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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?

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    ...In all of my math classes (I'm a math major), we talk about plotting points, but we graph functions. You can plot specific points in the graph of a function, but you don't just plot a function. Generally, plot, as a noun, refers to a set of points that may or may not be connected by a line, but that cannot be represented as a function. I don't know if there's some technical ground behind this, but if there is, one of us is being confused...2011-03-28
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    So you're saying plotting is discrete and graphing is continuous? (or at least piecewise continuous?)2011-03-28
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    That's my impression, but I have nothing to back it up except my own observations, which is why I left a comment instead of an answer.2011-03-28
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    There used to be graphics printers called *line plotters* which joined lines. *Graph* just means *draw* (as a verb) and I would take it to be more general than *plot*.2011-03-28
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    I just checked my ODEs book and it labels a "plot" of a sine wave. (which is continuous) Interestingly, it asks you to "sketch the graph of..." a continuous function. The book uses both terms in similar ways but I can't figure out how it chooses one over the other.2011-03-29
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    Also "figure", and occasionally "chart". Special case: "histogram" (used a lot in particle physics to mean a generic plot because of the (pernicious?) influence of [PAW](http://paw.web.cern.ch/paw/) and [Root](http://root.cern.ch/drupal/)).2011-03-29
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    There are so many other definitions of 'plot' and 'graph' that do not overlap, that it is hard to think of them as synonymous.2011-03-29
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    I don't really think they're synonymous. I'm just trying to figure out when to use one over the other.2011-03-29
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    @kitukwfyer, you should make it an official answer. This is the right difference. This is why electronic point printers are also called plotters. See also a plot of land: a small surface, a spot. You draw a graph from mathematical analysis and for empirical functions, you measure or compute points and you plot them.2011-03-29
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    @kitukwfyer - I'm only a physicist but we say plot, as in "plot y as a function of x" - in fact I can't think of when I would use graph as a verb. It might also be a UK/US thing?2011-03-29
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    @mgb: It's possible. I would understand "plot y as a function of x," but it sounds a little funny to me. I would use "graph" there preferentially. Still, I would understand it. It could also be a math/physics thing. My calc III professor never tires of warning us about the inherent dangers of using physicists' notation. XP I'm sure he'd be happy to extend it to language. :)2011-03-29
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    @kitukwfyer - come over to the dark side, we have cooler toys!2011-03-29
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    This has been migrated to Math as requested by the OP himself.2011-04-05
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    @RegDwight: awesome - thank you!2011-04-05
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    see also: "chart"2015-04-30

5 Answers 5

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As per @Alain Pannetier's advice, I'm making my comment into an answer. (To lazy to rephrase now, but I'll most likely try to make it pretty and more answer-y later.)

...In all of my math classes (I'm a math major), we talk about plotting points, but we graph functions. You can plot specific points in the graph of a function, but you don't just plot a function. Generally, plot, as a noun, refers to a set of points that may or may not be connected by a line, but that cannot be represented as a function. I don't know if there's some technical ground behind this, but if there is, one of us is being confused...

So, in short, "plot" is used for a finite set of points, while a "graph" is used for a function comprised of infinite points.

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    I'm not sure I believe this. (Particularly the "function comprised of infinite points" part.)2011-03-29
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    Also, there is such a thing as discrete functions. The [Mandelbrot set](http://en.wikipedia.org/wiki/Mandelbrot_set) is one example.2011-03-29
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    Yeah, I'm tired and writing a German essay while listening to Korean music. Who knows where my brain's at? Apparently not where I keep my definition of continuity. Editing, and probably not for the last time...Also, plotting all of the points of a function whose domain is all real numbers would take a while...2011-03-29
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we plot points, but we graph functions

this is useful guidance since: "plot" is to draw a graphic representation with respect to measurements or coordinates "graph" is to illustrate connections between several things by plotting dots and lines

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A graph in the sense of the object "a graph of a function" often has the specific definition, for a real-valued function $f$ of being the set of points $\{(x,y)|y=f(x)\}$. That is, graph can be used as a noun to mean literally the set of ordered pairs. It is interesting to note that this is also sometimes the definition of the function itself.

Beyond that one specific technical meaning, I do not think that (in the context of ODEs, calculus, or precalculus mathematics) there is a formal distinction between the two words. Informally, I use each word a bit differently.

The word graph in the sense of the action "graph a function" means to make a drawing of the set of points in the (noun) graph of the function. This sometimes has the connotation of a careful, precise drawing. In contrast, "sketch the graph" sometimes has the connotation of being a less-formal illustration showing key features of the graph of the function without necessarily being as precise.

The word plot in the sense of the object "a plot of a function" means a visual representation. There are other types of plots, such as scatter plots and line plots, that would not typically be referred to as graphs. I would say that graphs are a specific type of plot, but I'm not sure that's quite right.

The word plot in the sense of the action "plot the graph of a function" or "plot some points" typically means drawing with the connotation of precision. I generally would not use the phrasing "plot the graph."

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  • '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.

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Although I am not a native English speaker, I believe plotting refers to drawing a diagram of any kind. So you can plot a function, an architectural design , a software class diagram etc.

Graph is a more specific term that refers to plotting by using a system of axes. So while both are correct, graph is more precise.

To add to the confusion, there are also the notion of graph from graph theory. In my native language (Greek) we use graph only in relation with graph theory and grapheme for the drawing/plot/graph of a function. However, grapheme has a different meaning in English.

In conclusion, you could call a quick drawing on a blackboard that is not very precise a plot but not a graph. A precise drawing using axes can be called a graph and a plot.