TITLE: When should I use "graph" vs. "plot"?
QUESTION [22 upvotes]: My ODEs textbook uses both graph and plot but I can't figure out how it chooses one over the other.
From the book:

Sketch the graph of the solution in the x1x2-plane for t ≥ 0.
  [this one was referring to a continuous function]

Also from the book:

Plots of the solution and a tangent line approximation for the initial value problem (11).
  [this one referring to a continuous solution]

Is there a formal, to some degree, distinction between the two terms? When do I call it a plot and when do I call it a graph?

REPLY [2 votes]: 'Plot' (verb) has a more active, tangible connotation than 'graph' (verb), ie a person or machine placing ink on paper, a surveyor doing measurements and markings, a computer monitor radiating visible light producing some representation or figure, , and so on.  Obviously, 'plot' is associated with 'plotting devices' which evokes all sorts of technologies.  I would not be surprised if the word 'plot' has its roots in surveying, navigation, and astronomy, in that order.  
'Graph' (verb) has a more abstract intangible connotation than 'plot'.  When this word is used the resulting 'graph' (as a noun) is by design a representation of some mathematical object (see below).  And this object is the goal, not the representation itself.  For example, a teacher may graph $y = x^2$ on a dusty blackboard, but once he starts talking about the graph, it's usually not the chalk he's talking about anymore, he's talking about a mathematical object.
'Graph' (noun) is closely related to 'function'.  In mathematics, functions are often equated with their graphs, IE a function $f: X \rightarrow Y$ is just the choice of two sets $X$ and $Y$ and an appropriate subset of $f \subset X \times Y$, the latter being 'the graph' in the formal sense.  When visualized in the usual way, with elements from X as the independent variables, we can call these the traditional graphs.
Finally, from wikipedia: graphs, plots, and charts.  In the first link,  graphs are of the 'traditional' sort.  In the second link, a large number of plot types are given, including box plots and scatter plots.  However, in addition there are many graphs of the traditional sort, but with modified/scaled axes.  The entry on charts also has many examples which may be called either graphs or plots.