Graphing and Analyzing Linear Equations: Learn It 2

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.

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

  1. Make a table with one column labeled [latex]x[/latex], a second column labeled with the equation, and a third column listing the resulting ordered pairs.
  2. Enter [latex]x[/latex]–values down the first column using positive and negative values. Selecting the [latex]x[/latex]–values in numerical order will make the graphing simpler.
  3. Select [latex]x[/latex]–values that will yield [latex]y[/latex]–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.
Graph the equation [latex]y=2x - 1[/latex].We can begin by substituting a value for [latex]x[/latex] into the equation and determining the resulting value of [latex]y[/latex]. Each pair of [latex]x[/latex] and [latex]y[/latex]-values is an ordered pair that can be plotted. The table below lists values of [latex]x[/latex] from [latex]–3[/latex] to [latex]3[/latex] and the resulting values for [latex]y[/latex].

[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 these points from 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).

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.
Graph the equation [latex]y=-x+2[/latex].