GCSE Maths: Combinations of transformations and invariant points (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Combinations of transformations and invariant points
Sometimes a shape is moved by two or more transformations, one after the other. For example, “reflect A in the x-axis to give B, then rotate B through 90° about the origin to give C”. We work out each image in turn: the image of the first transformation is the object for the second. The order matters: reflecting and then rotating usually gives a different result from rotating and then reflecting.
A combination of transformations can often be replaced by a single transformation that does the whole job. Three facts are useful:
- Two reflections in parallel lines make a translation. The translation is perpendicular to the mirror lines and is twice the distance between them.
- Two reflections in lines that cross make a rotation about the point where the lines cross. The angle of the rotation is twice the angle between the mirror lines.
- Two rotations about the same centre make a rotation about that centre whose angle is the sum of the two angles. Two translations make a translation whose vector is the sum of the two vectors.
0.
Invariant points. A point that is unchanged by a transformation is called an invariant point. For a combination, test the single equivalent transformation. For example, a half turn about O has only one invariant point, O. A reflection has a whole line of invariant points (the mirror line). A translation has none. When you must find the invariant point of a combination, you can also look for the point whose image is itself.
Key ideas
- Do the transformations in the order given: the image of the first is the object of the second.
- Two reflections in parallel lines = a translation (twice the distance between the lines).
- Two reflections in lines that cross = a rotation about the crossing point (twice the angle between the lines).
- In general the order matters; for two translations it does not.
- An invariant point is its own image: the mirror line (reflection), the centre (rotation, enlargement); none for a translation.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: reflection in the line x = 1. Reflecting in a vertical line x = 1: the y-coordinate stays the same and the new x is 2 × 1 − x (the same distance on the other side of the line). Do this to every vertex of shape A.Shape B:
(−3, 1) → (2 × 1 − (−3), 1) = (5, 1)
(−1, 1) → (2 × 1 − (−1), 1) = (3, 1)
(−3, 4) → (2 × 1 − (−3), 4) = (5, 4) - Then do the second transformation: reflection in the line x = 4. Reflecting in a vertical line x = 4: the y-coordinate stays the same and the new x is 2 × 4 − x (the same distance on the other side of the line). Do this to every vertex of shape B.Shape C:
(5, 1) → (2 × 4 − 5, 1) = (3, 1)
(3, 1) → (2 × 4 − 3, 1) = (5, 1)
(5, 4) → (2 × 4 − 5, 4) = (3, 4) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (5, 1), (3, 1), (5, 4) then C (3, 1), (5, 1), (3, 4)
- Decide which type of transformation it is. A translation only slides the shape. If the shape had turned, its sides would no longer be parallel to the originals.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, C faces the same way as A, and every side is parallel to its original. So the transformation is a translation. - Pick a vertex and its image. Every point moves by the same amount, so one vertex is enough to find the move.(−3, 1) → (3, 1)
- Find the move across and the move up or down. Subtract the x-coordinates, then the y-coordinates (image − object). Positive means right/up, negative means left/down.Across: 3 − −3 = 6 (6 right)
Up/down: 1 − 1 = 0 (no move) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by 6
0 - Write the full description. We have found every piece of information needed for this transformation.Translation by 6
0
Check. Test another vertex: (−1, 1) + 6
0 = (5, 1), which is the matching vertex of C. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape A.Shape B:
(1, 1) → (1, 2 × 0 − 1) = (1, −1)
(4, 1) → (4, 2 × 0 − 1) = (4, −1)
(1, 3) → (1, 2 × 0 − 3) = (1, −3) - Then do the second transformation: reflection in the line x = 0. Reflecting in a vertical line x = 0: the y-coordinate stays the same and the new x is 2 × 0 − x (the same distance on the other side of the line). Do this to every vertex of shape B.Shape C:
(1, −1) → (2 × 0 − 1, −1) = (−1, −1)
(4, −1) → (2 × 0 − 4, −1) = (−4, −1)
(1, −3) → (2 × 0 − 1, −3) = (−1, −3) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −1), (4, −1), (1, −3) then C (−1, −1), (−4, −1), (−1, −3)
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, C is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. In a half turn, the centre is exactly half-way between a point and its image. Find the midpoint for two different vertices: they should be the same point.(1, 1) and (−1, −1): midpoint (0, 0)
(4, 1) and (−4, −1): midpoint (0, 0)
Both midpoints are the same, so the centre is (0, 0). - Find the angle and direction. From C the moves to the vertex and to its image point in exactly opposite directions, so it is a half turn. For a half turn the direction does not matter.From C to (1, 1): (1, 1); from C to (−1, −1): (−1, −1). Both signs have changed, so the angle is 180°.
- Write the full description. We have found every piece of information needed for this transformation.Rotation, 180°, centre (0, 0)
Check. Rotate (4, 1) about (0, 0) by 180°: we get (−4, −1), which matches the vertex (−4, −1) of C. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: rotation of 90° clockwise about (0, 0). Rotate about (0, 0): write each point as a move from the centre, turn the move using (a, b) → (b, −a), then add the centre back on. Do this to every vertex of shape A.Shape B:
(1, 1): move from centre (1, 1) → turned (1, −1) → (1, −1)
(4, 1): move from centre (4, 1) → turned (1, −4) → (1, −4)
(1, 3): move from centre (1, 3) → turned (3, −1) → (3, −1) - Then do the second transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape B.Shape C:
(1, −1) → (1, 2 × 0 − (−1)) = (1, 1)
(1, −4) → (1, 2 × 0 − (−4)) = (1, 4)
(3, −1) → (3, 2 × 0 − (−1)) = (3, 1) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −1), (1, −4), (3, −1) then C (1, 1), (1, 4), (3, 1)
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, but C is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Compare the coordinates of a vertex and its image. For the diagonal mirror lines y = x and y = −x the coordinates change in a special way.The coordinates have been swapped: (4, 1) became (1, 4).
- Use the midpoint as a check. The mirror line must pass through the midpoint of a point and its image.Midpoint of (4, 1) and (1, 4) = (2.5, 2.5), which lies on y = x.
- Write the equation of the mirror line. A reflection that swaps x and y is a reflection in the line y = x.Mirror line: y = x
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line y = x
Check. Reflect (1, 3) in y = x and we get (3, 1), which matches (3, 1) in C. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
1. B is mapped to C by a reflection in the line y = 0. Write down the coordinates of the vertices of C. A has vertices at (1, 2), (3, 2), (1, 5).
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next.
- Do the first transformation: translation by 2
1. Add 2 to every x and 1 to every y. Do this to every vertex of shape A.Shape B:
(1, 2) → (1 + 2, 2 + 1) = (3, 3)
(3, 2) → (3 + 2, 2 + 1) = (5, 3)
(1, 5) → (1 + 2, 5 + 1) = (3, 6) - Then do the second transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape B.Shape C:
(3, 3) → (3, 2 × 0 − 3) = (3, −3)
(5, 3) → (5, 2 × 0 − 3) = (5, −3)
(3, 6) → (3, 2 × 0 − 6) = (3, −6) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (3, 3), (5, 3), (3, 6) then C (3, −3), (5, −3), (3, −6)
Check. Shapes A and C are congruent (same size), and the vertex (1, 2) has gone to (3, −3): the transformations only move or flip the shape. ✓
Watch out. Do the transformations in the order given. Doing them the other way round usually gives a different answer.
0. Shape C is found by doing the translation by 4
0 first and then the rotation of 90° clockwise about (0, 0). Is shape C in the same place as shape B? Show your working.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Work out the final image for each order, then compare them.
- Do transformation 1 then transformation 2. First rotation of 90° clockwise about (0, 0), then translation by 4
0.After the first: (1, −1), (1, −3), (4, −1)
After the second: (5, −1), (5, −3), (8, −1)
Call this shape B. - Now do them in the opposite order. First translation by 4
0, then rotation of 90° clockwise about (0, 0).After the first: (5, 1), (7, 1), (5, 4)
After the second: (1, −5), (1, −7), (4, −5)
Call this shape C. - Compare B and C. If the order did not matter, B and C would be in exactly the same place.B: (5, −1), (5, −3), (8, −1)
C: (1, −5), (1, −7), (4, −5)
The vertices are different, so the shapes are in different places.
Check. Look at the picture: B and C are not on top of each other. ✓
Watch out. Many pairs of transformations give different results when the order is swapped. (Two translations are one exception: they always give the same result.)
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a rotation does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (0, 0).
- Test the centre. Apply the transformation to the centre.(0, 0) → (0, 0)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (0, 0) only.
Check. Test another point, (1, 1): it goes to (−1, −1), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Watch out
Do the transformations in the order stated, and always use the image of the first transformation as the object for the second, not the original shape. And a “single transformation” must be a reflection, rotation, translation or enlargement with all its detail.Exercise 21F Combinations and invariant points
Non-calculatorKey words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next.
- Do the first transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape A.Shape B:
(1, 1) → (1, 2 × 0 − 1) = (1, −1)
(4, 1) → (4, 2 × 0 − 1) = (4, −1)
(1, 3) → (1, 2 × 0 − 3) = (1, −3) - Then do the second transformation: reflection in the line x = 0. Reflecting in a vertical line x = 0: the y-coordinate stays the same and the new x is 2 × 0 − x (the same distance on the other side of the line). Do this to every vertex of shape B.Shape C:
(1, −1) → (2 × 0 − 1, −1) = (−1, −1)
(4, −1) → (2 × 0 − 4, −1) = (−4, −1)
(1, −3) → (2 × 0 − 1, −3) = (−1, −3) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −1), (4, −1), (1, −3) then C (−1, −1), (−4, −1), (−1, −3)
Check. The single transformation that does the whole job is: Rotation, 180°, centre (0, 0). Applying it to (1, 1) gives (−1, −1), which is the matching vertex of C. ✓
Watch out. Do the transformations in the order given. Doing them the other way round usually gives a different answer.
3, then reflect in the line y = x.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next.
- Do the first transformation: translation by 2
3. Add 2 to every x and 3 to every y. Do this to every vertex of shape A.Shape B:
(1, 1) → (1 + 2, 1 + 3) = (3, 4)
(4, 1) → (4 + 2, 1 + 3) = (6, 4)
(1, 3) → (1 + 2, 3 + 3) = (3, 6) - Then do the second transformation: reflection in the line y = x. Reflecting in y = x swaps the coordinates: (a, b) → (b, a). Do this to every vertex of shape B.Shape C:
(3, 4) → (4, 3)
(6, 4) → (4, 6)
(3, 6) → (6, 3) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (3, 4), (6, 4), (3, 6) then C (4, 3), (4, 6), (6, 3)
Check. Shapes A and C are congruent (same size), and the vertex (1, 1) has gone to (4, 3): the transformations only move or flip the shape. ✓
Watch out. Do the transformations in the order given. Doing them the other way round usually gives a different answer.
4.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next.
- Do the first transformation: rotation of 90° anticlockwise about (0, 0). Rotate about (0, 0): write each point as a move from the centre, turn the move using (a, b) → (−b, a), then add the centre back on. Do this to every vertex of shape A.Shape B:
(1, 1): move from centre (1, 1) → turned (−1, 1) → (−1, 1)
(4, 1): move from centre (4, 1) → turned (−1, 4) → (−1, 4)
(1, 3): move from centre (1, 3) → turned (−3, 1) → (−3, 1) - Then do the second transformation: translation by −1
4. Add −1 to every x and 4 to every y. Do this to every vertex of shape B.Shape C:
(−1, 1) → ((−1) + (−1), 1 + 4) = (−2, 5)
(−1, 4) → ((−1) + (−1), 4 + 4) = (−2, 8)
(−3, 1) → ((−3) + (−1), 1 + 4) = (−4, 5) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (−1, 1), (−1, 4), (−3, 1) then C (−2, 5), (−2, 8), (−4, 5)
Check. The single transformation that does the whole job is: Rotation, 90° anticlockwise, centre (−2.5, 1.5). Applying it to (1, 1) gives (−2, 5), which is the matching vertex of C. ✓
Watch out. Do the transformations in the order given. Doing them the other way round usually gives a different answer.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next.
- Do the first transformation: rotation of 180° about (0, 0). Rotate about (0, 0): write each point as a move from the centre, turn the move using (a, b) → (−a, −b), then add the centre back on. Do this to every vertex of shape A.Shape B:
(1, 1): move from centre (1, 1) → turned (−1, −1) → (−1, −1)
(4, 1): move from centre (4, 1) → turned (−4, −1) → (−4, −1)
(1, 3): move from centre (1, 3) → turned (−1, −3) → (−1, −3) - Then do the second transformation: reflection in the line x = 1. Reflecting in a vertical line x = 1: the y-coordinate stays the same and the new x is 2 × 1 − x (the same distance on the other side of the line). Do this to every vertex of shape B.Shape C:
(−1, −1) → (2 × 1 − (−1), −1) = (3, −1)
(−4, −1) → (2 × 1 − (−4), −1) = (6, −1)
(−1, −3) → (2 × 1 − (−1), −3) = (3, −3) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (−1, −1), (−4, −1), (−1, −3) then C (3, −1), (6, −1), (3, −3)
Check. Shapes A and C are congruent (same size), and the vertex (1, 1) has gone to (3, −1): the transformations only move or flip the shape. ✓
Watch out. Do the transformations in the order given. Doing them the other way round usually gives a different answer.
6
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape A.Shape B:
(1, 2) → (1, 2 × 0 − 2) = (1, −2)
(4, 2) → (4, 2 × 0 − 2) = (4, −2)
(1, 4) → (1, 2 × 0 − 4) = (1, −4) - Then do the second transformation: reflection in the line y = 3. Reflecting in a horizontal line y = 3: the x-coordinate stays the same and the new y is 2 × 3 − y. Do this to every vertex of shape B.Shape C:
(1, −2) → (1, 2 × 3 − (−2)) = (1, 8)
(4, −2) → (4, 2 × 3 − (−2)) = (4, 8)
(1, −4) → (1, 2 × 3 − (−4)) = (1, 10) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −2), (4, −2), (1, −4) then C (1, 8), (4, 8), (1, 10)
- Decide which type of transformation it is. A translation only slides the shape. If the shape had turned, its sides would no longer be parallel to the originals.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, C faces the same way as A, and every side is parallel to its original. So the transformation is a translation. - Pick a vertex and its image. Every point moves by the same amount, so one vertex is enough to find the move.(1, 2) → (1, 8)
- Find the move across and the move up or down. Subtract the x-coordinates, then the y-coordinates (image − object). Positive means right/up, negative means left/down.Across: 1 − 1 = 0 (no move)
Up/down: 8 − 2 = 6 (6 up) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by 0
6 - Write the full description. We have found every piece of information needed for this transformation.Translation by 0
6
Check. Test another vertex: (4, 2) + 0
6 = (4, 8), which is the matching vertex of C. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape A.Shape B:
(1, 1) → (1, 2 × 0 − 1) = (1, −1)
(4, 2) → (4, 2 × 0 − 2) = (4, −2)
(2, 4) → (2, 2 × 0 − 4) = (2, −4) - Then do the second transformation: reflection in the line y = x. Reflecting in y = x swaps the coordinates: (a, b) → (b, a). Do this to every vertex of shape B.Shape C:
(1, −1) → (−1, 1)
(4, −2) → (−2, 4)
(2, −4) → (−4, 2) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −1), (4, −2), (2, −4) then C (−1, 1), (−2, 4), (−4, 2)
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare the areas: 4 square units for A and 4 for C.
The size is unchanged, C is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.Distance2 of (1, 1) from (0, 0) = 12 + 12 = 2
Distance2 of (−1, 1) from (0, 0) = −12 + 12 = 2
They match, and the same works for the other vertices, so the centre is (0, 0). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (1, 1): (1, 1) (1 right, 1 up)
From C to (−1, 1): (−1, 1) (1 left, 1 up)
This matches (a, b) → (−b, a), so the turn is 90° anticlockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° anticlockwise, centre (0, 0)
Check. Rotate (4, 2) about (0, 0) by 90° anticlockwise: we get (−2, 4), which matches the vertex (−2, 4) of C. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: rotation of 90° clockwise about (0, 0). Rotate about (0, 0): write each point as a move from the centre, turn the move using (a, b) → (b, −a), then add the centre back on. Do this to every vertex of shape A.Shape B:
(1, 1): move from centre (1, 1) → turned (1, −1) → (1, −1)
(4, 1): move from centre (4, 1) → turned (1, −4) → (1, −4)
(1, 3): move from centre (1, 3) → turned (3, −1) → (3, −1) - Then do the second transformation: rotation of 180° about (0, 0). Rotate about (0, 0): write each point as a move from the centre, turn the move using (a, b) → (−a, −b), then add the centre back on. Do this to every vertex of shape B.Shape C:
(1, −1): move from centre (1, −1) → turned (−1, 1) → (−1, 1)
(1, −4): move from centre (1, −4) → turned (−1, 4) → (−1, 4)
(3, −1): move from centre (3, −1) → turned (−3, 1) → (−3, 1) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −1), (1, −4), (3, −1) then C (−1, 1), (−1, 4), (−3, 1)
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, C is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.Distance2 of (1, 1) from (0, 0) = 12 + 12 = 2
Distance2 of (−1, 1) from (0, 0) = −12 + 12 = 2
They match, and the same works for the other vertices, so the centre is (0, 0). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (1, 1): (1, 1) (1 right, 1 up)
From C to (−1, 1): (−1, 1) (1 left, 1 up)
This matches (a, b) → (−b, a), so the turn is 90° anticlockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° anticlockwise, centre (0, 0)
Check. Rotate (4, 1) about (0, 0) by 90° anticlockwise: we get (−1, 4), which matches the vertex (−1, 4) of C. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: reflection in the line x = 1. Reflecting in a vertical line x = 1: the y-coordinate stays the same and the new x is 2 × 1 − x (the same distance on the other side of the line). Do this to every vertex of shape A.Shape B:
(2, 2) → (2 × 1 − 2, 2) = (0, 2)
(5, 2) → (2 × 1 − 5, 2) = (−3, 2)
(4, 4) → (2 × 1 − 4, 4) = (−2, 4)
(2, 4) → (2 × 1 − 2, 4) = (0, 4) - Then do the second transformation: reflection in the line y = 1. Reflecting in a horizontal line y = 1: the x-coordinate stays the same and the new y is 2 × 1 − y. Do this to every vertex of shape B.Shape C:
(0, 2) → (0, 2 × 1 − 2) = (0, 0)
(−3, 2) → (−3, 2 × 1 − 2) = (−3, 0)
(−2, 4) → (−2, 2 × 1 − 4) = (−2, −2)
(0, 4) → (0, 2 × 1 − 4) = (0, −2) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (0, 2), (−3, 2), (−2, 4), (0, 4) then C (0, 0), (−3, 0), (−2, −2), (0, −2)
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, C is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. In a half turn, the centre is exactly half-way between a point and its image. Find the midpoint for two different vertices: they should be the same point.(2, 2) and (0, 0): midpoint (1, 1)
(5, 2) and (−3, 0): midpoint (1, 1)
Both midpoints are the same, so the centre is (1, 1). - Find the angle and direction. From C the moves to the vertex and to its image point in exactly opposite directions, so it is a half turn. For a half turn the direction does not matter.From C to (2, 2): (1, 1); from C to (0, 0): (−1, −1). Both signs have changed, so the angle is 180°.
- Write the full description. We have found every piece of information needed for this transformation.Rotation, 180°, centre (1, 1)
Check. Rotate (5, 2) about (1, 1) by 180°: we get (−3, 0), which matches the vertex (−3, 0) of C. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
0. B is mapped to C by a reflection in the line x = 0. Describe fully the single transformation that maps A onto C. A has vertices at (1, 1), (3, 1), (1, 4).
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: translation by 2
0. Add 2 to every x and 0 to every y. Do this to every vertex of shape A.Shape B:
(1, 1) → (1 + 2, 1 + 0) = (3, 1)
(3, 1) → (3 + 2, 1 + 0) = (5, 1)
(1, 4) → (1 + 2, 4 + 0) = (3, 4) - Then do the second transformation: reflection in the line x = 0. Reflecting in a vertical line x = 0: the y-coordinate stays the same and the new x is 2 × 0 − x (the same distance on the other side of the line). Do this to every vertex of shape B.Shape C:
(3, 1) → (2 × 0 − 3, 1) = (−3, 1)
(5, 1) → (2 × 0 − 5, 1) = (−5, 1)
(3, 4) → (2 × 0 − 3, 4) = (−3, 4) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (3, 1), (5, 1), (3, 4) then C (−3, 1), (−5, 1), (−3, 4)
- Decide which type of transformation it is. A reflection flips the shape over. All other transformations of this type keep the shape facing the same way.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, but C is the mirror image of A: if you go round the corners in the same order, one shape is clockwise and the other is anticlockwise. So the transformation is a reflection. - Pick vertices and their images, and find the midpoints. A mirror line is exactly half-way between an object point and its image, so find the midpoint of a point and its image.(1, 1) and (−3, 1): midpoint = ((1 + −3) ÷ 2, (1 + 1) ÷ 2) = (−1, 1)
(3, 1) and (−5, 1): midpoint = ((3 + −5) ÷ 2, (1 + 1) ÷ 2) = (−1, 1) - Write the equation of the mirror line. The midpoints all have x = −1. The points are in a vertical line, so the mirror line is the vertical line x = −1.Mirror line: x = −1
- Write the full description. We have found every piece of information needed for this transformation.Reflection in the line x = −1
Check. Reflect (3, 1) in x = −1 and we get (−5, 1), which matches (−5, 1) in C. ✓
Watch out. A reflection needs the equation of the mirror line. Writing just “reflection” or “reflect in the x-axis” for a line like x = 3 is wrong.
−3. Describe fully the single transformation that maps A onto C. A has vertices at (2, 2), (4, 2), (2, 5).
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: enlargement with scale factor 2 and centre (0, 0). Enlarge about (0, 0): multiply each move from the centre by 2, then add the centre back on. Do this to every vertex of shape A.Shape B:
(2, 2): move from centre (2, 2) × 2 = (4, 4) → (4, 4)
(4, 2): move from centre (4, 2) × 2 = (8, 4) → (8, 4)
(2, 5): move from centre (2, 5) × 2 = (4, 10) → (4, 10) - Then do the second transformation: translation by −3
−3. Add −3 to every x and −3 to every y. Do this to every vertex of shape B.Shape C:
(4, 4) → (4 + (−3), 4 + (−3)) = (1, 1)
(8, 4) → (8 + (−3), 4 + (−3)) = (5, 1)
(4, 10) → (4 + (−3), 10 + (−3)) = (1, 7) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (4, 4), (8, 4), (4, 10) then C (1, 1), (5, 1), (1, 7)
- Decide which type of transformation it is. Reflections, rotations and translations never change the size of a shape. If the size changes, it must be an enlargement.Compare a side: it is 3 long in A and 6 long in C.
The sides have changed length but the shape is the same, so the transformation is an enlargement. - Find the size of the scale factor. Scale factor = length on the image ÷ matching length on the object.A side of A is 3 and the matching side of C is 6.
6 ÷ 3 = 2 - Decide whether it is positive or negative. C is the right way up and on the same side of the centre, so the scale factor is positive. It is larger than A.Scale factor = 2
- Find the centre of enlargement. Draw a straight line through a vertex of the object and the matching vertex of the image. Do the same for a second vertex. The centre is where the lines cross. (The lines are found here from their equations.)Line through (2, 2) and (1, 1): y = x
Line through (4, 2) and (5, 1): y = −x + 6
They cross at (3, 3), so C = (3, 3). - Write the full description. We have found every piece of information needed for this transformation.Enlargement, scale factor 2, centre (3, 3)
Check. Enlarge (2, 2) from (3, 3) with scale factor 2: (3 + 2 × −1, 3 + 2 × −1) = (1, 1), which is the matching vertex of C. ✓
Watch out. An enlargement needs both the scale factor and the centre. Do not forget that a fractional scale factor still means “enlargement”.
0. Shape C is found by doing the translation by 3
0 first and then the reflection in the line y = x. Is shape C in the same place as shape B? Show your working.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Work out the final image for each order, then compare them.
- Do transformation 1 then transformation 2. First reflection in the line y = x, then translation by 3
0.After the first: (2, 2), (2, 4), (3, 4), (3, 3), (5, 3), (5, 2)
After the second: (5, 2), (5, 4), (6, 4), (6, 3), (8, 3), (8, 2)
Call this shape B. - Now do them in the opposite order. First translation by 3
0, then reflection in the line y = x.After the first: (5, 2), (7, 2), (7, 3), (6, 3), (6, 5), (5, 5)
After the second: (2, 5), (2, 7), (3, 7), (3, 6), (5, 6), (5, 5)
Call this shape C. - Compare B and C. If the order did not matter, B and C would be in exactly the same place.B: (5, 2), (5, 4), (6, 4), (6, 3), (8, 3), (8, 2)
C: (2, 5), (2, 7), (3, 7), (3, 6), (5, 6), (5, 5)
The vertices are different, so the shapes are in different places.
Check. Look at the picture: B and C are not on top of each other. ✓
Watch out. Many pairs of transformations give different results when the order is swapped. (Two translations are one exception: they always give the same result.)
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Work out the final image for each order, then compare them.
- Do transformation 1 then transformation 2. First rotation of 90° anticlockwise about (1, 1), then reflection in the line x = 0.After the first: (1, 1), (0, 4), (−2, 2)
After the second: (−1, 1), (0, 4), (2, 2)
Call this shape B. - Now do them in the opposite order. First reflection in the line x = 0, then rotation of 90° anticlockwise about (1, 1).After the first: (−1, 1), (−4, 2), (−2, 4)
After the second: (1, −1), (0, −4), (−2, −2)
Call this shape C. - Compare B and C. If the order did not matter, B and C would be in exactly the same place.B: (−1, 1), (0, 4), (2, 2)
C: (1, −1), (0, −4), (−2, −2)
The vertices are different, so the shapes are in different places.
Check. Look at the picture: B and C are not on top of each other. ✓
Watch out. Many pairs of transformations give different results when the order is swapped. (Two translations are one exception: they always give the same result.)
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a translation does. A translation moves every point by the same vector, and the vector is not zero.Every point moves by 6
0. - Test a point. See if its image is the same as the point.(0, 0) → (6, 0), which is different.
- Write the answer. No point stays where it is.There are no invariant points.
Check. For a point to stay put we would need to add 0 to both coordinates, but we add 6 and 0. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a rotation does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (0, 0).
- Test the centre. Apply the transformation to the centre.(0, 0) → (0, 0)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (0, 0) only.
Check. Test another point, (1, 1): it goes to (−1, −1), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a rotation does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (1, 1).
- Test the centre. Apply the transformation to the centre.(1, 1) → (1, 1)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (1, 1) only.
Check. Test another point, (2, 2): it goes to (0, 0), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
−3
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Think about what the transformation does to each point, then find which points stay where they are.
- Say what an invariant point is. An invariant point is a point that does not move when the transformation is done.Invariant = unchanged
- Think about what a enlargement does. Every point is moved (turned or stretched) about the centre, except the centre itself: it is at distance 0 from the centre, so it cannot move.The centre is (3, 3).
- Test the centre. Apply the transformation to the centre.(3, 3) → (3, 3)
- Write the answer. Only the centre is invariant (the scale factor is not 1, and the turn is not 0° or 360°).Invariant point: (3, 3) only.
Check. Test another point, (4, 4): it goes to (5, 5), which is different, so it is not invariant. ✓
Watch out. Do not confuse “invariant point” with “image”. An invariant point is its own image.
0.
Method. Follow a general point (a, b) through both reflections. A reflection in x = c sends a to 2c − a.
- First reflection, in x = p. The x-coordinate becomes 2p − a and y does not change.(a, b) → (2p − a, b)
- Second reflection, in x = q. The new x is 2q minus the old x.(2p − a, b) → (2q − (2p − a), b)
- Simplify. Take care with the brackets: −(2p − a) = −2p + a.2q − 2p + a = a + 2(q − p)
- Conclude. Every point moves 2(q − p) to the right and 0 up, whatever a and b are.The single transformation is a translation by 2(q − p)
0
Check. With p = 1 and q = 4: 2(4 − 1) = 6, which matches the translation by 6
0 seen in this section. ✓
Watch out. The translation is twice the distance between the mirror lines, not the distance from the shape to the lines.
Method. Follow the point (a, b) through the two reflections and compare with the rule for a half turn.
- Reflect in the x-axis. The sign of the y-coordinate changes.(a, b) → (a, −b)
- Reflect in the y-axis. The sign of the x-coordinate changes.(a, −b) → (−a, −b)
- Compare with a half turn about the origin. (a, b) → (−a, −b) is the rule for a rotation of 180° about the origin.The two give the same result for every point, so the combination is a half turn.
Check. Test with (3, 1): (3, 1) → (3, −1) → (−3, −1), and a half turn of (3, 1) gives (−3, −1). ✓
Watch out. A general point (a, b) turns an example into a proof.
2.
Method. Work out the final position of the point for each order. If they differ, order matters.
- Reflect in the x-axis, then translate. The reflection changes the sign of y.(2, 1) → (2, −1) → (2 + 3, −1 + 2) = (5, 1)
- Translate, then reflect in the x-axis. Translate first, then change the sign of y.(2, 1) → (2 + 3, 1 + 2) = (5, 3) → (5, −3)
- Compare. (5, 1) is not the same as (5, −3).The order does matter, so Jo is wrong.
Check. The point is a different place for each order, so one example is enough to disprove Jo. ✓
Watch out. One counter-example is enough to show that a statement is not always true.
Method. Follow (a, b) through each reflection using the diagonal-mirror rules.
- Reflect in y = x. Swap the coordinates.(a, b) → (b, a)
- Reflect in y = −x. Swap the coordinates and change both signs.(b, a) → (−a, −b)
- Identify. (a, b) → (−a, −b) is a half turn about the origin. The mirror lines cross at O at 90°, and 2 × 90° = 180°.Rotation, 180°, centre (0, 0)
Check. Test (3, 1): (3, 1) → (1, 3) → (−3, −1). ✓
Watch out. The angle of the rotation is twice the angle between the mirror lines.
Key words. A combination of transformations means doing one transformation and then another. A point that does not move is an invariant point.
Method. Carry out the transformations one at a time, in the order given, using the image of one as the object for the next. Then compare the final image with the original to find the single transformation that does the whole job.
- Do the first transformation: reflection in the line y = 0. Reflecting in a horizontal line y = 0: the x-coordinate stays the same and the new y is 2 × 0 − y. Do this to every vertex of shape A.Shape B:
(1, 1) → (1, 2 × 0 − 1) = (1, −1)
(4, 1) → (4, 2 × 0 − 1) = (4, −1)
(1, 3) → (1, 2 × 0 − 3) = (1, −3) - Then do the second transformation: reflection in the line y = x. Reflecting in y = x swaps the coordinates: (a, b) → (b, a). Do this to every vertex of shape B.Shape C:
(1, −1) → (−1, 1)
(4, −1) → (−1, 4)
(1, −3) → (−3, 1) - Plot the shapes. Joining the image points in order gives each new shape. The picture shows the whole journey.B (1, −1), (4, −1), (1, −3) then C (−1, 1), (−1, 4), (−3, 1)
- Decide which type of transformation it is. A rotation turns the shape. Compare with a translation, where all sides stay parallel to the originals.Compare a side: it is 3 long in A and 3 long in C.
The size is unchanged, C is not flipped, but its sides are not parallel to the originals: the shape has turned. So the transformation is a rotation. - Find the centre of rotation. The centre is the same distance from a point and from its image. The origin is the first place to try. It works if the squared distances from the origin to the point and its image match.Distance2 of (1, 1) from (0, 0) = 12 + 12 = 2
Distance2 of (−1, 1) from (0, 0) = −12 + 12 = 2
They match, and the same works for the other vertices, so the centre is (0, 0). - Find the angle and direction. Write the move from C to a vertex, and the move from C to its image. A quarter turn changes (a, b) in a special way: clockwise gives (b, −a) and anticlockwise gives (−b, a).From C to (1, 1): (1, 1) (1 right, 1 up)
From C to (−1, 1): (−1, 1) (1 left, 1 up)
This matches (a, b) → (−b, a), so the turn is 90° anticlockwise. - Write the full description. We have found every piece of information needed for this transformation.Rotation, 90° anticlockwise, centre (0, 0)
Check. Rotate (4, 1) about (0, 0) by 90° anticlockwise: we get (−1, 4), which matches the vertex (−1, 4) of C. ✓
Watch out. Give three things for a rotation: the angle, the direction (clockwise or anticlockwise) and the centre. A 180° turn does not need a direction.
Method. The two mirror lines cross at (2, 1), at right angles, so the combination should be a rotation of 180° about their crossing point. Check with a point.
- Find where the lines cross. The lines are x = 2 (vertical) and y = 1 (horizontal).They cross at (2, 1).
- Reflect (5, 3) in x = 2. The new x is 2 × 2 − 5 = −1.(5, 3) → (−1, 3)
- Reflect (−1, 3) in y = 1. The new y is 2 × 1 − 3 = −1.(−1, 3) → (−1, −1)
- Check the half turn. For a half turn about (2, 1), the centre is the midpoint of (5, 3) and its image (−1, −1).Midpoint = ((5 + (−1)) ÷ 2, (3 + (−1)) ÷ 2) = (2, 1)
- Write the answer. The lines are at 90°, so the angle is 2 × 90° = 180°.Rotation, 180°, centre (2, 1)
Check. Distance check: (5, 3) is 3 right and 2 up from (2, 1); (−1, −1) is 3 left and 2 down. ✓
Watch out. The centre of a rotation made by two reflections is where the mirror lines cross.