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# Section 3.5: What is a Graph?

A graph is not a picture. Rather, a graph is a set of ordered pairs of numbers. When you plot a graph’s ordered pairs in a coordinate system you get a visual representation of the graph. The representation of a graph you see results from merging the conventions of the coordinate system with the set of ordered pairs being plotted. The graph of y = 2x + 1 is the set of ordered pairs (x,y) such that x and y are real numbers, and y = 2x + 1. The pairs (0.1, 1.2), (0.4, 1.8), (2.7, 6.4) are all in the graph of y = 2x + 1.

In general, the graph of a mathematical statement involving two variables x and y is the set

$$\{(x,y) \text{ such that the values of }x \text{ and } y\text{ make the statement true}\}.$$

A mathematical statement's displayed graph is the statement's graph displayed within the conventions of a coordinate system. Even though we shouldn't, we will often use the terms graph and displayed graph interchangeably. Just keep in mind that a graph's appearance depends upon the coordinate system in which it is displayed.

Figure 3.5.1 shows the graph of y = 2x + 1 displayed in two different coordinate systems. The graph of y = 2x + 1 appears as a line in a rectangular (Cartesian) coordinate system. It appears as a spiral in a polar coordinate system. The graph (set of ordered pairs) is the same in both cases, but in different coordinate systems the graph of y = 2x + 1 appears differently.

Figure 3.5.1. Two plots of the graph of y = 2x + 1. Left: Plotted in a rectangular coordinate system. Right: Plotted in a polar coordinate system. The same graph (set of ordered pairs) appears differently in different coordinate systems.