Coordinate geometry deals with graphing (or plotting) and analyzing points, lines, and areas on the coordinate plane (coordinate graph). Each point on a number line is assigned a number. In the same way, each point in a plane is assigned a pair of numbers. These numbers represent the placement of the point relative to two intersecting lines. In coordinate graphs (see Figure 1), two perpendicular number lines are used and are called coordinate axes. One axis is horizontal and is called the x‐axis. The other is vertical and is called the y‐axis. The point of intersection of the two number lines is called the origin and is represented by the coordinates (0, 0).
Figure 1. An x‐y coordinate graph.
Each point on a plane is located by a unique ordered pair of numbers called the coordinates. Some coordinates are noted in Figure 2.
Figure 2. Graphing or plotting coordinates.
Notice that on the x‐axis numbers to the right of 0 are positive and to the left of 0 are negative. On the y‐axis, numbers above 0 are positive and below 0 are negative. Also, note that the first number in the ordered pair is called the x‐coordinate, or abscissa, and the second number is the y‐coordinate, or ordinate. The x‐coordinate shows the right or left direction, and the y‐coordinate shows the up or down direction.
The coordinate graph is divided into four quarters called quadrants. These quadrants are labeled in Figure 3.
Figure 3. Coordinate graph with quadrants labeled.
Notice the following:

In quadrant I, x is always positive and y is always positive.

In quadrant II, x is always negative and y is always positive.

In quadrant III, x and y are both always negative.

In quadrant IV, x is always positive and y is always negative.
Graphing equations on the coordinate plane
To graph an equation on the coordinate plane, find the coordinate by giving a value to one variable and solving the resulting equation for the other value. Repeat this process to find other coordinates. (When giving a value for one variable, you could start with 0, then try 1, and so on.) Then graph the solutions.
Example 1
Graph the equation x + y = 6.
Using a simple chart is helpful.
Now plot these coordinates as shown in Figure 4.
Figure 4. Plotting of coordinates (0,6), (1,5), (2,4)
Notice that these solutions, when plotted, form a straight line. Equations whose solution sets form a straight line are called linear equations. Complete the graph of x + y = 6 by drawing the line that passes through these points (see Figure 5).
Figure 5. The line that passes through the points graphed in Figure 4.
Equations that have a variable raised to a power, show division by a variable, involve variables with square roots, or have variables multiplied together will not form a straight line when their solutions are graphed. These are called nonlinear equations.
Example 2
Graph the equation y = x ^{2} + 4.
Use a simple chart.
x 
y 
2 
8 
1 
5 
0 
4 
1 
5 
2 
8 
Now plot these coordinates as shown in Figure 6.
Notice that these solutions, when plotted, do not form a straight line.
These solutions, when plotted, give a curved line (nonlinear). The more points plotted, the easier it is to see and describe the solutions set.