Keys, compass, and data
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If you can illustrate numerical information in a pictorial form then that is a graph. So there are infinitely many kinds of graphs. One of my favourites illustrates Napoleon's invasion of Russia and his retreat. See link for more.
The answer depends on what information is graphed. There are distance-time graphs, velocity-time graphs, speed-time graphs, acceleration-time graphs.
Mathematicians use graphs to illustrate information.
Graphs are a convenient way to impart information to some people, and graph paper simplifies the process of producing graphs.
Yes