Algebra II

Equations can be graphed on a set of coordinate axes. The location of every point on a graph can be determined by two coordinates, written as an ordered pair, ( x,y). These are also known as Cartesian coordinates, after the French mathematician Rene Descartes, who is credited with their invention. If the slope and intercept (you'll get to them shortly) or the coordinates of two points on a linear graph are known, the equation of the line can be determined.

Following are terms with which you should be familiar:

• Coordinates of a point. 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 called the coordinates of the point.

• x‐axis; y‐axis. To locate points in a plane, two perpendicular lines are used: a horizontal line called the x‐axis and a vertical line called the y‐axis.

• Coordinate plane. The x‐axis, the y‐axis, and all the points in their plane are called a coordinate plane.

• Ordered pairs. Every point in a coordinate plane is named by a pair of numbers whose order is important. This pair of numbers, written in parentheses and separated by a comma, is the ordered pair for the point.

• Origin. The point of intersection of the x‐axis and the y‐axis is called the origin. The ordered pair for the origin is (0,0).

• x‐coordinate. The number to the left of the comma in an ordered pair is the x‐coordinate of the point and indicates the amount of movement parallel to the x‐axis from the origin. The movement is to the right if the number is positive and to the left if the number is negative.

• y‐coordinate . The number to the right of the comma in an ordered pair is the y‐coordinate of the point and indicates the amount of movement perpendicular to the x‐axis. The movement is above the x‐axis if the number is positive and below the x‐axis if the number is negative.

• Quadrants. The x‐axis and y‐axis separate the coordinate plane into four regions called quadrants. The upper right quadrant is quadrant I, the upper left quadrant is quadrant II, the lower left quadrant is quadrant III, and the lower right quadrant is quadrant IV. Notice that, as shown in Figure 1,

  • 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 is always negative and y is always negative (–,–)

  • in quadrant IV, x is always positive and y is always negative (+,–)

The point associated with an ordered pair of real numbers is called the graph of the ordered pair.