![]() 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. 'Graph' (noun) is closely related to 'function'. 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. ![]() And this object is the goal, not the representation itself. When this word is used the resulting 'graph' (as a noun) is by design a representation of some mathematical object (see below). 'Graph' (verb) has a more abstract intangible connotation than 'plot'. I would not be surprised if the word 'plot' has its roots in surveying, navigation, and astronomy, in that order. Obviously, 'plot' is associated with 'plotting devices' which evokes all sorts of technologies. '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. I generally would not use the phrasing "plot the graph." 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 would say that graphs are a specific type of plot, but I'm not sure that's quite right. There are other types of plots, such as scatter plots and line plots, that would not typically be referred to as graphs. The word plot in the sense of the object "a plot of a function" means a visual representation. 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. This sometimes has the connotation of a careful, precise drawing. 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. Informally, I use each word a bit differently. It is interesting to note that this is also sometimes the definition of the function itself.īeyond 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. ![]() That is, graph can be used as a noun to mean literally the set of ordered pairs. 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 $\$.
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