## Graphing Integers: Overview

Graphing on the **coordinate plane** is a way to visualize relationships between two quantities. René Descartes, a French mathematician and philosopher, discovered this unique way to combine algebra and geometry. He recognized that by drawing two perpendicular number lines called **axes**, and labeling them with positive and negative numbers, he could locate any point on the plane with * x-* and

*coordinates. The diagram below shows the different parts of the coordinate plane.*

*y-*The point where the *x-*axis and *y-*axis intersect is called the **origin**. Notice that *x-*values to the right of the *y-*axis are positive and *x-*values to the left of the *y-*axis are negative. Similarly, *y-*values above the *x-*axis are positive and *y-*values below the *x-*axis are negative. The axes divide the plane into four **quadrants,** which are numbered starting in the upper right-hand quadrant and moving counterclockwise as I, II, III, and IV.

An **ordered pair** of numbers, *(x, y)* is used to locate a point on the plane. The first number in an ordered pair is called the *x-*coordinate and represents a location on the *x-*axis. The second number is called the *y-*coordinate and represents a location on the *y-*axis. To locate a point on the coordinate plane, do the following:

(1) | Start at the origin. |

(2) | The first number is the x-coordinate. If it is positive, move to the right the appropriate number of units. If it is negative, move to the left the appropriate number of units. |

(3) | The second number is the y-coordinate. If it is positive, move up the appropriate number of units. If it is negative, move down the appropriate number of units. |

Look at the coordinate grid below. The ordered pair for point *A* is (3, ^{-}2). To locate point *A,* we move three units to the right and two units down. Also shown on the coordinate plane are points *B* (^{-}3, 5), *C* (^{-}1, ^{-}4), *D* (0, ^{-}3), *E* (2, 4), and *F* (4, 0).

Notice that points in Quadrant I have a positive *x-* and a positive *y-*coordinate. Points in Quadrant II have a negative *x-*coordinate and a positive *y-*coordinate. Points in Quadrant III have a negative *x-*coordinate and a negative *y-*coordinate, and points in Quadrant IV have a positive *x-*coordinate and a negative *y-*coordinate.

Students can connect points on coordinate grids to form geometric shapes. Use Learning Tool 20 in the *Learning Tools Folder.* You can use these shapes to introduce transformations to students. A **transformation** is a change in the position of a plane figure. Three kinds of transformations are **translation,** sliding a figure a given distance in a given direction; **reflection,** flipping a figure across a given line; and **rotation,** turning a figure about a given point. See examples below.

Translation |
Reflection |
Rotation |

One way to represent relationships between pairs of numbers is by using an equation such as *x* − 2 = *y.* A second way to represent a relationship is by making a **function table** like the one below.

x |
x − 2 = y |
y | (x, y) |

^{-}3 |
^{-}3 − 2 = ^{-}5 |
^{-}5 |
(^{-}3, ^{-}5) |

^{-}2 |
^{-}2 − 2 = ^{-}4 |
^{-}4 |
(^{-}2, ^{-}4) |

^{-}1 |
^{-}1 − 2 = ^{-}3 |
^{-}3 |
(^{-}1, ^{-}3) |

0 | 0 − 2 = ^{-}2 |
^{-}2 |
(0, ^{-}2) |

1 | 1 − 2 = ^{-}1 |
^{-}1 |
(1, ^{-}1) |

2 | 2 − 2 = 0 | 0 | (2, 0) |

3 | 3 − 2 = 1 | 1 | (3, 1) |

This table can then be used to create the graph of the relationship. By plotting the points from the table above, we can see the relationship that exists between the *x-* and *y-* values that satisfy the equation *x* − 2 = *y.*

We can now see that the pattern of points lies along a straight line. By connecting the points with a solid line and indicating that it can be extended in both directions, we have created the graph of *x* − 2 = *y.* A **linear function** is a function with a straight-line graph.