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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 [latex]y=2x - 1[/latex]. 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.

[latex]x[/latex] [latex]y=2x - 1[/latex] [latex]\left(x,y\right)[/latex]
[latex]-3[/latex] [latex]y=2\left(-3\right)-1=-7[/latex] [latex]\left(-3,-7\right)[/latex]
[latex]-2[/latex] [latex]y=2\left(-2\right)-1=-5[/latex] [latex]\left(-2,-5\right)[/latex]
[latex]-1[/latex] [latex]y=2\left(-1\right)-1=-3[/latex] [latex]\left(-1,-3\right)[/latex]
[latex]0[/latex] [latex]y=2\left(0\right)-1=-1[/latex] [latex]\left(0,-1\right)[/latex]
[latex]1[/latex] [latex]y=2\left(1\right)-1=1[/latex] [latex]\left(1,1\right)[/latex]
[latex]2[/latex] [latex]y=2\left(2\right)-1=3[/latex] [latex]\left(2,3\right)[/latex]
[latex]3[/latex] [latex]y=2\left(3\right)-1=5[/latex] [latex]\left(3,5\right)[/latex]
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 [latex]y=-x+2[/latex] by plotting points.

Solution

First, we construct a table similar to the one below. Choose x values and calculate y.
[latex]x[/latex] [latex]y=-x+2[/latex] [latex]\left(x,y\right)[/latex]
[latex]-5[/latex] [latex]y=-\left(-5\right)+2=7[/latex] [latex]\left(-5,7\right)[/latex]
[latex]-3[/latex] [latex]y=-\left(-3\right)+2=5[/latex] [latex]\left(-3,5\right)[/latex]
[latex]-1[/latex] [latex]y=-\left(-1\right)+2=3[/latex] [latex]\left(-1,3\right)[/latex]
[latex]0[/latex] [latex]y=-\left(0\right)+2=2[/latex] [latex]\left(0,2\right)[/latex]
[latex]1[/latex] [latex]y=-\left(1\right)+2=1[/latex] [latex]\left(1,1\right)[/latex]
[latex]3[/latex] [latex]y=-\left(3\right)+2=-1[/latex] [latex]\left(3,-1\right)[/latex]
[latex]5[/latex] [latex]y=-\left(5\right)+2=-3[/latex] [latex]\left(5,-3\right)[/latex]
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: [latex]y=\frac{1}{2}x+2[/latex].
[latex]x[/latex] [latex]y=\frac{1}{2}x+2[/latex] [latex]\left(x,y\right)[/latex]
[latex]-2[/latex] [latex]y=\frac{1}{2}\left(-2\right)+2=1[/latex] [latex]\left(-2,1\right)[/latex]
[latex]-1[/latex] [latex]y=\frac{1}{2}\left(-1\right)+2=\frac{3}{2}[/latex] [latex]\left(-1,\frac{3}{2}\right)[/latex]
[latex]0[/latex] [latex]y=\frac{1}{2}\left(0\right)+2=2[/latex] [latex]\left(0,2\right)[/latex]
[latex]1[/latex] [latex]y=\frac{1}{2}\left(1\right)+2=\frac{5}{2}[/latex] [latex]\left(1,\frac{5}{2}\right)[/latex]
[latex]2[/latex] [latex]y=\frac{1}{2}\left(2\right)+2=3[/latex] [latex]\left(2,3\right)[/latex]
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

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