At least in undergraduate algebra, we can discuss the properties of algebraic structures and their elements without losing generality with notation such as let $G$ be a group and $g\in G$. In using this notation we can manipulate the symbols and determine properties that hold for algebraic structures with certain conditions.
When talking about graphs and certain properties they have in analysis, it seems more difficult to create this sort of notation given that the moment you put pencil to paper and draw you immediately lose generality by giving the graph a certain shape.
Given this roadblock, how do you approach learning analysis concepts which make more sense visually, yet do not have some arbitrary representation?
Put simply, for me at least, it depends on the behaviour I'm investigating! For many cases I simply draw a nice, curvy, cubic-like graph, and often this is enough. You shouldn't get too hung up on the lack of generality when trying to gain intuition about things. The most important thing to ask is: why stop at one picture? If you draw one graph and don't understand the behaviour, play around, try some radically different types of picture and see if any new insights come to light. The point of the picture should be to allow you to understand the specific examples presented so that you may more easily apply the rule in general.
A related concept arose between my friends when we studied topology. With so many different topological spaces to choose from how do you ever nail anything down in pictures? The answer is, more often than not, to simply draw a dull, vaguely shaped (usually contractible!) blob. Despite not illustrating any features worthy of interest, this is often enough as a base to investigate certain ideas (such as coning, suspension etc.). Of course once you become interested in homotopy, perhaps you start considering a blob with some nonzero genus, so that you can see more interesting behaviour of loops and such.
In general it comes down to a tradeoff between faithfully representing the concept on the page, and actually ending up with a simple and easy to look at diagram. But I would say that even a too-specific diagram is far better than no picture at all, as long as one is careful to keep the proviso in mind that things need not be exactly that way.