Transformation of Graphs using Matrices

By now, we are probably familiar with transformations of graphs. This involves changing the coordinate points on a graph according to a set of rules. Using various rules, we can rotate, translate, dilate, and reflect our graphs. But can we accomplish this with matrices instead of our usual algebraic methods? Let''s find out:

Translating graphs using matrices

As we may recall, a translation involves moving an image across a graph from one location to another. Translations do not affect the size, shape, or orientation of the images they affect.

You may have encountered translations in the past, and we have learned that we can translate graphs both vertically and horizontally.

$y=f\left(x\right)+a$ translates our graph up by units

$y=f\left(x\right)-a$ translates our graph down by units

$y=f\left(x+a\right)$ translates our graph units to the left

$y=f\left(x-a\right)$ translates our graph units units to the right

But what happens if we use matrices to translate our graph instead?

Let''s start with the coordinates of a triangle:

• $T\left(2,-1\right)$
• $R\left(4,3\right)$
• $I\left(-3,-2\right)$

Can we move this triangle 5 units to the left and 2 units up using matrices?

Let''s start by writing our coordinates in matrix format, with the top row representing our x-values and the bottom row representing our y-values:

$\left[\begin{array}{ccc}2& 4& -3\\ -1& 3& -2\end{array}\right]$

If we want to translate the triangle 5 units to the left, we need to decrease each x-coordinate by 5.

If we want to translate the triangle 2 units up, we need to increase each y-coordinate by 2.

Using these values, we can create a translation matrix. This matrix simply contains all of the translations that we need to make, with elements in the appropriate rows for x and y:

$\left[\begin{array}{ccc}-5& -5& -5\\ 2& 2& 2\end{array}\right]$

Now we can add this translation matrix to our original matrix for our triangle coordinates:

$\left[\begin{array}{ccc}2& 4& -3\\ -1& 3& -2\end{array}\right]+\left[\begin{array}{ccc}-5& -5& -5\\ 2& 2& 2\end{array}\right]=\left[\begin{array}{ccc}-3& -1& -8\\ 1& 5& 0\end{array}\right]$

This last resulting matrix represents our coordinate points for the translated triangle. Remember, all we need to do is add the elements that occupy the same position in their respective matrices.

We are left with the following coordinates:

• $T\left(-3,1\right)$
• $R\left(-1,5\right)$
• $I\left(-8,0\right)$

Let''s take a look at the graph of our translated triangle compared to the original image:

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Translating graphs adds a new layer of complexity to normal translation operations, and this can catch students off guard during classroom lectures. Fortunately, students have the opportunity to revisit these potentially confusing topics during 1-on-1 tutoring sessions. They can take this opportunity to ask various questions that their teacher might not have had time to address during class time. Tutors can also choose a pace that feels productive yet manageable for your student. Speak with our Educational Directors today, and rest assured: Varsity Tutors will pair your student with a suitable tutor.