Curve, as a word for graph, in math, is misleading, as a label for count pairs, on many levels. First, it is because it refers to a visual impression, since many of mathematical functions represented have the form of curvy line, of the concept. What it fails to capture and signify is that the “curve” represents pairs of numbers, i.e. pairs of counts. Function, in mathematics, is a set of pairs. It represents counts you have paired for this or that reason. Function is not a formula. Formula is only a rule. Functions is more of a map, or the most precisely, it is a set of paired numbers.
Then, how we ended up with the “curve” word? Some genius came up with the idea to consider counts of length, i.e. magnitude of length, number that is obtained by measuring length, length of line. The count obtained by measuring the length of a line will be the same as the count obtained say, by counting cars, measuring mass, temperature, stating the price of goods, or speed, or how many trains go through the station during the day, or any other counts that you can think of. Thus, showing the line on the graph, and knowing that the next step is measuring its length of that line, and matching that count (obtained from length) with the count of another object will give you the representation of the quantity you are interested in, as depicted in Fig 1.
Fig 1. Matching the counts.
Each point on the curve represents a pair of lengths, namely x and y. This is the most important property of the curve, that its points represent the pairs of lengths. The shape of curve is not there, in most of the cases, to be considered aesthetically. The curve shape tells you the relation between the paired lengths, and that’s the most important information you can get by visually inspecting the curve of the graph in mathematics.
Fig 2. Another view of matching the counts and using a graph for respresentation.
So, the major conclusions follow. Function in math is a defined pair of counts. Functions is not a formula. The other good word for function is mapping, map, between two or more numbers. Pairing numbers is another better word for function. Curve in math graph is a visual representation of paired lengths. Lengths of the curve, line are equivalent to the numbers, counts obtained from other sources, for instance, by measurements, agreements, counting, picking the number.
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