (it was 2 left of the line, so it ends 2 right of it).
Then reflect in $x = 7$ : $(6, 1) \to (8, 1)$ .

That is a move of 6 to the right, which matches $2 \times (7 - 4) = 6$ . Check with a second point before you trust the answer.

Reflections that are not parallel

Two reflections in lines that cross give a rotation about the crossing point, through twice the angle between the lines. Reflecting in the $x$ -axis and then the $y$ -axis is a rotation of $180^\circ$ about the origin.

Key rules and formulas

One after the other
Transform the object to get A′, then transform A′ to get A″. The order matters — rotating then translating is usually not the same as translating then rotating.
Two parallel mirrors
$reflect in x = a, then x = b \Rightarrow translation by (2 (b - a) 0)$
Two reflections in parallel lines give a translation of twice the distance between the lines, in the direction from the first line to the second.
Congruence
Reflections, rotations and translations never change size, so any combination of them gives a congruent image. Include an enlargement and the image is only similar.

Worked examples

Worked example 1

Rotate then translate

Triangle ABC is rotated

9 0^{\circ}

anticlockwise about O, then translated by

(0 - 4)

. The single equivalent transformation turns out to be a rotation. How do you find it?

Try it yourself first, then reveal the solution one step at a time.

Worked example 2

Two reflections

A shape is reflected in

x = 1

and then in

x = 5

. What single transformation is this?

Try it yourself first, then reveal the solution one step at a time.

Practice: combining transformations

8 questions

Eight questions. Follow one point through each step — it is the quickest check.

Question 1 of 8Fill the gaps
A shape is reflected in $x = 1$ and then in $x = 5$ . Give the single equivalent translation vector.
top Gap a, bottom Gap b
Question 2 of 8Step by step
Follow a point.
Point $(1, 2)$ is reflected in $x = 4$ , then in $x = 6$ .
1. After the first reflection, what is the $x$ -coordinate?
  Step 1 answer
Question 3 of 8Multiple choice
Reflecting in $x = 5$ and then in $x = 6$ is equivalent to which single transformation?

Question 4 of 8Complete the table

Is the final image congruent to the original?

First	Then	Congruent? (yes/no)
Reflection in y = x	Rotation 90° clockwise	Row 1, Congruent? (yes/no)
Translation	Reflection in y = 3	Row 2, Congruent? (yes/no)
Enlargement, scale factor 2	Translation	Row 3, Congruent? (yes/no)

Question 5 of 8Multiple choice
A shape is reflected in the $x$ -axis and then in the $y$ -axis. What single transformation does the same?
Question 6 of 8Fill the gaps
Triangle ABC is translated by $(- 5 - 5)$ , then by $(17)$ . Give the single translation from ABC to the final image.
top Gap a, bottom Gap b
Question 7 of 8Fill the gaps
Following on: what translation takes the final image back to ABC?
top Gap a, bottom Gap b
Question 8 of 8True or falseChallenge
Rotating a shape then translating it always gives the same result as translating it then rotating it (same rotation and translation).