GCSE Maths: Translation (Foundation)
Part of the GCSE Maths transformations and similarity chapter (Foundation). Chapter overview · Higher 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.
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.
3. Write down the coordinates of the vertices of the image B. A has vertices at (1, 1), (3, 1), (1, 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.2
3 means 2 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 + 2, 1 + 3) = (3, 4)
(3, 1) → (3 + 2, 1 + 3) = (5, 4)
(1, 4) → (1 + 2, 4 + 3) = (3, 7) - 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 (3, 4), (5, 4), (3, 7).
Check. Undo the translation: take 2
3 away from (3, 4) and we get back to (3 − 2, 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 (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.