Graphing Equations by Plotting Points

We can plot a set of points to represent an equation. When such an equation contains both an x variable and a y variable, it is called an equation in two variables. Its graph is called a graph in two variables. Any graph on a two-dimensional plane is a graph in two variables.

Suppose we want to graph the equation y=2x1. We can begin by substituting a value for x into the equation and determining the resulting value of y. Each pair of x– and y-values is an ordered pair that can be plotted. The table below lists values of x from –3 to 3 and the resulting values for y.

x y=2x1 (x,y)
3 y=2(3)1=7 (3,7)
2 y=2(2)1=5 (2,5)
1 y=2(1)1=3 (1,3)
0 y=2(0)1=1 (0,1)
1 y=2(1)1=1 (1,1)
2 y=2(2)1=3 (2,3)
3 y=2(3)1=5 (3,5)

We can plot the points in the table. The points for this particular equation form a line, so we can connect them. This is not true for all equations.

This is a graph of a line on an x, y coordinate plane. The x- and y-axis range from negative 8 to 8. A line passes through the points (-3, -7); (-2, -5); (-1, -3); (0, -1); (1, 1); (2, 3); and (3, 5).

Figure 6

Note that the x-values chosen are arbitrary, regardless of the type of equation we are graphing. Of course, some situations may require particular values of x to be plotted in order to see a particular result. Otherwise, it is logical to choose values that can be calculated easily, and it is always a good idea to choose values that are both negative and positive. There is no rule dictating how many points to plot, although we need at least two to graph a line. Keep in mind, however, that the more points we plot, the more accurately we can sketch the graph.

How To: Given an equation, graph by plotting points.

  1. Make a table with one column labeled x, a second column labeled with the equation, and a third column listing the resulting ordered pairs.
  2. Enter x-values down the first column using positive and negative values. Selecting the x-values in numerical order will make the graphing simpler.
  3. Select x-values that will yield y-values with little effort, preferably ones that can be calculated mentally.
  4. Plot the ordered pairs.
  5. Connect the points if they form a line.

Example 2: Graphing an Equation in Two Variables by Plotting Points

Graph the equation y=x+2 by plotting points.

Solution

First, we construct a table similar to the one below. Choose x values and calculate y.

x y=x+2 (x,y)
5 y=(5)+2=7 (5,7)
3 y=(3)+2=5 (3,5)
1 y=(1)+2=3 (1,3)
0 y=(0)+2=2 (0,2)
1 y=(1)+2=1 (1,1)
3 y=(3)+2=1 (3,1)
5 y=(5)+2=3 (5,3)

Now, plot the points. Connect them if they form a line.

This image is a graph of a line on an x, y coordinate plane. The x-axis includes numbers that range from negative 7 to 7. The y-axis includes numbers that range from negative 5 to 8. A line passes through the points: (-5, 7); (-3, 5); (-1, 3); (0, 2); (1, 1); (3, -1); and (5, -3).

Figure 7

Construct a table and graph the equation by plotting points: y=12x+2.

x y=12x+2 (x,y)
2 y=12(2)+2=1 (2,1)
1 y=12(1)+2=32 (1,32)
0 y=12(0)+2=2 (0,2)
1 y=12(1)+2=52 (1,52)
2 y=12(2)+2=3 (2,3)
This is an image of a graph on an x, y coordinate plane. The x and y-axis range from negative 5 to 5. A line passes through the points (-2, 1); (-1, 3/2); (0, 2); (1, 5/2); and (2, 3).

Figure 8