The two types of pencil are distinguished by what their lines have in common and by what they differ in.
Remark
The fundamental difference: in a proper pencil all the lines have a point in common (the support) and differ in their slope; in an improper pencil all the lines have the same slope and differ in their position.
On the left a proper pencil: all the lines rotate about the support . On the right an improper pencil: all the lines are parallel and translate keeping the same slope.
Links
Topics: Pencils of lines
Concepts: Slope · Pencil of lines · Improper pencil · Proper pencil · Line · Support of the pencil
Functions: Line
Skills: Analytic geometry · Plotting a graph