5I.3 - Using matrices to perform transformations
We can extend our knowledge of transformations using mapping notation, to transformations using matrices:
Transformation: |
Mapping notation |
Corresponding matrix |
|
\(x'=1x+0y\) \(y'=0x+ay\) |
\[\begin{bmatrix} 1 & 0\\ 0 & a \end{bmatrix}\] |
|
\(x'=nx+0y\) \(y'=0x+1y\) |
\[\begin{bmatrix} n & 0\\ 0 & 1 \end{bmatrix}\] |
|
\(x'=1x+0y\) \(y'=0x-1y\) |
\[\begin{bmatrix} 1 & 0\\ 0 & -1 \end{bmatrix}\] |
|
\(x'=-1x+0y\) \(y'=0x+1y\) |
\[\begin{bmatrix} -1 & 0\\ 0 & 1 \end{bmatrix}\] |
|
\(x'=x+h\) \(y'=y+k\) |
\[\begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} h\\ k \end{bmatrix}\] |
|
\(x'=0x+1y\) \(y'=1x+0y\) |
\begin{bmatrix} 0 & 1\\ 1 & 0 \end{bmatrix} |
In general, a transformation using matrices will be presented in the form: \[T\left ( \begin{bmatrix}
x\\
y
\end{bmatrix} \right )
=
\begin{bmatrix}
n & 0\\
0 & a
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
+
\begin{bmatrix}
h\\
k
\end{bmatrix}\] Evaluating the matrix expression above gives:
x\\
y
\end{bmatrix} \right )
=
\begin{bmatrix}
n & 0\\
0 & a
\end{bmatrix}
\begin{bmatrix}
x\\
y
\end{bmatrix}
+
\begin{bmatrix}
h\\
k
\end{bmatrix}\] Evaluating the matrix expression above gives:
- \(x'=nx+h\)
- \(y'=ay+k\)
- Solve the matrix equation describing the transformation for \(x'\) and \(y'\).
- Solve the equation for the non-dashed terms, \(x\) and \(y\).
- Substitute the equivalent expressions (involving \(x'\) and \(y'\)) into the original rule.
- Solve the substituted expression in Step 3 for \(y\).
- Inspect the matrix equation describing the transformation and interpret the transformations using the table above.
- Evaluate the matrix equation to find \(x'=nx+h\) and \(y'=ay+k\) and use your knowledge of mapping notation to describe the transformation.
- Find the equation of the image graph and inspect the equation.
Transformations of points on the plane:
5I.3 - Example 1: Transforming a point on the plane using matrices
By using a suitable matrix equation, find the image of the point \((2,3)\) under a translation of \(4\) units in the positive \(x\)-direction and \(3\) units in the negative \(y\)-direction.
|
5I.3 - Example 1: Video solution
5I.3 - Example 1: Practice
Question 1: ABC Question 2: ABC 5I.3 - Example 1: Solutions
Question 1: ABC Question 2: ABC |
5I.3 - Example 2: Transforming a point on the plane using matrices
By using a suitable matrix equation, find the image of the point \((3,-1)\) when it is reflected in the \(y\)-axis.
|
5I.3 - Example 2: Video solution
5I.3 - Example 2: Practice
Question 1: ABC Question 2: ABC 5I.3 - Example 2: Solutions
Question 1: ABC Question 2: ABC |
5I.3 - Example 3: Transforming a point on the plane using matrices
By using a suitable matrix equation, find the image of the point \((2,5)\) under the following transformations:
|
5I.3 - Example 3: Video solution
5I.3 - Example 3: Practice
Question 1: ABC Question 2: ABC 5I.3 - Example 3: Solutions
Question 1: ABC Question 2: ABC |
Transformations of graphs using matrices:
5I.3 - Example 4: Transformations of graphs using matrices
The transformation \(T:R^2\rightarrow R^2\) is defined by \[T\left ( \begin{bmatrix}
x\\ y \end{bmatrix} \right ) = \begin{bmatrix} 1 & 0\\ 0 & -2 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} -4\\ 1 \end{bmatrix}\] The curve \(y=\frac{1}{x}\) undergoes the transformation \(T\). Find the equation of the image graph. |
5I.3 - Example 5: Transformations of graphs using matrices
The transformation \(T:R^2\rightarrow R^2\) is defined by \[T\left ( \begin{bmatrix}
x\\ y \end{bmatrix} \right ) = \begin{bmatrix} 2 & 0\\ 0 & -3 \end{bmatrix} \begin{bmatrix} x\\ y \end{bmatrix} + \begin{bmatrix} 3\\ -1 \end{bmatrix}\]The image of the curve \(y=x^2-5\) under the transformation \(T\) has the equation \(y=ax^2+bx+c\). Find the values of \(a\), \(b\) and \(c\). |
5I.3 - Example 5: Video solution
5I.3 - Example 5: Practice
Question 1: ABC Question 2: ABC 5I.3 - Example 5: Solutions
Question 1: ABC Question 2: ABC |