# Transformation of Graphs using Matrices - Translation

When studying pre-calculus material, we commonly have to work with matrices. We can use matrices when performing a transformation on a graph. In particular, we might want to slide a two-dimensional shape around a coordinate plane without altering it in any other way. When we move a shape around without changing any other elements such as the orientation, shape, and size, we call this a translation.

Matrices are a highly important mathematical topic with many tangible applications. If we're encrypting sensitive data that we want to keep secure, we can use matrices. Matrices are valuable if we want to store large amounts of data, so they are commonly used throughout computer science. In addition, we can use matrices in physics when studying quantum systems. The potential uses of matrices are practically boundless. Before we learn how to translate shapes using matrices, it's wise for us to brush up on some related topics. These include compatible matrices and inverse matrices.

## How to translate a graph using matrices

Recall that a matrix is an array of numbers. We can use matrices to represent the coordinates of a shape. We do this by arranging the matrix so that the x-values are in the first row and the y-values are in the second row. For example, imagine we have a $\u2206ABC$ . The coordinates of each vertex are $A\left(2,2\right)$ , $B\left(2,0\right)$ , and $C\left(3,-2\right)$ . We can represent this set of numbers in matrix form like so:

Notice that each coordinate can be represented as a single column in this matrix. Let's say we want to move our $\u2206ABC$ five units to the left and three units up.

Since the x- values are represented in the top row, we will subtract five from each of these entries to shift our graph five units left. Since the y-coordinates are represented by the entries in the bottom row, we need to add three to each of these entries to shift our graph upwards by three units. This shift can be represented by the matrix:

We refer to this as the translation matrix. Recall that we can add matrices together as long as they have the same dimensions. If we add these matrices together, the resulting matrix should give us the coordinates of our shifted matrix.

Remember that to add two matrices, we simply add each corresponding element like so:

Therefore, the coordinates of our modified $\u2206{A}^{\prime}{B}^{\prime}{C}^{\prime}$ will be ${A}^{\prime}\left(-3,5\right)$ , ${A}^{\prime}\left(-7,3\right)$ , and ${C}^{\prime}\left(-2,1\right)$ .

Let's look at another example. This time we'll graph it to see what the shift looks like on a coordinate plane.

If we have a $\u2206TRI$ , where $T\left(2,-1\right)$ , $R\left(4,3\right)$ , and $I\left(-3,-2\right)$ , use matrices to shift it five units to the left and two units up.

First, we'll lay these coordinates out in the form of a matrix.

To shift this triangle five units left, we need to subtract five from each x-coordinate. To shift it upwards by two units, we must add two to each y-value. This means that our transformation matrix will look like this:

Now we simply add these two matrices together to calculate the coordinates of ${T}^{\prime}{R}^{\prime}{I}^{\prime}$ .

Therefore, the coordinates of ${T}^{\prime}{R}^{\prime}{I}^{\prime}$ are ${T}^{\prime}\left(-3,1\right)$ , ${T}^{\prime}\left(-1,5\right)$ , ${T}^{\prime}\left(-8,0\right)$ . We have a graphical depiction of this translation below. It's a good idea to get into the habit of checking our answers on graph paper. This is helpful for catching any mistakes that we might have made in our calculations.

## Practice translating a graph using matrices

1. If we have a $\u2206XYZ$ where the coordinates are $X\left(-4,5\right)$ , $Y\left(-4,1\right)$ , and $Z\left(-1,1\right)$ , use matrices to shift the graph three units right and four units down.

We'll first express the coordinates in the form of a matrix.

Since we're shifting the shape three units right, we need to insert threes for each x-coordinate in our translation matrix. Since we're shifting it four units down, each y-coordinate in the translation matrix will be -4. Our translation matrix will therefore be:

Add these two matrices together.

The coordinates of ${X}^{\prime}{Y}^{\prime}{Z}^{\prime}$ will be ${X}^{\prime}\left(-1,1\right)$ , ${Y}^{\prime}\left(-1,-3\right)$ , ${Z}^{\prime}\left(2,-3\right)$ .

2. Given the $\u2206ABC$ where the coordinates are $A\left(-2,2\right)$ , $B\left(1,4\right)$ , and $C\left(2,1\right)$ , use matrices to shift the shape two units right and three units up.

We express the coordinates in matrix form.

The translation matrix will be

Add these two together.

Therefore, the coordinates of ${A}^{\prime}{B}^{\prime}{C}^{\prime}$ are ${A}^{\prime}\left(0,5\right)$ , ${B}^{\prime}\left(3,7\right)$ , ${C}^{\prime}\left(4,4\right)$ .

3. Given a square $SQUA$ where the coordinates are $S\left(-2,2\right)$ , $Q\left(4,2\right)$ , $U\left(-2,-2\right)$ , $A\left(4,-2\right)$ , use matrices to translate the square three units left and two units down.

First, we'll put the coordinates in matrix form.

The translation matrix will be

Now add the matrices together.

The translated coordinates of ${S}^{\prime}{Q}^{\prime}{U}^{\prime}{A}^{\prime}$ are ${S}^{\prime}\left(-5,0\right)$ , ${Q}^{\prime}\left(1,0\right)$ , ${U}^{\prime}\left(-5,-4\right)$ , and ${A}^{\prime}\left(1,-4\right)$ .

4. Given a pentagon $PENTA$ where the coordinates are $P\left(0,1\right)$ , $E\left(2,-1\right)$ , $N\left(1,-3\right)$ , $T\left(-1,-3\right)$ , $A\left(-2,-1\right)$ , use matrices to translate the pentagon four units right and four units down.

We will first put the coordinates in matrix form.

The translation matrix will be

Now we'll add the two matrices together.

Therefore, the coordinates of ${P}^{\prime}{E}^{\prime}{N}^{\prime}{T}^{\prime}{A}^{\prime}$ are ${P}^{\prime}\left(4,-3\right)$ , ${E}^{\prime}\left(6,-5\right)$ , ${N}^{\prime}\left(5,-7\right)$ , ${T}^{\prime}\left(3,-7\right)$ , and ${A}^{\prime}\left(2,-5\right)$ .

5. If we have a $\u2206XYZ$ where $X\left(1,4\right)$ , $Y\left(-2,0\right)$ , and $Z\left(2,-2\right)$ , use matrices to translate it six units to the right and two units up.

Express the coordinates in the form of a matrix.

The translation matrix will be

Add the matrices together.

The coordinates of ${X}^{\prime}{Y}^{\prime}{Z}^{\prime}$ are ${X}^{\prime}\left(7,6\right)$ , ${Y}^{\prime}\left(4,2\right)$ , and ${Z}^{\prime}\left(8,0\right)$ .

6. If we have a $\u2206TRI$ with the coordinates $T\left(3,5\right)$ , $R\left(3,3\right)$ , and $I\left(-2,2\right)$ , use matrices to translate it four units left and six units down.

Express the coordinates in matrix form.

The translation matrix will be

Now we add them together.

Therefore, the coordinates of ${T}^{\prime}{R}^{\prime}{I}^{\prime}$ are ${T}^{\prime}\left(-1,-1\right)$ , ${R}^{\prime}\left(-1,-3\right)$ , and ${I}^{\prime}\left(-6,-4\right)$ .

## Topics related to the Transformation of Graphs using Matrices - Translation

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## Receive further assistance translating graphs using matrices

It's evident from all the above calculations that translating graphs with matrices can be a laborious and sometimes confusing process. It never hurts to reach out for assistance if you feel that you're starting to get confused. Did you know that tutoring has been shown to help most students make improvements in their academic performance? When you reach out to Varsity Tutors, one of our helpful Educational Directors will respond to any questions you have about tutoring. When talking to you, they will get an idea of your unique situation and select a tutor who is well-equipped to address any problems you're facing. Additionally, an Educational Director will put together a personalized learning plan. You and your tutor can change this learning plan whenever you like.

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