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 $y=2x - 1$ . 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.

$x$	$y=2x - 1$	$\left(x,y\right)$
$-3$	$y=2\left(-3\right)-1=-7$	$\left(-3,-7\right)$
$-2$	$y=2\left(-2\right)-1=-5$	$\left(-2,-5\right)$
$-1$	$y=2\left(-1\right)-1=-3$	$\left(-1,-3\right)$
$0$	$y=2\left(0\right)-1=-1$	$\left(0,-1\right)$
$1$	$y=2\left(1\right)-1=1$	$\left(1,1\right)$
$2$	$y=2\left(2\right)-1=3$	$\left(2,3\right)$
$3$	$y=2\left(3\right)-1=5$	$\left(3,5\right)$

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.

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

$x$	$y=-x+2$	$\left(x,y\right)$
$-5$	$y=-\left(-5\right)+2=7$	$\left(-5,7\right)$
$-3$	$y=-\left(-3\right)+2=5$	$\left(-3,5\right)$
$-1$	$y=-\left(-1\right)+2=3$	$\left(-1,3\right)$
$0$	$y=-\left(0\right)+2=2$	$\left(0,2\right)$
$1$	$y=-\left(1\right)+2=1$	$\left(1,1\right)$
$3$	$y=-\left(3\right)+2=-1$	$\left(3,-1\right)$
$5$	$y=-\left(5\right)+2=-3$	$\left(5,-3\right)$

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: $y=\frac{1}{2}x+2$ .

$x$	$y=\frac{1}{2}x+2$	$\left(x,y\right)$
$-2$	$y=\frac{1}{2}\left(-2\right)+2=1$	$\left(-2,1\right)$
$-1$	$y=\frac{1}{2}\left(-1\right)+2=\frac{3}{2}$	$\left(-1,\frac{3}{2}\right)$
$0$	$y=\frac{1}{2}\left(0\right)+2=2$	$\left(0,2\right)$
$1$	$y=\frac{1}{2}\left(1\right)+2=\frac{5}{2}$	$\left(1,\frac{5}{2}\right)$
$2$	$y=\frac{1}{2}\left(2\right)+2=3$	$\left(2,3\right)$

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

The Rectangular Coordinate Systems and Graphs