Matrix Translation of a Figure

Background

Previously, we've seen how to rotate, reflect, and dilate a figure using matrices. This last lesson in this series will cover the final matrix transformation, matrix translation.

Before we continue, it's important that we review how to represent a function with a matrix. We put the ordered pair vertically in the matrix. Say we're given the points, $(x,y)$, $(a,b)$, and $(1, 2)$. We would represent these points in matrix form like so:

$$\begin{bmatrix} x&a&1 \\ y&b&2 \\ \end{bmatrix}$$

As we can see, the coordinate points are vertically paired in the matrix.

Matrix Translation

A translation is when we move a figure on a graph from one place to another. We can also call a translation a "shift", because we are shifting the function over by a given number of units. The important part about translations is that the size and orientation of the function don't change, only its location on the graph.

To translate a function use matrices, we add or subtract the amount of units that we want to move a function to each component in the matrix. If we have a triangle with coordinates $(a,b)$, $(c,d)$, and $(e,f)$, and we want to shift the triangle $1$ unit down and $2$ units to the left, our matrices would look like:

$$\begin{bmatrix} a&c&e \\ b&d&f \\ \end{bmatrix} \rightarrow \begin{bmatrix} a-2&c-2&e-2 \\ b-1&d-1&f-1 \\ \end{bmatrix}$$

Let's look at an example!

Example

Use matrices to shift the square with points $(2,3)$, $(2,1)$, $(0,1)$, and $(0,3)$ left by $3$ units.

Solution

The matrix for the given square looks like:

$$\begin{bmatrix} 2&2&0&0 \\ 3&1&1&3 \\ \end{bmatrix}$$

It doesn't really matter what order the pairs are in, as long as the respective $x$ and $y$ coordinates are paired together!

The points of our squared are graphed,

Now, we'll shift the $x$ values left by $3$ units.

$$\begin{bmatrix} 2-3&2-3&0-3&0-3 \\ 3&1&1&3 \\ \end{bmatrix} \rightarrow \begin{bmatrix} -1&-1&-3&-3 \\ 3&1&1&3 \\ \end{bmatrix}$$

This new matrix tells us that the points for our new translated square are: $(-1,3)$, $(-1,1)$, $(-3, 1)$, and $(-3,3)$. Graphed, this new square looks like:

(Video) Matrix Translations

by statslectures

This video by statslectures walks you through an example of a matrix translation.

Summary

We start with a graphed triangle with the three points, $(1,7)$, $(1,2)$, and $(6,2)$. Graph this on a coordinate system.

We will now arrange this points in a vertex matrix:

$$\begin{bmatrix} x \\ y \\ \end{bmatrix} = \begin{bmatrix} 1&1&6 \\ 7&2&2 \\ \end{bmatrix}$$

We will then use matrix addition and subtraction to translate the image. Translation is when we move a figure without changing its size or shape. If we want to move this triangle two units to the left and two units down, we would first subtract $2$ from each of the $x$ values, then we subtract $2$ from each of the $y$ values. You don't need a transformation matrix here. Our new matrix looks like:

$$\begin{bmatrix} -1&-1&4 \\ 5&0&0 \\ \end{bmatrix}$$

Now, we have our three new points: $(-1,5)$, $(-1, 0)$, and $(4,0)$.

Matrix Translation of a Figure

The last geometric transformation we will be talking about is the matrix translation of a figure.

To do a matrix translation of a figure using matrices, you will simply add or subtract the constant value to the entries in the matrix. If you want to shift a figure horizontally, then you'll only add/subtract the constant from the $y$-values. Similarly, if you want to shift a figure vertically, then you'll only add/subtract the constant from the $x$-values.