Transformation of Graphs using Matrices - Dilation

One common task that we learn when studying geometry is how to dilate a graph. However, we can do the same task with matrices. Matrices are highly versatile tools that can represent a broad range of mathematical structures. For instance, matrices are used heavily in animation and computer graphics.

How to dilate a graph using matrices

Dilation is one of the transformations that we can perform on a two-dimensional shape. Dilation involves changing the size of the shape. In order to perform dilation on a shape, we need a center point and a scale factor. The scale factor is essentially the ratio by which a figure is reduced or enlarged. Let''s say we have a triangle $∆DEF$ , where $D$ is at the point $\left(-3,1\right)$ , $E$ at $\left(2,4\right)$ , and $F$ at $\left(5,-4\right)$ . We want to dilate this figure by a scale factor of two to get $∆D^{\prime }{E}^{\prime }{F}^{\prime }$

The points of each vertex can be inserted into a matrix like so.

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

As we can see, each column in this matrix represents an individual coordinate in the triangle. Since we want to dilate this figure by a scale factor of two, we need to use scalar multiplication. All this involves is multiplying each matrix entry by the scale factor.

$2\times \left[\begin{array}{ccc}-3& 2& 5\\ 1& 4& -4\end{array}\right]=\left[\begin{array}{ccc}2\times -3& 2\times 2& 2\times 5\\ 2\times 1& 2\times 4& 2\times -4\end{array}\right]=\left[\begin{array}{ccc}-6& 4& 10\\ 2& 8& -8\end{array}\right]$

Therefore, the vertices of $∆D^{\prime }{E}^{\prime }{F}^{\prime }$ are $D^{\prime }\left(-6,2\right),E^{\prime }\left(4,8\right),F^{\prime }\left(10,-8\right)$

Let''s try another one, but this time we''ll scale it down. If we have a triangle $∆ABC$ where $A\left(-8,-4\right)$ , $B\left(8,4\right)$ , and $C\left(8,-4\right)$ , let's dilate it by a scale factor of $\frac{1}{4}$ .

We''ll put these coordinates in the form of a matrix, then we''ll multiply each entry by the scale factor.

$\left[\begin{array}{ccc}-8& 8& 8\\ -4& 4& -4\end{array}\right]$

$\frac{1}{4}\times \left[\begin{array}{ccc}-8& 8& 8\\ -4& 4& -4\end{array}\right]=\left[\begin{array}{ccc}\frac{1}{4}\times -8& \frac{1}{4}\times 8& \frac{1}{4}\times 8\\ \frac{1}{4}\times -4& \frac{1}{4}\times 4& \frac{1}{4}\times -4\end{array}\right]=\left[\begin{array}{ccc}-2& 2& 2\\ -1& 1& -1\end{array}\right]$

Therefore, the new coordinates are $A^{\prime }\left(-2,-1\right),B^{\prime }\left(2,1\right),C^{\prime }\left(2,-1\right)$ .

Practice dilating graphs using matrices

Now that we know how to dilate an object using a matrix, let''s try some practice problems. It may be tempting to look at the answers ahead of time, but it''s a good idea to give the problem a try before doing so.

1. If we have a triangle $∆XYZ$ where $X\left(-6,6\right)$ , $Y\left(6,3\right)$ , and $Z\left(3,-6\right)$ , use a matrix to dilate the graph by a factor of $\frac{1}{3}$ .
   

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

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

2. If we have a square $ABCD$ where $A\left(-1,2\right)$ , $B\left(2,2\right)$ , $C\left(-1,-2\right)$ , and $D\left(2,-2\right)$ , use a matrix to dilate the graph by a factor of two.
   

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

   The dilated coordinates will be $A^{\prime }\left(-2,4\right),B^{\prime }\left(4,4\right),C^{\prime }\left(-2,-4\right),D^{\prime }\left(4,-4\right)$ .

3. If we have a triangle $∆DEF$ where $D\left(-1,4\right)$ , $E\left(-3,0\right)$ , and $F\left(1,2\right)$ , use a matrix to dilate the graph by a factor of three.
   

   $3\left[\begin{array}{ccc}-1& -3& 1\\ 4& 0& 2\end{array}\right]=\left[\begin{array}{ccc}-3& -9& 3\\ 12& 0& 6\end{array}\right]$

   The dilated coordinates of $∆DEF$ will be $D^{\prime }\left(-3,12\right),E^{\prime }\left(-9,0\right),F^{\prime }\left(3,6\right)$ .

4. If we have a square $ABCD$ where $A\left(-6,6\right)$ , $B\left(6,6\right)$ , $C\left(-6,-3\right)$ , and $D\left(6,-3\right)$ , use a matrix to dilate the graph by a factor of $\frac{1}{3}$ .
   

   $\frac{1}{3}\left[\begin{array}{cccc}-6& 6& -6& 6\\ 6& 6& -3& -3\end{array}\right]=\left[\begin{array}{cccc}-2& 2& -2& 2\\ 2& 2& -1& -1\end{array}\right]$

   The dilated coordinates will be $A^{\prime }\left(-2,2\right),B^{\prime }\left(2,2\right),C^{\prime }\left(-2,-1\right),D^{\prime }\left(2,-1\right)$ .

5. If we have a triangle $∆XYZ$ where $X\left(-1,2\right)$ , $Y\left(2,2\right)$ , and $X\left(1,-1\right)$ , use a matrix to dilate the graph by a factor of three.
   

   $3\left[\begin{array}{ccc}-1& 2& 1\\ 2& 2& -1\end{array}\right]=\left[\begin{array}{ccc}-3& 6& 3\\ 6& 6& -3\end{array}\right]$

   The dilated coordinates will be $X^{\prime }\left(-3,6\right),Y^{\prime }\left(6,6\right),Z^{\prime }\left(3,-3\right)$ .

6. If we have a triangle $∆ABC$ where $A\left(-1,3\right)$ , $B\left(1,1\right)$ , and $C\left(-1,-2\right)$ , use a matrix to dilate the graph by a factor of two.
   

   $2\left[\begin{array}{ccc}-1& 1& -1\\ 3& 1& -2\end{array}\right]=\left[\begin{array}{ccc}-2& 2& -2\\ 6& 2& -4\end{array}\right]$

   The dilated coordinates will be $A^{\prime }\left(-2,6\right),B^{\prime }\left(2,2\right),C^{\prime }\left(-2,-4\right)$ .

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