Geometric transformations
Master geometric transformations for Year 11 Specialist Mathematics in Queensland (QCAA). Reflections, rotations about the origin, enlargements and one-way stretches are each represented by a 2×2 matrix, and multiplying that matrix by a point gives its image.
You will learn the standard transformation matrices, how to apply them to points and polygons, and how to recognise the transformation a given matrix represents — the matrix picture of geometry that leads on to composing transformations and using inverses and determinants later in the course.
Theory
Geometric transformations move every point of the plane, and the standard reflections, rotations and stretches about the origin are each represented by a 2×2 matrix in Year 11 Specialist Mathematics (QCAA, Queensland). Multiplying that matrix by a point’s column vector gives the image, and the columns of the matrix are the images of the base vectors. This page covers the standard matrices and how to apply and recognise them.
A geometric transformation maps each point \((x,y)\) of the plane to an image \((x',y')\). The transformations in this topic are linear and fix the origin, so each one is represented by a single 2×2 matrix \(A\), and the image of a point is found by \(\begin{pmatrix}x'\\y'\end{pmatrix}=A\begin{pmatrix}x\\y\end{pmatrix}\).
The columns of \(A\) are the images of the base vectors \(i=(1,0)\) and \(j=(0,1)\). This one fact lets you both write the matrix for a described transformation and recognise the transformation a given matrix represents.
The standard transformations are the reflections in the \(x\)-axis, the \(y\)-axis and the line \(y=x\); the rotations of \(90^\circ\), \(180^\circ\) and \(270^\circ\) about the origin \(O\); the enlargement (dilation) of factor \(k\); and the one-way stretches parallel to an axis.
To transform a polygon, apply the matrix to every vertex in turn and join the images in the same order.
The image of a point under the transformation with matrix \(A\) is:
The standard matrices (columns = images of \(i\) and \(j\)):
| Transformation | Matrix | Rule \((x,y)\mapsto\) |
|---|---|---|
| Reflection in the \(x\)-axis | \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\) | \((x,-y)\) |
| Reflection in the \(y\)-axis | \(\begin{pmatrix}-1&0\\0&1\end{pmatrix}\) | \((-x,y)\) |
| Reflection in \(y=x\) | \(\begin{pmatrix}0&1\\1&0\end{pmatrix}\) | \((y,x)\) |
| Rotation \(90^\circ\) anticlockwise | \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\) | \((-y,x)\) |
| Rotation \(180^\circ\) | \(\begin{pmatrix}-1&0\\0&-1\end{pmatrix}\) | \((-x,-y)\) |
| Rotation \(270^\circ\) anticlockwise | \(\begin{pmatrix}0&1\\-1&0\end{pmatrix}\) | \((y,-x)\) |
| Enlargement, factor \(k\) | \(\begin{pmatrix}k&0\\0&k\end{pmatrix}\) | \((kx,ky)\) |
| Stretch \(\parallel x\)-axis, factor \(k\) | \(\begin{pmatrix}k&0\\0&1\end{pmatrix}\) | \((kx,y)\) |
| Stretch \(\parallel y\)-axis, factor \(k\) | \(\begin{pmatrix}1&0\\0&k\end{pmatrix}\) | \((x,ky)\) |
How to apply a transformation to a point
- Write the matrix \(A\) for the transformation (use the columns = images of \(i\) and \(j\), or the standard-matrix table).
- Set up the product \(\begin{pmatrix}x'\\y'\end{pmatrix}=A\begin{pmatrix}x\\y\end{pmatrix}\), with the point as a column on the right.
- Multiply row by column: the top entry is \(ax+by\) and the bottom entry is \(cx+dy\).
- Read off the image \((x',y')\); for a polygon, repeat for every vertex and join the images in order.
Write the reflection matrix, then multiply it by the point (keep \(x\), negate \(y\)):
| \(\begin{pmatrix}x'\\y'\end{pmatrix}\) | \(=\) | \(\begin{pmatrix}1&0\\0&-1\end{pmatrix}\begin{pmatrix}5\\-2\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}(1)(5)+(0)(-2)\\(0)(5)+(-1)(-2)\end{pmatrix}\) | |
| \(=\) | \(\begin{pmatrix}5\\2\end{pmatrix}\) |
The image is \((5,2)\).
The \(90^\circ\) anticlockwise rotation has matrix \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\); apply it:
| \(\begin{pmatrix}x'\\y'\end{pmatrix}\) | \(=\) | \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\begin{pmatrix}2\\3\end{pmatrix}\) |
| \(=\) | \(\begin{pmatrix}(0)(2)+(-1)(3)\\(1)(2)+(0)(3)\end{pmatrix}\) | |
| \(=\) | \(\begin{pmatrix}-3\\2\end{pmatrix}\) |
The image is \((-3,2)\).
Read off where the base vectors go — the columns are the images of \(i\) and \(j\):
| \(i=\begin{pmatrix}1\\0\end{pmatrix}\) | \(\mapsto\) | \(\begin{pmatrix}0\\1\end{pmatrix}\) |
| \(j=\begin{pmatrix}0\\1\end{pmatrix}\) | \(\mapsto\) | \(\begin{pmatrix}1\\0\end{pmatrix}\) |
| \((x,y)\) | \(\mapsto\) | \((y,x)\) |
Swapping the coordinates is exactly the rule for a reflection in the line \(y=x\).
It is a reflection in the line \(y=x\).
Reflection in \(y=x\) swaps the coordinates of every vertex, \((x,y)\mapsto(y,x)\):
| \(P(1,1)\) | \(\mapsto\) | \(P'(1,1)\) |
| \(Q(4,1)\) | \(\mapsto\) | \(Q'(1,4)\) |
| \(R(4,3)\) | \(\mapsto\) | \(R'(3,4)\) |
The image triangle has vertices \((1,1)\), \((1,4)\) and \((3,4)\).
Common pitfalls
Frequently asked questions
How do you find the image of a point under a transformation matrix?
Write the point as a column vector and multiply it on the left by the matrix: \(\begin{pmatrix}x'\\y'\end{pmatrix}=A\begin{pmatrix}x\\y\end{pmatrix}\). The top entry is \(ax+by\) and the bottom entry is \(cx+dy\).
How do you write the matrix for a given transformation?
Work out where the base vectors \(i=(1,0)\) and \(j=(0,1)\) go. Those two images are the first and second columns of the matrix.
What is the matrix for a reflection in the \(y\)-axis?
It is \(\begin{pmatrix}-1&0\\0&1\end{pmatrix}\), which negates \(x\) and keeps \(y\), so \((x,y)\mapsto(-x,y)\).
What is the matrix for a \(90^\circ\) anticlockwise rotation about the origin?
It is \(\begin{pmatrix}0&-1\\1&0\end{pmatrix}\), which sends \((x,y)\) to \((-y,x)\).
How do you transform a whole polygon?
Apply the matrix to every vertex separately, then join the image points in the same order to get the image polygon.
What is the difference between an enlargement and a one-way stretch?
An enlargement \(\begin{pmatrix}k&0\\0&k\end{pmatrix}\) scales both coordinates by \(k\); a one-way stretch scales only one coordinate, e.g. \(\begin{pmatrix}k&0\\0&1\end{pmatrix}\) stretches parallel to the \(x\)-axis.