A linear transformation can rotate, scale, shear or reflect vectors about the origin, but ordinary translation is not linear because it does not preserve the zero vector.

An affine transformation adds a translation to a linear transformation:

$$\mathbf{x}'=A\mathbf{x}+\mathbf{t}.$$

Here $A$ controls the linear part and $\mathbf{t}$ shifts every point by the same amount.

Affine transformations preserve straight lines and parallelism. They do not necessarily preserve lengths or angles: uniform rotation does, while scaling and shear do not.

Points and displacement vectors behave differently under translation. A point moves by $\mathbf{t}$, while a displacement between two points transforms only by $A$ because the translation cancels.

Graphics uses affine transformations to place, orient and scale objects without changing the coordinates stored in the object's own local model.