Often, to get an idea of the behavior of an equation we will make a picture that represents the solutions to the equations. A graph is simply a picture of the solutions to an equation. Before we spend much time on making a visual representation of an equation, we first have to understand the basis of graphing. Following is an example of what is called the coordinate plane.
The plane is divided into four sections by a horizontal number line (x-axis) and a vertical number line (y-axis). Where the two lines meet in the center is called the origin. This center origin is where x = 0 and y = 0. As we move to the right the numbers count up from zero, representing x = 1, 2, 3…
To the left the numbers count down from zero, representing x = − 1, − 2, − 3 . Similarly, as we move up the number count up from zero, y = 1, 2, 3… and as we move down count down from zero, y = − 1, − 2, − 3. We can put dots on the graph which we will call points. Each point has an “address” that defines its location. The first number will be the value on the x − axis or horizontal number line. This is the distance the point moves left/right from the origin. The second number will represent the value on the y − axis or vertical number line. This is the distance the point moves up/down from the origin. The points are given as an ordered pair (x, y).
Plotting Points on the Coordinate Plane
Just as we can give the coordinates for a set of points, we can take a set of points and plot them on the plane.
Plotting Points from a Table of Values
The main purpose is not to plot random points, but rather to give a picture of solutions to an equation. We may have an equation such as y = 2x – 3. We may be interested in what types of solutions are possible in this equation. We can visualize the solution by making a graph of possible x and y combinations that make this equation a true statement. We will have to start by finding possible x and y combinations. We will do this using a table of values.
Intercepts
Most lines have two kinds of intercepts: x-intercepts, and y-intercepts. These are the locations where the line crosses, or intercepts, one of the axes on the coordinate plane.
x-intercept: the location on a graph where a line or curve intersects the x-axis: (x, 0)
y-intercept: the location on a graph where a line or curve intersects the y-axis: (0, y)
The graph below shows a line's x-intercept and y-intercept:
Slope
The slope of a line describes its steepness. Slope can be positive, negative, or zero. For now, we will focus on positive and negative slopes. To determine if the slope of a line is positive or negative, "read" the graph left to right. If the line increases, or goes up, then the slope is positive. If the line decreases, or goes down, the slope is negative.
Here are the graphs of two lines: one with a positive slope, and one with a negative slope.
Source: Adapted from "Beginning and Intermediate Algebra" by Tyler Wallace, an open source textbook available at: http://wallace.ccfaculty.org/book/book.html
the location on a graph where a line or curve intersects the x-axis: (x, 0)
the location on a graph where a line or curve intersects the y-axis: (0, y)