Intermediate Algebra/The Coordinate (Cartesian) Plane

Coordinate Plane

 * A Coordinate Plane (Also referred to as a Cartesian Plane, after René Descartes) is a 2-dimensional plane with a horizontal axis and a vertical axis. Both axes extend to infinity, but in graphs only segments of them are drawn (and sometimes arrows are used to indicate the infinite length, such as in the picture to the right). The horizontal axis is known as the x-axis and the vertical axis is known as the y-axis. The point where they intersect is known as the origin.

Coordinates of a Point

 * When a point is plotted on a coordinate plane, its coordinates can be found. The x-coordinate is the point's distance from the y axis, and the y-coordinate is the distance from the x-axis. When the coordinates are integer numbers, they can be easily found on a graph by looking at the numbers on the axes. Together, the coordinates can be expressed as an ordered pair . The ordered pair (4,3) represents a point for squares to the right of the origin and three squares upward from it. On the x-axis, numbers increase toward the right and decrease toward the left. On the y-axis, numbers decrease going downward and increase going upward.

Plotting an Equation on a Coordinate Plane

 * It is also possible to represent an equation on a coordinate plane by plotting multiple ordered pairs. In the case of a linear equation in the form $$y=mx+b$$, the x-coordinate for a given point will be equal to the x variable at that point and likewise the y-coordinate will be equal to the y variable at that point.


 * The simplest way to produce a graph of a linear equation is to solve for y at arbitrary values of x and plot the results. In the case of the equation $$y=2x+4$$, if you were graphing for x-coordinates from 0 to 5, you would find the value of y in that equation for all integer values of x from 0 to 5.


 * $$(2*0)+4=4$$


 * $$(2*1)+4=6$$


 * $$(2*2)+4=8$$


 * $$(2*3)+4=10$$


 * $$(2*4)+4=12$$


 * $$(2*5)+4=14$$


 * The resulting ordered pairs are (0,4),(1,6),(2,8),(3,10),(4,12) and (5,14). All linear equations have an infinite number of possible ordered pairs, but for graphing, a finite number will suffice. Plot these coordinate pairs on a sheet of graph paper (Count off the length of the x-coordinate to to right then count the length of the y-coordinate upward if need be). Once all of the points are plotted, draw a line connecting them. This line represents all the values of the equation $$y=2x+4$$ from $$x=0$$ to $$x=5$$.