# 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:

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

Transformation of Graphs using Matrices - Rotations

Scalar Multiplication of Matrices

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## Pair your student with a tutor who understand how to reflect graphs with matrices

Reflections with matrices follow the same basic principles that students are familiar with. But there are a few important changes -- and these changes can easily catch students out. A solid choice is to revisit these new concepts alongside a tutor during 1-on-1 sessions. Tutors can answer questions you or your student didn''t have time to ask during class. They can also try a range of different explanations until a student finally "gets it." Speak with our Educational Directors today to learn more, and rest assured: Varsity Tutors will pair you student with an excellent tutor.

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