Grappling with Graphs in Algebra

The cartoons in the newspaper often show a worried businessperson pointing to a graph full of ups and downs — usually the punch line involves a huge drop in sales. As entertaining as these cartoons may be, they also cut right to a major usefulness of graphs. Graphs give instant information on what the lowest value is and what the highest value is. They give information on trends, patterns, and the current status. Another great application is to put two graphs in the same picture to compare them.

In the simplest terms, a graph in algebra is drawn on a coordinate plane — two lines that cross one another at right angles to form four sections or quadrants. The two lines, or axes (pronounced ax-eez), are number lines usually marked with the integers (positive and negative whole numbers and zero). The positives go upward on the vertical axis and to the right on the horizontal axis. Figure 1 shows a coordinate plane. The line going left and right — the horizontal line — is the x-axis and the vertical line going up and down is the y-axis.

The little marks on the axes are called tick marks. They are all uniformly spaced (like the tick-tocks of a clock are the same time apart) and are usually labeled with the integers, negative to positive, left to right, and downward to upward, with zero in the middle — at the point where the axes meet.

The four quadrants are numbered I, II, III, and IV, with capital Roman numerals starting with the upper right quadrant and going counter-clockwise. The reason for this is simple: It's tradition. Table1 illustrates the quadrants and their names and positions.

Table 1: Quadrants

Making a point

You do the type of point finding needed to do graphing when you find the whereabouts of Peoria, Illinois, at G7 on a road atlas. You move your finger so it's down from the G and across from the 7. Graphing in algebra is just a bit different because numbers replace the letters, and you start in the middle at the point of origin.

Points are dots on a piece of paper or blackboard that represent positions or places with respect to the axes — vertical and horizontal lines — of a graph. The coordinates of a point tell you where it is in the graph.

The axes of an algebraic graph are usually labeled with integers, but they can be labeled with any rational numbers, as long as the numbers are the same distance apart from each other, such as the one-quarter distance between 1/4, 1/2, and 3/4,

The points you put in the graph don't have to be integers or line up with the axes. They can be in the spaces between the lines. That way, they can represent any number. The points in the spaces are just estimates of numbers, though. You could tell from the graph that a point is between 2 and 3, but you'd have a harder time deciding whether that point is 2.5 or 2.6.

Ordering pairs, or coordinating coordinates

To actually put a point in a graph, you need information on where to put that point. That's where ordered pairs come in.

An ordered pair is a set of two numbers called coordinates that are written inside a parenthesis with a comma separating them. Some examples are: (2, 3), (-1, 4), and (5, 0). This particular notation indicates that the order matters: The first number, or x-coordinate, tells you the point's position with respect to the x-axis — how far to the left or right from the point of origin — and the second number, or y-coordinate, tells you the point's position with respect to the y-axis — how far up or down from the point of origin. For example, the point for the ordered pair (3, 2) is three moves to the right of the origin, and two moves up from there. Look at Figure 2 to see where the points are for several ordered pairs.

Everything starts at the origin — the intersection of the two axes. The ordered pair for the origin is (0,0). The numbers in this ordered pair tell you that the point didn't go left, right, up, or down. Its position is at the starting place.

Notice that point (2,0) lies right on the x-axis. Whenever 0 is a coordinate within the ordered pair, then the point must be located on an axis.

Table 1 shows the names of the quadrants, their positions in the coordinate plane, and characteristics of coordinate points in the various quadrants.

The actual graphing on the coordinate plane involves placing dots for points in their correct position. Sometimes the points are sitting there, all alone, like an unused page in a child's "connect the dots" book. When the dots get connected, they sometimes form lines, U-shaped curves, or even more exciting pictures. It depends on the equation that the points come from.