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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, according to the conventions of that 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 u and v* is the set of ordered pairs

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

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

It is customary that the first coordinate in a pair is a value of the relationship's independent variable and the second a value of the relationships dependent variable. But this is only customary. There might not be an independent or dependent variable, such as in $x^2-5xy+y^3=3$ (see Figure 3.5.1, left) or conventions might be violated (see Figure 3.5.1, right). In Figure 3.5.1 (right), the graph of $x^2-5xy+y^3=3$ is displayed within a rectangular coordinate system, but values of $x$ are located on the vertical axis and values of $y$ are located on the horizontal axis.

*Figure 3.5.1. Graph of $x^2-5xy+y^3$ displayed in Cartesian (rectangular) coordinates in two ways. Neither $x$ nor $y$ can be taken as an independent variable, and values of $x$ and $y$ are placed on axes according to two different conventions.*

Figure 3.5.2 shows the graph of $v=2u+1$
displayed in two different coordinate systems. On the left, the graph of $v =
2u + 1$ appears as a line in a rectangular (Cartesian) coordinate
system because it is conventional in rectangular coordinates that values of the independent variable (in this case *u*) are located on the horizontal axis and values of the dependent variable (in this case *v*) are located on the vertical axis.

Figure 3.5.2 (right) shows the graph of $v=2u+1$ in a polar coordinate system. It appears as a spiral because, in polar coordinates, values of *v* give the radius of a circle centered at the pole and values of *u* give a direction from the pole. The graph
(set of ordered pairs) is the same in both cases, but in different
coordinate systems the graph of $v=2u+1$ appears
differently.

*Figure 3.5.2. Two displays of the graph of
$v=2u+1$. Left: Displayed in a rectangular coordinate system.
Right: Displayed in a polar coordinate system. The same graph (set of
ordered pairs) appears differently in different coordinate systems.*

As we explain more fully in Section 3.9, the coordinates of a point $(u,v)$ when displayed in Cartesian coordinates are read from left-to-right. In the polar system, thanks to Isaac Newton, a point's undisplayed coordinates $(u,v)$ are read from right-to-left to display them in polar coordinates. Coordinates are thus written $(r,\theta)$, where the value of *r* (the dependent variable) is the radius of a circle centered at the pole and the value of *\theta* (the independent variable) is a direction from the pole. Points displayed in polar coordinates therefore seem to have their coordinates reversed from Cartesian conventions, wherein the first coordinate of a displayed coordinate pair is the relationship's independent variable.

Of course, you can depart from any convention in a coordinate system as long as you state clearly how you departed from it.