GCSE · Maths · AQA · Spec 8300 · Higher

Combinations of transformations and invariance (Higher)

Reflect a triangle in the x-axis, then reflect that image in the y-axis. Two flips, so why does it look as if the triangle has just been spun round?

Two reflections, one fixed point

Reflect in the x-axis, then in the y-axis

Turn about O 0°. Ring to O 2.0. AC 3.0. A″C″ 3.0. Angle C 45°. Angle C″ 45°. Relationship: Reflecting in the x-axis and then in the y-axis does the same job as one rotation of 180° about O. Lengths and angles are invariant; position changes. O is the only point that ends where it started.

Turn about O0°Ring to O2.0AC3.0A″C″3.0Angle C45°Angle C″45°Drag the ring from A round O until it sits on A″

Reflecting in the x-axis and then in the y-axis does the same job as one rotation of 180° about O. Lengths and angles are invariant; position changes. O is the only point that ends where it started.

Triangle ABC is reflected in the x-axis to give A′B′C′ (faint), and A′B′C′ is reflected in the y-axis to give A″B″C″ (bold). The ring starts on A: drag it round O and watch the turn angle. Going A → B → C is anticlockwise, and so is A″ → B″ → C″. The middle image runs the other way round.

What do you expect?

Reflect, then translate

Point P is at (1, 2). It is reflected in the y-axis and then translated by the column vector (4 over 0), which means 4 right and 0 up.

Before you work it out, which of these is closest to what you think?

Doing the translation first would land P in the same place. It's the same two moves either way.The points on the y-axis stay put, because reflecting in the y-axis doesn't move them.The single move that does the same job must be a translation, because the last step was a translation.I need to do both steps in the order given, then compare where the points end up.

How sure are you?

I'm sureI think soI'm guessing

Doing the translation first would land P in the same place. It's the same two moves either way.

This is a common misconception

Try both. Reflect then translate: (1, 2) → (−1, 2) → (3, 2). Translate then reflect: (1, 2) → (5, 2) → (−5, 2). Same two moves, different finishing points. Some pairs do give the same result in either order, but you have to check that. Never assume it.

The points on the y-axis stay put, because reflecting in the y-axis doesn't move them.

This is a common misconception

The reflection leaves (0, 2) where it is, but then the translation moves it 4 right to (4, 2). A point is invariant only if it ends where it started after both steps. Here that happens on the line x = 2: (2, 5) → (−2, 5) → (2, 5). Neither step fixes that line by itself, but the combination does.

The single move that does the same job must be a translation, because the last step was a translation.

This is a common misconception

Follow a few points: (1, 2) ends at (3, 2) and (0, 5) ends at (4, 5). Every point (x, y) ends at (4 − x, y), the same distance from x = 2 but on the other side. So the combination is a reflection in the line x = 2. The last step doesn't decide the kind of transformation.

I need to do both steps in the order given, then compare where the points end up.

This matches the evidence

Yes. That's how you find everything: P ends at (3, 2), the combination is a reflection in x = 2, and the points on x = 2 are invariant. In the other order, the same working gives a reflection in x = −2 instead.

The order, the kind of transformation and the invariant points all belong to the combination. Work them out from where the points actually end up.

Describe the combination as one move

Problem

Triangle T has vertices A(1, 1), B(3, 1) and C(1, 2). T is rotated 90° clockwise about the origin, then translated by the column vector (3 over −1). Describe fully the single transformation that maps T onto its final image, and say what is invariant.

Spot the slip

Find the line where it goes wrong

Shape Q is translated by the column vector (−3 over 2) and then reflected in the x-axis. Where does the vertex (4, 1) end up?

A student's answer — which line goes wrong?

Step 1 is the translation and step 2 is the reflection, so translate first.
The vector (−3 over 2) moves (4, 1) to (4 + 2, 1 − 3) = (6, −2).
Reflecting in the x-axis keeps x and changes the sign of y: (6, −2) goes to (6, 2).
So the vertex ends at (6, 2).

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

Do the moves in order, then compare the first shape with the last.

What you need to know

Apply a combination one step at a time, in the order given. The image from step 1 is the object for step 2.
Rotations, reflections and translations keep every length and angle, so the final image is congruent to the object. Its position, the direction it faces and the order its vertices run round can change.
An invariant point ends exactly where it started after the whole combination. Check it for the combination, not for one step.
A translation is written as a column vector (a over b): a is the move across (positive means right) and b is the move up or down (positive means up).

The big picture

To combine transformations, apply them one at a time in the order given, then compare the original shape with the final image. Rotations, reflections and translations never change lengths or angles, so the final image is congruent to the object; its position, the direction it faces and the way its vertices run round can change. The order can change the result, and the combination can be one transformation of a different kind. An invariant point ends where it started after the whole combination. A translation is written as a column vector: top number across, bottom number up or down.

Key points

1The order of the steps can change the final image, so follow the order given.

2A combination can do the same job as one transformation of a different kind. Two reflections in the axes give a rotation of 180° about the origin.

3To describe a single transformation fully, give a rotation its angle, direction and centre, a reflection its mirror line, and a translation its column vector.

4Every point on the mirror line is invariant under a reflection. The centre is the only invariant point of a rotation (other than a full turn). A translation by a non-zero vector has no invariant points.

5One reflection reverses the order the vertices run round. A second reflection reverses it back.

6Two translations combine into one translation: add the top numbers, then add the bottom numbers.

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Worked example

Problem

A shape is translated by the column vector (2 over −5) and then by (−6 over 3). Describe fully the single transformation that has the same effect, and say whether any point is invariant.

⚠ Watch out

Answering 'describe fully the single transformation' by listing the steps again, or by adding the question's column vector to a rotation. The answer is one transformation: a rotation needs its angle, direction and centre, a reflection needs its mirror line, and a translation needs its own column vector.

🧠

Memory hook

Step by step, then stand back. Do the moves in order, then compare the first shape with the last to name the one move, and look for the point that never moved.

✓

Check yourself

Reflect in x = 1, then in x = 3. What single transformation is that? Check: (0, 0) → (2, 0) → (4, 0). Every point moves 4 right: translation by (4 over 0).

Flashcards

(7)

Under any combination of rotations, reflections and translations, what is always invariant?

Every length and every angle, so the final image is congruent to the object.

What makes a point invariant under a combination of transformations?

It ends exactly where it started after all the steps, in the order given.

What does the column vector (a over b) tell you?

a is the move across (positive right, negative left) and b is the move up or down (positive up, negative down).

How do you combine two translations into one?

Add the top numbers and add the bottom numbers. The result is a single translation by that vector.

What must you give to describe a rotation fully?

The angle, the direction (clockwise or anticlockwise) and the centre of rotation.

Which points are invariant under a single reflection?

Every point on the mirror line, and no others.

How can you tell from the final image that a combination has not flipped the shape?

The vertices still run round in the same direction as the object, for example both anticlockwise.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

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Reflect, then translate

Find the line where it goes wrong

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