What are forms of affine transformations? Please help me tackle the below question:


  Explain THREE forms of affine transformations using relevant examples for each case.


The answers I want to see are scaling, rotation and reflection.
Note: I have answered before the edit from trichoplax and I thought you were searching for other transformations other than the one you mentioned. The informations below are still useful so I will keep the answer here, but it does not directly answer your question.



Affine transformations (surprise!) map affine spaces to affine spaces. An affine space is substantially a vector space where you can establish an origin and define points as tuples of their coordinates. This is far from a formal definition, if you are interested in one I can edit the answer. 

Now an affine transformation $T$ transform points into points and must preserve affine combinations: 

$T(\lambda_1 P_1 + \lambda_2 P_2 + ...  +  \lambda_n P_n) = \lambda_1 T(P_1) + \lambda_2 T(P_2) + ... + \lambda_n T(P_n)  $   

Given $\sum_{i}^{n} \lambda_i = 1 $

However these transformations can't be arbitrary as the following must be preserved: 


Parallelism is preserved. This means that if you transform parallel lines they remain parallel after the affine transformation.
Relative ratios are preserved. This means that if you have $R = (1-\beta) P + \beta Q$  then $T(R) = (1 - \beta) T(P) + \beta T(Q)$ 
Addition between vector and points are preserved. Meaning that $T( P + \vec{v} ) = T(P) + T(\vec{v})$


With these properties in mind you can come up with a very big number of affine transforms yourself. A couple of obvious ones other the ones you mentioned are: 

Translation 

$$T_vp = 
\begin{bmatrix}
1&amp;0&amp;0&amp;v_x\\0&amp;1&amp;0&amp;v_y\\0&amp;0&amp;1&amp;v_z\\0&amp;0&amp;0&amp;1\end{bmatrix}
\begin{bmatrix}
p_x+v_x\\p_x+v_y\\p_z+v_z\\1\end{bmatrix}=p+v$$

That moves a point into a specific direction by a specific amount specified by a displacement vector. 

Shearing 


image from wikipedia

That is a transform that displace all points in a given direction by an amount that is proportional to their perpendicular distance to a line parallel to that direction. 

For example the transform matrix for an horizontal shear in a 2D space is given by: 

$$
\begin{bmatrix}x'\\y'\end{bmatrix} = \begin{bmatrix}x+my\\y\end{bmatrix} = 
\begin{bmatrix}1&amp;m\\0&amp;1\end{bmatrix}\begin{bmatrix}x\\y\end{bmatrix}$$



Again, these two are just example, the important information you should really keep is the definition (and properties) of an affine transform; with that in mind it shouldn't be too hard to recognize an affine transform.  Also note that combining affine transforms will give you an affine transform!