GCSE Maths: Translation (Higher)
Part of the GCSE Maths transformations and similarity chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Translation
A translation slides a shape from one place to another. The shape is not turned, not flipped and not resized: every point moves the same distance in the same direction. We describe a translation with a column vector, which is two numbers written one above the other in brackets.
−2: 4 to the right and 2 down. Every vertex moves the same way.
In a column vector a
b:
- the top number is the move across (the x-direction): positive means right, negative means left;
- the bottom number is the move up or down (the y-direction): positive means up, negative means down.
So −3
2 means 3 left and 2 up. To translate a point, add the top number to its x-coordinate and the bottom number to its y-coordinate. For example, (5, 1) translated by −3
2 is (5 − 3, 1 + 2) = (2, 3).
Describing a translation. Choose any vertex and its image, then work out image − object for the x-coordinates (top number) and for the y-coordinates (bottom number). To undo a translation, use the opposite vector: the inverse of 4
−2 is −4
2.
Key ideas
- A translation slides a shape. The image is congruent to the object and faces the same way.
- A column vector a
b means a across (right if positive) and b up/down (up if positive). - Image point = object point + vector. Vector = image − object.
- The vector that undoes a translation is the opposite vector (change both signs).
- A translation has no invariant points.
3. Write down the coordinates of the vertices of the image B. A has vertices at (1, 1), (4, 1), (1, 3).
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.4
3 means 4 right, 3 up. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(1, 1) → (1 + 4, 1 + 3) = (5, 4)
(4, 1) → (4 + 4, 1 + 3) = (8, 4)
(1, 3) → (1 + 4, 3 + 3) = (5, 6) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (5, 4), (8, 4), (5, 6).
Check. Undo the translation: take 4
3 away from (5, 4) and we get back to (5 − 4, 4 − 3) = (1, 1). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
−4. Write down the coordinates of the vertices of the image B. A has vertices at (5, 4), (7, 4), (7, 5), (6, 5), (6, 7), (5, 7).
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.−5
−4 means 5 left, 4 down. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(5, 4) → (5 − 5, 4 − 4) = (0, 0)
(7, 4) → (7 − 5, 4 − 4) = (2, 0)
(7, 5) → (7 − 5, 5 − 4) = (2, 1)
(6, 5) → (6 − 5, 5 − 4) = (1, 1)
(6, 7) → (6 − 5, 7 − 4) = (1, 3)
(5, 7) → (5 − 5, 7 − 4) = (0, 3) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (0, 0), (2, 0), (2, 1), (1, 1), (1, 3), (0, 3).
Check. Undo the translation: take −5
−4 away from (0, 0) and we get back to (0 + 5, 0 + 4) = (5, 4). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- 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 B.
The size is unchanged, B 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, 1) → (4, −3)
- 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: 4 − 1 = 3 (3 right)
Up/down: −3 − 1 = −4 (4 down) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by 3
−4 - Write the full description. We have found every piece of information needed for this transformation.Translation by 3
−4
Check. Test another vertex: (4, 1) + 3
−4 = (7, −3), which is the matching vertex of B. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. The vector is the image coordinates minus the object coordinates.
- Find the move across. Image x minus object x. A negative answer means left.−1 − 2 = −3 (3 left)
- Find the move up or down. Image y minus object y. A negative answer means down.1 − 5 = −4 (4 down)
- Write the column vector. Across on the top, up/down underneath.−3
−4
Check. Add the vector to P: (2 − 3, 5 − 4) = (−1, 1) = P′. ✓
Watch out. Take away the object from the image, not the other way round.
2. B is then translated by −1
4 to give shape C. Write down the single vector that maps A to C.
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. Two translations one after the other are the same as one translation whose vector is the sum of the two vectors.
- Add the top numbers. The total move across is the sum of the two moves across.3 + (−1) = 2
- Add the bottom numbers. The total move up is the sum of the two moves up.2 + 4 = 6
- Write the single vector. 2
6
Check. Start at (0, 0): move 3
2 to (3, 2), then −1
4 to (2, 6), which is the same as one move of 2
6. ✓
Watch out. You can add vectors in either order: translations are one of the few transformations where the order does not matter.
Watch out
Do not mix up the two numbers. The top number is across (x); the bottom number is up/down (y). A column vector with a negative top number moves the shape to the left, and a negative bottom number moves it down.Exercise 21C Translation
Non-calculator2
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.3
2 means 3 right, 2 up. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(2, 2) → (2 + 3, 2 + 2) = (5, 4)
(5, 2) → (5 + 3, 2 + 2) = (8, 4)
(2, 4) → (2 + 3, 4 + 2) = (5, 6) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (5, 4), (8, 4), (5, 6).
Check. Undo the translation: take 3
2 away from (5, 4) and we get back to (5 − 3, 4 − 2) = (2, 2). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
0
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.−4
0 means 4 left. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(2, 2) → (2 − 4, 2 + 0) = (−2, 2)
(5, 2) → (5 − 4, 2 + 0) = (1, 2)
(2, 4) → (2 − 4, 4 + 0) = (−2, 4) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (−2, 2), (1, 2), (−2, 4).
Check. Undo the translation: take −4
0 away from (−2, 2) and we get back to (−2 + 4, 2 − 0) = (2, 2). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
−5
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.0
−5 means 5 down. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(2, 2) → (2 + 0, 2 − 5) = (2, −3)
(5, 2) → (5 + 0, 2 − 5) = (5, −3)
(2, 4) → (2 + 0, 4 − 5) = (2, −1) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (2, −3), (5, −3), (2, −1).
Check. Undo the translation: take 0
−5 away from (2, −3) and we get back to (2 − 0, −3 + 5) = (2, 2). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
4
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.−3
4 means 3 left, 4 up. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(2, 2) → (2 − 3, 2 + 4) = (−1, 6)
(5, 2) → (5 − 3, 2 + 4) = (2, 6)
(2, 4) → (2 − 3, 4 + 4) = (−1, 8) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (−1, 6), (2, 6), (−1, 8).
Check. Undo the translation: take −3
4 away from (−1, 6) and we get back to (−1 + 3, 6 − 4) = (2, 2). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
−4. Write down the coordinates of the vertices of the image B. A has vertices at (1, 1), (4, 1), (3, 3), (1, 3).
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.3
−4 means 3 right, 4 down. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(1, 1) → (1 + 3, 1 − 4) = (4, −3)
(4, 1) → (4 + 3, 1 − 4) = (7, −3)
(3, 3) → (3 + 3, 3 − 4) = (6, −1)
(1, 3) → (1 + 3, 3 − 4) = (4, −1) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (4, −3), (7, −3), (6, −1), (4, −1).
Check. Undo the translation: take 3
−4 away from (4, −3) and we get back to (4 − 3, −3 + 4) = (1, 1). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
−5. Write down the coordinates of the vertices of the image B. A has vertices at (6, 5), (8, 5), (8, 6), (7, 6), (7, 8), (6, 8).
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.−4
−5 means 4 left, 5 down. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(6, 5) → (6 − 4, 5 − 5) = (2, 0)
(8, 5) → (8 − 4, 5 − 5) = (4, 0)
(8, 6) → (8 − 4, 6 − 5) = (4, 1)
(7, 6) → (7 − 4, 6 − 5) = (3, 1)
(7, 8) → (7 − 4, 8 − 5) = (3, 3)
(6, 8) → (6 − 4, 8 − 5) = (2, 3) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (2, 0), (4, 0), (4, 1), (3, 1), (3, 3), (2, 3).
Check. Undo the translation: take −4
−5 away from (2, 0) and we get back to (2 + 4, 0 + 5) = (6, 5). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
−5. Write down the coordinates of the vertices of the image B. A has vertices at (1, 5), (4, 6), (2, 8).
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. To translate a shape by a column vector, add the top number to every x-coordinate and the bottom number to every y-coordinate.
- Read the column vector. The top number is the move across (x direction): positive means right, negative means left. The bottom number is the move up or down (y direction): positive means up, negative means down.6
−5 means 6 right, 5 down. - Add the vector to every vertex. Every point of the shape slides the same way, so we add the same numbers to every x and every y.(1, 5) → (1 + 6, 5 − 5) = (7, 0)
(4, 6) → (4 + 6, 6 − 5) = (10, 1)
(2, 8) → (2 + 6, 8 − 5) = (8, 3) - Plot the image points and join them to make shape B. A translation does not turn, flip or resize the shape. The image looks exactly like the object, just in a new place.B has vertices at (7, 0), (10, 1), (8, 3).
Check. Undo the translation: take 6
−5 away from (7, 0) and we get back to (7 − 6, 0 + 5) = (1, 5). ✓
Watch out. The top number is across (x) and the bottom number is up/down (y). Do not swap them, and watch the signs: a negative top number means left.
4
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Add 3 to every x and 4 to every y.Translation by 3
4 - Apply it to the point. Follow the rule using the coordinates of the point.(2, −1) → (2 + 3, (−1) + 4) = (5, 3)
- Write the coordinates of the image. The image of P is called P'.P' = (5, 3)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
2
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Add −5 to every x and 2 to every y.Translation by −5
2 - Apply it to the point. Follow the rule using the coordinates of the point.(2, −1) → (2 + (−5), (−1) + 2) = (−3, 1)
- Write the coordinates of the image. The image of P is called P'.P' = (−3, 1)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
−6
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Add 0 to every x and −6 to every y.Translation by 0
−6 - Apply it to the point. Follow the rule using the coordinates of the point.(2, −1) → (2 + 0, (−1) + (−6)) = (2, −7)
- Write the coordinates of the image. The image of P is called P'.P' = (2, −7)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
−3
Key words. A transformation changes the position or size of a shape. The original shape is the object; the new shape is the image.
Method. Apply the transformation to the coordinates of P.
- Remember what this transformation does. Add −2 to every x and −3 to every y.Translation by −2
−3 - Apply it to the point. Follow the rule using the coordinates of the point.(2, −1) → (2 + (−2), (−1) + (−3)) = (0, −4)
- Write the coordinates of the image. The image of P is called P'.P' = (0, −4)
Check. Look at the picture: P and P' are placed as the transformation requires. ✓
Watch out. Take care with minus signs: write the new coordinates carefully, one at a time.
2
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- 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 B.
The size is unchanged, B 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, 1) → (6, 3)
- 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: 6 − 1 = 5 (5 right)
Up/down: 3 − 1 = 2 (2 up) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by 5
2 - Write the full description. We have found every piece of information needed for this transformation.Translation by 5
2
Check. Test another vertex: (4, 1) + 5
2 = (9, 3), which is the matching vertex of B. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
3
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- 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 B.
The size is unchanged, B 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.(7, 1) → (1, 4)
- 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 − 7 = −6 (6 left)
Up/down: 4 − 1 = 3 (3 up) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by −6
3 - Write the full description. We have found every piece of information needed for this transformation.Translation by −6
3
Check. Test another vertex: (10, 1) + −6
3 = (4, 4), which is the matching vertex of B. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
−4
Key words. A translation slides a shape without turning or flipping it. A column vector gives the move: the top number is across (x), the bottom number is up or down (y).
Method. Decide which type of transformation maps A onto B, then find the extra details each type needs (a line, a centre, an angle, a vector or a scale factor).
- 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 B.
The size is unchanged, B 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, 5) → (5, 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: 5 − 1 = 4 (4 right)
Up/down: 1 − 5 = −4 (4 down) - Write the move as a column vector. The across move goes on top and the up/down move goes underneath.Translation by 4
−4 - Write the full description. We have found every piece of information needed for this transformation.Translation by 4
−4
Check. Test another vertex: (3, 5) + 4
−4 = (7, 1), which is the matching vertex of B. ✓
Watch out. Write the vector as across on top, up/down underneath, and take image − object, not the other way round.
4
Method. Vector = image − object.
- Across. Image x minus object x.4 − 1 = 3
- Up/down. Image y minus object y.6 − 2 = 4
- Write the vector. 3
4
Check. (1 + 3, 2 + 4) = (4, 6). ✓
Watch out. Image minus object.
0
Method. Vector = image − object.
- Across. Image x minus object x.2 − 5 = −3
- Up/down. Image y minus object y.3 − 3 = 0
- Write the vector. −3
0
Check. (5 − 3, 3 + 0) = (2, 3). ✓
Watch out. A bottom number of 0 means no vertical move.
−5
Method. Vector = image − object.
- Across. Image x minus object x.1 − (−2) = 1 + 2 = 3
- Up/down. Image y minus object y.−1 − 4 = −5
- Write the vector. 3
−5
Check. (−2 + 3, 4 − 5) = (1, −1). ✓
Watch out. Subtracting a negative number adds.
−3 to give shape B. What is the column vector of the translation that takes B back to A?
3
Method. To undo a move, go the opposite way: use the same numbers with the signs changed.
- Write the original move. B is 4 right and 3 down from A.4
−3 = 4 right, 3 down - Reverse it. To get back we go 4 left and 3 up.−4
3
Check. Add the two vectors: (4 + (−4), −3 + 3) = (0, 0), so there is no overall move. ✓
Watch out. The inverse vector has both signs changed, not just one.
3 moves a shape 3 to the right and 2 up.” Explain what is wrong with Tom’s answer.
Method. Read the column vector in the right order: the top number is the x move and the bottom number is the y move.
- Read the top number. The top number is 2: that is the move across. Positive means right.2 to the right
- Read the bottom number. The bottom number is 3: that is the move up. Positive means up.3 up
- Say what Tom did. Tom swapped the numbers.Correct: 2 right and 3 up
Check. (0, 0) translated by 2
3 is (2, 3): 2 across and 3 up. ✓
Watch out. Across first (like x before y).
2 and then by −5
1. Work out where the robot finishes, and the single translation vector from the start to the finish.
3
Method. Do the first move, then the second move, and compare the finish with the start.
- First move. Add the vector to the start point.(1 + 3, 1 + 2) = (4, 3)
- Second move. Add the second vector to the new point.(4 − 5, 3 + 1) = (−1, 4)
- Find the single vector. Finish minus start.(−1 − 1, 4 − 1) = (−2, 3), which is −2
3
Check. Add the two vectors: (3 + (−5), 2 + 1) = (−2, 3). ✓
Watch out. Add the vectors, or follow each move in turn; both give the same answer.
Method. First find the vector of the translation, then add it to the new point.
- Find the vector. Image minus object.(7 − 2, −3 − 1) = (5, −4), which is 5
−4 - Add the vector to (−1, 4). Every point moves by the same vector.(−1 + 5, 4 − 4) = (4, 0)
Check. Subtract the vector from (4, 0): (4 − 5, 0 + 4) = (−1, 4). ✓
Watch out. The vector comes from the first pair of points only; use it for the second point.
2 followed by a translation by −1
4 gives the same final position as the translations done in the opposite order. Start at (0, 0).
Method. Work out the finish for each order.
- Order 1. (0, 0) → 3
2 → (3, 2) → −1
4 → (2, 6) - Order 2. (0, 0) → −1
4 → (−1, 4) → 3
2 → (2, 6) - Compare. Both finish at the same point.Both finish at (2, 6).
Check. 3 + (−1) = 2 and 2 + 4 = 6 whichever order we add. ✓
Watch out. Adding numbers can be done in either order, which is why the order of two translations does not matter.
0, which means no move at all
Method. A translation moves every point by the same vector. Think about which vectors leave a point where it started.
- Consider a point. For a point to be mapped to itself, we add 0 to its x-coordinate and 0 to its y-coordinate.Vector = 0
0 - Consider the shape. If every vertex does not move, then the shape does not move.The translation is “do nothing”.
- Conclude. If the vector is not zero, every point moves, and a bounded shape cannot return to the same place.Only the zero vector maps a shape onto itself.
Check. Try 1
0 on a square: the square moves one unit to the right, so it is not in the same place. ✓
Watch out. A reflection or rotation can map a symmetric shape onto itself; a translation cannot.