Coordinate Geometry

Cartesian Coordinate System

The Cartesian coordinate system, also known as the coordinate plane, is used to graph lines, circles, parabolas, points, and other mathematical objects.

The power of the coordinate plane lies in the use of ordered pairs. The ordered pair (5,-2) refers to the point which has an x value of 5 and a y value of -2. Stated differently, the pairing is in the form (x, y). If two or more points are connected, a line or curve is formed.

The following terms are used when interacting with coordinate planes:

• X-Axis - The horizontal line running through the center of the graph from left to right.
• Y-Axis - The vertical line running through the center of the graph from bottom to top.
• Ordered Pair - The means of identifying a point through its coordinates. The proper notation is: (X, Y) where (0, 0) is the intersection of the x and y-axis.
  For example, point A is at (2, 4) since it is horizontally 2 units to the right of the center and it is vertically 4 units above the center.
  Similarly, point C is at (-6, -2) since it is horizontally 6 units to the left of the center and it is vertically 2 units below the center.
  Point E: (0, 0)
  Point B: (-7 , 5)
  Point D: (2, -1)
• Origin - The point in the center of the coordinate plane where the x and y-axis intersect (0, 0). The origin is point E in this graph.

Quadrants

Each coordinate plane is divided up into four quadrants, labeled below. (Note: Some graphs only show one quadrant. In this case, the other quadrants still exist, but they are merely not shown).

In the first quadrant, both x and y are positive while in the second quadrant x is negative and y is positive. The chart below depicts the sign of x and y [denoted (X, Y)].

• Quadrant I: (+, +)
• Quadrant II: (-, +)
• Quadrant III: (-, -)
• Quadrant IV: (+, -)

Slope

One property of a line is its slope, which is a measure of the steepness of the line. Every line has a slope defined by rise over run (i.e., the amount the line rises vertically over the amount the line runs horizontally). Rise over run refers the change of the rise (y values) of any two points on the line over the change in the run (x values) of the same two points on the line. For example:

Slope Categories

There are four types of slope: positive, negative, zero, and undefined.

• Blue Line - Positive Slope
• Red Line - Negative Slope
• Green Line - Slope of Zero
• Brown Line - Undefined Slope