An affine transformation of the Euclidean plane (which is the subject of this material) is a geometric transformation that preserves the ratio of three collinear points, unless it displays them in a single common point. As a result, this transformation preserves straight lines and parallelism. The affine transformation $f$ of a plane, which maps point $X$ to point $X1$, can be represented as $$f:\,X1 = \left(\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}\right)X + \left(\begin{matrix} b_{1} \\ b_{2} \end{matrix}\right),$$ where $A = \left(\begin{matrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{matrix}\right)$ is an invertible matrix and $\vec{b}=\left(\begin{matrix} b_{1} \\ b_{2} \end{matrix}\right)$ is the vector of translation. The following dynamic applet allows us, through the transformations of a cat (it's name is Kit) photograph, to observe the influence of the elements of the matrix $A$ and the vector $\vec{b}$ on the form of the resulting affine transformation.
Implementation in JSXGraph
The affine transformation is implemented in the JSXGraph applet
through the Transformation element, specifically its generic type (see https://jsxgraph.org/docs/symbols/Transformation.html),
which is given by a matrix form in homogeneous coordinates as
$$\left(\begin{matrix}
a & b & c \\
d & e & f \\
g & h & i
\end{matrix}\right)\cdot
\left(\begin{matrix}
z \\
x \\
y
\end{matrix}\right).$$
In order to obtain a representation of just affine transformations, this general
matrix notation has to be reduced to a form
$$\left(\begin{matrix}
1 & 0 & 0 \\
b_{1} & a_{11} & a_{12} \\
b_{2} & a_{21} & a_{22}
\end{matrix}\right)\cdot
\left(\begin{matrix}
1 \\
x \\
y
\end{matrix}\right).$$
where the role of each element is clear from its labeling and from what has been stated above.
Task: Affine transformations of the Euclidean plane include also plane isometries and homothety. Try to use the sliders to set all the variable elements of $A$ and $\vec{b}$ so that you gradually create matrix representations of selected representatives of all Euclidean plane isometries and of a homothety, following their effect on the image.