Video: Translations on a Coordinate Plane

In this video, we will learn how to apply the concept of translation to a point or a geometric shape and determine the coordinates of a translated image on the 𝑥𝑦-plane.

14:16

Video Transcript

In this video, we will learn how to apply the concept of translation to a point or a geometric shape and determine the coordinates of a translated image on the 𝑥𝑦-plane.

A translation is a particular type of transformation which simply moves or shifts an object to a different position. The object remains the same size and in the same orientation. That’s the same way up. So, the image looks exactly the same as the original object, just in a different place.

We describe translations by describing the horizontal and vertical distances that the object needs to move. For example, we might describe a translation as being 11 units right and five units down. The convention is that we always describe the horizontal movement first and then the vertical movement, in the same way that we always give the 𝑥-value first and then the 𝑦-value when writing a pair of coordinates.

In our first example, we’ll see how to describe in words the translation of a single point on a coordinate grid.

The coordinates of point 𝐴 and its image 𝐴 prime after a translation are illustrated in the graph below. Describe this translation in words.

The first thing we need to be really clear on is which is the original point and which is the image of the point after translation. The information in the question tells us that point 𝐴 is the original object and point 𝐴 prime is the image. And this is usual. We often use prime notation to describe the image of a point after it has been translated.

We know that point 𝐴 has simply been translated. So, in order to describe this translation in words, we just need to work out the number of units that point 𝐴 has been translated by in each direction. We can see that point 𝐴 has been translated to the right, first of all, and then in an upwards direction. As the scale used on each axis is one square for one unit, we can simply count the squares to determine how many units the object has been translated in each direction.

Point 𝐴 has been translated 12 units to the right and four units up. Remember, we always give the horizontal movement first followed by the vertical movement. So, we can describe this translation in words as a translation 12 units right and four units up.

Now, to avoid the risk of making mistakes when counting a lot of squares on a diagram like this one, we could also work this out by subtracting the coordinates of 𝐴 from the coordinates of 𝐴 prime. For example, to work out the number of units moved horizontally to the right, we can subtract the 𝑥-coordinate of 𝐴 from the 𝑥-coordinate of 𝐴 prime, seven minus negative five, which is the same as seven plus five, giving 12.

In the same way, to work out the vertical movement, we can subtract the 𝑦-coordinate of 𝐴 from the 𝑦-coordinate of 𝐴 prime, one minus negative three, which is equal to four. Our answer to the problem is 12 units right and four units up.

In our next example, we’ll see how we can find the image of a single point following a translation.

The point three, five has been translated three right and three down. What are the coordinates of the image?

Now, there’s no need to draw a detailed diagram involving a coordinate grid for this question, but we can at least draw a sketch of what’s happening. We start with the point three, five. And we translate it three units to the right and three units down to give its image point, whose coordinates we need to find. Well, a translation three units right, first of all, means that the 𝑥-coordinate of the point will increase by three. So, the new 𝑥-coordinate of the image will be the original 𝑥-coordinate, three, plus three.

A translation of three units down means that the 𝑦-coordinate of the point, which was originally five, will decrease by three. So, the 𝑦-coordinate of the image will be five minus three. That gives the coordinates of the image point after this translation as six, two.

So, we’ve seen how to translate a single point. In our next example, we’ll see how to perform a translation of a shape drawn on a coordinate grid.

Describe the translation from shape A to shape B.

First, let’s be entirely clear about which shape is our object and which is our image. The translation takes us from shape A to shape B. So, shape A is the object, and shape B is the image. We can see that this movement then will be to the left horizontally and up vertically. To work out the number of units A has been translated by in each direction, we can pick a pair of corresponding corners or vertices on the two shapes and then count the number of units moved.

Looking horizontally, because remember we always describe the horizontal movement first, we can see that shape A has been translated by seven units, and it’s to the left. The vertical movement is one unit, and it is upwards. So, using words, we can describe the translation from shape A to shape B as seven units left and one unit up. We could also pick another pair of corresponding vertices on the two shapes to check our answer if we wish.

Now, in all the examples we’ve seen so far, we’ve described each of the translations using words, for example, three units right and two units up. There is actually some mathematical shorthand we can use to describe translations. And it’s called a column vector.

A column vector looks a little bit like a coordinate, but instead of the two values being next to each other, they are one on top of the other. The first number describes the horizontal movement, the number of units that the object has been translated to the right. And the second number describes the vertical movement, the number of units that the object has been translated up.

So, in the case of our example of three units right and two units up, we could describe this more simply using the column vector three, two to indicate a translation three units right and two units up. If that’s the case then, how would we describe a translation two units left and six units up? Well, the vertical part is straight forward. But what about two units left?

We said in our definition that the first number in our column vector needs to be the number of units right. Well, we can think of it like this. Moving two units to the left is the same as moving negative two units to the right. The convention is that we take translations to the right to be in the positive direction as this is the direction in which the values increase. If an object is moving to the left though, we can express this using a negative value. So, the translation two units left and six units up could be written using the column vector negative two, six.

In the same way, we can see how to treat a translation in which the vertical movement is down, such as a translation of five units right and 10 units down. The horizontal part, the first number in our column vector, will be five. And a translation of 10 units down can also be thought of as a translation of negative 10 units up. So, we can write the second number in our column vector as negative 10. Once again, it’s important to remember that translations to the right and up are thought of as being positive because these are the directions in which the numbers increase.

If an object is moving to the left or down, then we would write this part of the column vector using a negative value. We may also see column vectors written using square brackets. And the meaning is exactly the same. In our final example, we’ll consider how we can describe translations using this vector notation.

Shape A has been translated to shape B and then to shape C. Write a vector to represent the translation from shape A to shape B. Write a vector to represent the translation from shape B to shape C. Write a vector to represent the translation from shape C to shape A.

We know that a translation is simply a shift or movement of an object from one position to another. In each part of this question, we were asked to describe these translations using vectors. We recall that the convention when writing this vector is to write the number of units right, first of all, followed by the number of units up. Let’s consider, then, the translation from shape A to shape B first of all. And we’ll use a pair of corresponding corners or vertices on the two shapes.

Considering the horizontal movement, first of all, and counting the squares, we can see that shape A has been translated five units to the left. Moving five units to the left is the same as moving negative five units to the right. So, the first number in our column vector to describe the horizontal translation is negative five. Vertically, we then see that shape A has been translated two units down. And a translation two units down is equivalent to a translation of negative two units up. So, we can fill in the second number in our column vector. The vector that represents the translation from shape A to shape B then is the column vector negative five, negative two. And the negatives indicate that the translation is to the left and down.

Now, let’s consider the vector that represents the translation from shape B to shape C. We can use a different pair of corresponding vertices this time if we wish. We can see that the general direction of movement or translation from shape B to shape C is to the right and then down. This means that we’re expecting a positive number for the first number in our column vector and a negative number for the second.

Looking at the horizontal movement, first of all, we can see that this vertex moves three units to the right. So, we can express this using positive three. Looking vertically, we can see that the shape is translated four units down. So, we express this as negative four. Remember, translation of negative four units up is equivalent to a translation of four units down. So, the vector that represents the translation from shape B to shape C is the column vector three, negative four.

Finally, we need to write the vector that represents the translation from shape C to shape A. And we’ll pick a different pair of corresponding vertices to use again this time. The direction of movement here is to the right and up. So, both values in our column vector will be positive. Looking horizontally, first of all, we see that shape C is translated two units to the right. So, the first value in our column vector is two. And then, looking vertically, we see that shape C is translated six units up. So, the vector that represents the translation from shape C to shape A is the column vector two, six.

Now, we can observe something interesting here. Which is that we could find the vector that would describe the translation from shape A to shape C by adding together the vector that describes the translation from A to B and then the vector that describes the translation from B to C. If we did this, then adding the component parts of our two vectors together, we’d find that the vector that represents the translation from A to C is negative two, negative six. Which we can see is the exact negative of the vector that represents the translation from C to A.

The reason for this is that these two translations are at the exact same distances but in the opposite direction. To go from shape A to shape C, we’re translating the object left and down. Whereas to go from shape C to shape A, we’re translating right and up. We have our three answers for this problem though. The three vectors are negative five, negative two; three, negative four; and two, six.

Let’s review some of the key points that we’ve seen in this video. Firstly, a translation simply moves or shifts an object to a different position. The object and its image will always be the same size and shape. We can say that they are congruent to one another. And they’ll always be in the same orientation, which means they’re the same way up.

One way that we can describe translations is using words. For example, we could describe the translation that takes an object to its image as a translation by three units right and two units down. We can also describe translations using column vectors. We write the number of units moved horizontally to the right first and then the number of units moved vertically up second. So, the translation three units right and two units down could be expressed as three, negative two.

The important thing to remember here is that the directions of right and up are taken to be positive. So, if a translation is to the left or down, then this can be represented in a column vector using a negative value.

Using these principles, we can now perform translations of either single points or geometric shapes drawn on coordinate grids. And we can describe translations using either words or column vectors.

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