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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, 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 = 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 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.