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Introduction to Cartesian Coordinates—Number Lines

Trigonometry involves angles and their relationships to distances. All of these relationships arise from the characteristics of a circle, and can be defined on the basis of the graph of a circle in the Cartesian plane.

The Cartesian plane, also called the rectangular coordinate plane or rectangular coordinates, consists of two number lines that intersect at a right angle. This makes it possible to graph equations that relate one variable to another. Most such graphs look like lines or curves.

Two Perpendicular Number Lines

Figure 1-1 illustrates the simplest possible set of rectangular coordinates. Both number lines have uniform increments. That is, the points on the axes that represent consecutive integers are always the same distance apart. The two number lines intersect at their zero points. The horizontal (or east/west) axis is called the x axis; the vertical (or north/south) axis is called the y axis.

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

Ordered Pairs, Abscissa, and Ordinate

Ordered Pairs

Figure 1-2 shows two points plotted in rectangular coordinates. Points are denoted as ordered pairs in the form ( x , y ) in which the first number represents the value on the x axis and the second number represents the value on the y axis. The word “ordered” means that the order in which the numbers are listed is important. For example, the ordered pair (3.5,5.0) is not the same as the ordered pair (5.0,3.5), even though both pairs contain the same two numbers.

In ordered-pair notation, there is no space after the comma, as there is in the notation of a set or sequence. When denoting an ordered pair, it is customary to place the two numbers or variables together right up against the comma.

Fig. 1-2. Two points, plotted in rectangular coordinates.

In Fig. 1-2, two points are shown, one with an abscissa of 3.5 and an ordinate of 5.0, and the other with an abscissa of –5.2 and an ordinate of –4.7.

Abscissa And Ordinate

In most sets of coordinates where the axes are labeled x and y , the variable y is called the dependent variable (because its value “depends” on the value of x ), and the variable x is called 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 .

Relations and Functions

Relations

Mathematical relationships, technically called relations , between two variables x and y can be written in such a way that y is expressed in terms of x . The following are some examples of relations denoted in this form:

y = 5

y = x + 1

y = 2 x

y = x 2

Some Simple Graphs

Figure 1-3 shows how the graphs of the above equations look on the Cartesian plane. Mathematicians and scientists call such graphs curves , even if they are straight lines.

Fig. 1-3. Graphs of four simple functions. See text for details.

The graph of y = 5 (curve A) is a horizontal line passing through the point (0,5) on the y axis. The graph of y = x + 1 (curve B) is a straight line that ramps upward at a 45° angle (from left to right) and passes through the point (0,1) on the y axis. The graph of y = 2x (curve C) is a straight line that ramps upward more steeply, and that passes through the origin. The graph of y = x 2 (curve D) is known as a parabola. In this case the parabola rests on the origin, opens upward, and is symmetrical with respect to the y axis.

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