Transformation of Graphs using Matrices - Reflection

Reflections are some of the most interesting transformations we can carry out on a graph. As the name implies, this creates a "mirror image" of our preimage, and it can be useful in a number of different circumstances. But can we reflect images using matrices instead of our usual algebraic methods? Let''s find out:

Reflecting graphs using matrices

As we may recall, a reflection is a type of translation that involves "flipping" an image. An easy example of a reflection is picking up a paper star, flipping it over, and putting it back down on a table. Unlike a few other transformations, reflection does not alter an image''s size, shape, or orientation.

• For a vertical reflection, $f\left(x\right)$ becomes $-f\left(x\right)$
• For a horizontal reflection, $f\left(x\right)$ becomes $f\left(-x\right)$

We also know that a reflection maps every point of a figure to an image across a line of symmetry.

But what if we wanted to reflect our image using matrices instead?

We use three reflection matrices to accomplish this:

$\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$ -- Reflection over the x-axis.

$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$ -- Reflection over the y-axis.

$\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right]$ -- Reflection over the line $y=x$

We always place these reflection matrices on the left for our multiplication operations.

Practicing our reflections using matrices

Let''s say we have a pentagon with the following coordinates:

• $A\left(2,4\right)$
• $B\left(4,3\right)$
• $C\left(4,0\right)$
• $D\left(2,-1\right)$
• $E\left(0,2\right)$

Can we use matrices to reflect this pentagon over the y-axis?

We can start by writing out our coordinate points in matrix form. Note that the x-coordinates go in row 1, while the y-coordinates go in row 2.

$\left[\begin{array}{ccccc}2& 4& 4& 2& 0\\ 4& 3& 0& -1& 2\end{array}\right]$

Our next step is to select the correct reflection matrix. Recall that in order to reflect over the y-axis, we need to use this reflection matrix:

$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]$

Now let''s put these two matrices together into a multiplication equation -- remembering to put our reflection matrix on the left:

$\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]\left[\begin{array}{ccccc}2& 4& 4& 2& 0\\ 4& 3& 0& -1& 2\end{array}\right]=\left[\begin{array}{ccccc}-2& -4& -4& -2& 0\\ 4& 3& 0& -1& 2\end{array}\right]$

This last resulting matrix represents the coordinates of the reflected image. Here are those coordinates:

• $A^{\prime \prime }\left(-2,4\right)$
• $B^{\prime \prime }\left(-4,3\right)$
• $C^{\prime \prime }\left(-4,0\right)$
• $D^{\prime \prime }\left(-2,-1\right)$
• $E^{\prime \prime }\left(0,2\right)$

What does this reflected pentagon look like compared to the original image? Let''s take a look:

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