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A graph is not a picture. Rather, *a graph is a set of ordered pairs
of numbers*. When you plot a graphâ€™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*
= 2*x* + 1 is the set of ordered pairs (*x*,*y*) such
that *x* and *y* are real numbers, and *y* =
2*x* + 1. The pairs (0.1, 1.2), (0.4, 1.8), (2.7, 6.4) are all in
the graph of *y* = 2*x* + 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* = 2*x* + 1
displayed in two different coordinate systems. The graph of *y* =
2*x* + 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* = 2*x* + 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.*