The Cartesian Plane Help

Introduction to the Cartesian Plane 

The Cartesian plane, also called the rectangular coordinate plane or rectangular coordinates, is defined by two number lines that intersect at a right angle. This makes it possible to pictorially render equations that relate one variable to another. You should have a knowledge of middle-school algebra before tackling this chapter. Upon casual observation, some of the equations in this chapter look a little complicated, but nothing here goes beyond middle-school algebra.

Figure 6-1 illustrates the simplest possible set of rectangular coordinates. Both number lines have equal increments. This means that on either axis, points corresponding to consecutive integers are the same distance apart, no matter where on the axis we look. The two number lines intersect at their zero points. The horizontal (right-and-left) axis is called the x axis; the vertical (up-and-down) axis is called the y axis .

Fig. 6-1 . The Cartesian plane is defined by two number lines that intersect at right angles.

Ordered Pairs As Points

Figure 6-2 shows two specific points, called P and Q , plotted on the Cartesian plane. The coordinates of point P are (−5, −4), and the coordinates of point Q are (3,5). Any given point on the plane can be denoted as an ordered pair in the form (x,y) , determined by the numerical values at which perpendiculars from the point intersect the x and y axes. In Fig. 6-2, the perpendiculars are shown as horizontal and vertical dashed lines. When denoting an ordered pair, it is customary to place the two numbers or variables together right up against the comma. There is no space after the comma.

Fig. 6-2 . Two points P and Q , plotted in rectangular coordinates, and a third point R , important in finding the distance d between P and Q.

The word “ordered” means that the order in which the numbers are listed is important. The ordered pair (7,2) is not the same as the ordered pair (2,7), even though both pairs contain the same two numbers. In this respect, ordered pairs are different than mere sets of numbers. Think of a highway, which consists of a northbound lane and a southbound lane. If there is never any traffic on the highway, it doesn’t matter which lane (the one on the eastern side or the one on the western side) is called “northbound” and which is called “southbound.” But when there are cars and trucks on that road, it makes a big difference! The untraveled road is like a set; the traveled road is like an ordered pair.

Abscissa, Ordinate, And Origin

In any graphing scheme, there is at least one independent variable and at least one dependent variable . As the name suggests, the value of the independent variable does not “depend” on anything; it just “happens.” The value of the dependent variable is affected by the value of the independent variable.

The independent-variable coordinate (usually x ) of a point on the Cartesian plane is called the abscissa , and the dependent-variable coordinate (usually y) is called the ordinate . The point (0,0) is called the origin. In Fig. 6-2, point P has an abscissa of −5 and an ordinate of −4, and point Q has an abscissa of 3 and an ordinate of 5.

Fig. 6-2 . Two points P and Q , plotted in rectangular coordinates, and a third point R , important in finding the distance d between P and Q.