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 Depdendent Variable

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 Dependent Variable

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# The Cartesian Coordinate System

Suppose we know the sum of two numbers is 6.

We can represent this situation with the equation x + y = 6.

There infinitely many possibilities for x and y. For example:

 1 and 51 + 5 = 6 -4 and 10-4 + 10 = 6 0 and 60 + 6 = 6

We can visualize this relationship between x and y using two number lines, one for x and the other for y.

First, draw a horizontal number line.

This is usually called the x-axis and is labeled with the variable x.

Next, draw a vertical number line perpendicular to the x-axis.

The two number lines should intersect at their zeros.

The vertical number line is usually called the y-axis and is labeled with the variable y.

The point of intersection, the zero of each number line, is called the origin.

To make it easier to locate a point, we draw a rectangular grid as a background.

The axes and the grid define a flat surface called the xy-plane.

The number lines and grid form a rectangular coordinate system. We typically use x and y for the variables, so a rectangular coordinate system is often called an xy-coordinate system. The French mathematician Rene Descartes (1596-1650) is credited with developing this type of coordinate system, so it is also referred to as the Cartesian coordinate system.

The x- and y-axes divide the plane into four regions called quadrants. We label these with Roman numerals I, II, III, and IV in a counter-clockwise direction beginning in the upper right.

 Quadrant I II III IV Sign of x positive negative negative positive Sign of ypositive positive negative  negative

A point on an axis does not lie in a quadrant.

Example 1

State the quadrant in which each labeled point lies.

 a. A b. B c. C d. D e. E

Solution

a. Quadrant II. Point A lies in the upper left quadrant.

b. Quadrant IV. Point B lies in the lower right quadrant.

c. None. Points on either axis do not lie in a quadrant.

d. Quadrant III. Point D lies in the lower left quadrant.

e. Quadrant I. Point E lies in the upper right quadrant.