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Integer Translations, Paths Between Points

Understand how to move points from one location to another on a coordinate grid.

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Integer Translations, Paths Between Points

License: CC BY-NC 3.0

Reggie is taking art and has been given a picture of two people sitting in a room with a cat on the floor. His task is to redraw the picture but with the cat by the woman's feet, next to the sewing basket, instead of where it currently is. How can he figure out how to reorganize this image?

In this concept, you will learn how to translate points from one place to another on a coordinate plane.

Translating Points on a Coordinate Plane

coordinate grid is a grid in which points are graphed. It usually has two or more intersecting lines which divide a plane into quadrants, and in which ordered pairs, or coordinates, are defined. It usually has four quadrants, or sections, to it. 

The origin is the place where the two lines intersect. Its coordinates are defined as (0,0).

The x-axis is the line running from left to right that has the numbers defined on it and is usually labeled with an "x". The x-coordinate of an ordered pair is found with relation to it. All the points located on the x-axis have a y-coordinate of 0.

The y-axis is the central line that runs up-down and is labeled with a "y". Y-coordinates are plotted in reference to this axis. Again, all the x-coordinates of points located on the y-axis are 0.

An ordered pair is a list of two numbers in parenthesis, separated by a comma like this: (5,-3). It tells where a point is located on the coordinate plane. The first number is the x-coordinate. It tells you where to go on the x-axis. If it is positive, you go to the right. If it is negative, you go to the left. The second number is the y-coordinate. It tells you where to go on the y-axis. If it is positive, you go up. If it is negative, you go down.

The  vertex of a shape is the place where two sides of the shape come together. In general, when a shape is defined inside of a coordinate plane, it is defined by the vertices, and then the lines are drawn to connect them.

A translation is when a figure or a point is slid on a coordinate grid from one place or another without changing its overall orientation. 

In the figure above, the original point \begin{align*}A\end{align*}A is located at (-2, 3). Then it is translated to (4,1). \begin{align*}A'\end{align*}A is the new point. What was the overall translation?

First, figure out how many units \begin{align*}A\end{align*}A moved on the \begin{align*}x\end{align*}x axis

To find out, count the difference between the original point and the final point

In this case, it is +6 units.

Next, find how many units \begin{align*}A\end{align*}A moved on the \begin{align*}y\end{align*}y axis.

If you count, you can see that it moved -2 units (remember that negative 'y' means down).

Then, put them together.

The translation is (6, -2).


Example 1

Earlier, you were given a problem about Reggie and his painted cat.

First, he lightly sketches a coordinate grid over the picture. 

Next, he determines that the cat basically fits in a rectangle and is made up of a circle and an oval. He lightly sketches a box around the original cat on the coordinate plane. 

Then, he finds the vertices of the original cat rectangle. They seem to be about (-11, -12), (-5, -12), (-11, -15), (-5, -15).

Then, he decides he wants the cat to go up 1 and over 9. 

Then, he adds 9 to the x-coordinate of all of the vertices and adds 1 to the y-coordinates. This yields a new box with vertices (-2, -11), (4, -11), (-2, -14), (4, -14).

Finally, he draws this new box, and he sketches the cat shapes within it until it looks like a cat again. He erases the boxes, and his new cat is happy in its new place.

Example 2

Calculate the following transformation numerically, and use a graph to check your answer. 

A point is plotted at \begin{align*}(-2,1)\end{align*}(2,1) and then translated \begin{align*}(3,-4)\end{align*}(3,4), what are the coordinates of the translated point?

First, add the x-coordinate of the original point to the x-coordinate of the transformation.

-2 + 3 = 1

Next, add the y-coordinate of the original point to the y-coordinate of the transformation.

1 + (-4) = -3.

Then, write the resultant ordered pair.

The answer is (1,-3).

Finally, check your answer by graphing the original point and physically carrying out the transformation.

Example 3

If the original point is located at (3,5) and is moved three units across and four units up, what are the coordinates of the final point?

First, figure out the x translation.

"Across" is in the x-direction. Because it isn't otherwise stated, you can assume it is to the right, which is positive.

Next, add the x-translation to the initial x-coordinate to find the new x-coordinate.

3 + 3 = 6

Then, figure out the y translation.

It is four units up.

Then, add that to the initial y-coordinate.

4 + 5 = 9

Finally, write the resultant coordinate.

The answer is \begin{align*}(6,9)\end{align*}(6,9).

Example 4

If the initial point is (-5,4) and the final point is (2,7) write the translation.

First, find the x-translation.

-5 + ___ = 2 . In order to get from -5 to 2, the point must move 7 units to the right.

Next, find the y-translation.

4 + ___ = 7 . In order to get from 4 to 7, the point must move up 3 units.

Then, write the translation in words.

The answer is 7 units to the right and 3 units up.

Example 5

If a certain point is translated 2 units to the left and 4 units down, it is at the final point (-2, 6). What was the location of the original point?

First, find the original x-coordinate by working backwards.

2 units to the left means it is translated -2 on the x-axis. ___ + -2 = -2 The only coordinate that makes this work is 0.

Next, find the original y-coordinate the same way.

4 units down means it is translated -4 on the y-axis. ___ + (-4) = 6 The original y-coordinate must be 10 to make this work.

Then, write the original coordinate as an ordered pair.

The answer is (0,10)

Finally, draw a graph with both the beginning and final point to check your work.


Write each translation in coordinate notation.

  1. \begin{align*}A\end{align*}A to \begin{align*}A'\end{align*}A
  2. \begin{align*}B\end{align*}B to \begin{align*}B'\end{align*}B
  3. \begin{align*}C\end{align*}C to \begin{align*}C'\end{align*}C
  4. \begin{align*}D\end{align*}D to \begin{align*}D'\end{align*}D
  5. \begin{align*}E\end{align*}E to \begin{align*}E'\end{align*}E
  6. \begin{align*}F\end{align*}F to \begin{align*}F'\end{align*}F
  7. \begin{align*}G\end{align*}G to \begin{align*}G'\end{align*}G
  8. \begin{align*}H\end{align*}H to \begin{align*}H'\end{align*}H
  9. \begin{align*}H\end{align*}H to \begin{align*}E\end{align*}E
  10. \begin{align*}E\end{align*}E to \begin{align*}D\end{align*}D
  11. \begin{align*}E'\end{align*}E to \begin{align*}G\end{align*}G
  12. \begin{align*}E'\end{align*}E to \begin{align*}C\end{align*}C
  13. \begin{align*}B\end{align*}B to \begin{align*}A'\end{align*}A
  14. \begin{align*}B\end{align*}B to \begin{align*}G'\end{align*}G
  15. \begin{align*}C'\end{align*}C to \begin{align*}D'\end{align*}D

Review (Answers)

To see the Review answers, open this PDF file and look for section 11.17. 





The x-axis is the horizontal axis in the coordinate plane, commonly representing the value of the input or independent variable.

y axis

The y-axis is the vertical number line of the Cartesian plane.


The coordinates of a point represent the point's location on the Cartesian plane. Coordinates are written in ordered pairs: (x, y).


Latitude is a coordinate that specifies the north-south location of a point on the Earth's surface.


Longitude is a coordinate that specifies the east-west location of a point on the Earth's surface.

Ordered Pair

An ordered pair, (x, y), describes the location of a point on a coordinate grid.


The origin is the point of intersection of the x and y axes on the Cartesian plane. The coordinates of the origin are (0, 0).


A quadrant is one-fourth of the coordinate plane. The four quadrants are numbered using Roman Numerals I, II, III, and IV, starting in the top-right, and increasing counter-clockwise.

Image Attributions

  1. [1]^ License: CC BY-NC 3.0

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