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Translations intro

Learn what translations are and how to perform them in our interactive widget.
To see what a translation is, please grab the point and move it around.
Nice! You translated the point. In geometry, a translation moves a thing up and down or left and right.
Here, try translating this segment by dragging it from the middle, not the endpoints:
Notice how the segment's direction and length stayed the same as you moved it. Translations only move things from one place to another; they don't change their size, arrangement, or direction.
Now that we've got a basic understanding of what translations are, let's learn how to use them on the coordinate plane.

Translations on the coordinate plane

Coordinates allow us to be very precise about the translations we perform.
Without coordinates, we could say something like, "We get start color #ca337c, B, prime, end color #ca337c by translating start color #11accd, B, end color #11accd down and to the right."
A pre-image point labeled B at the top left of the space. The image point labeled B prime is down and to the right of the pre-image.
But that's not very precise. If we use a coordinate grid, we can say something more exact: "We get start color #ca337c, B, prime, end color #ca337c by translating start color #11accd, B, end color #11accd by 5 units to the right and 4 units down."
Point B is translated five units to the right and four units down to form B prime.
More compactly, we can describe this as a translation by open angle, 5, comma, minus, 4, close angle.
The negative sign in front of the 4 tells us the vertical shift is downwards instead of upwards. Similarly, a translation to the left is indicated by the first value being negative.

Pre-images and images

For any transformation, we have the pre-image figure, which is the figure we are performing the transformation upon, and the image figure, which is the result of the transformation. For example, in our translation, the pre-image point was start color #11accd, B, end color #11accd and the image point was start color #ca337c, B, prime, end color #ca337c.
Note that we indicated the image by start color #ca337c, B, prime, end color #ca337c, pronounced B prime. It is common, when working with transformations, to use the same letter for the image and the pre-image, simply adding the "prime" suffix to the image.

Let's try some practice problems

Problem 1

Each unit in the grid equals 1.
Draw the image of the line segment under a translation by open angle, 2, comma, minus, 3, close angle.

Problem 2

Each unit in the grid equals 1.
Draw the image of the circle after a translation by open angle, minus, 7, comma, minus, 1, close angle.

Challenge problem

What translation maps point start color #11accd, C, end color #11accd to point start color #ca337c, C, prime, end color #ca337c?
2 points on a coordinate plane. The x- and y- axes scale by one. The pre-image point is labeled C. The image point labeled C prime is located left one unit from the pre-image and up four units.
Choose 1 answer:
Choose 1 answer:

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