6.1 The rectangular coordinate systems and graphs  (Page 2/21)

Plotting points in a rectangular coordinate system

Plot the points $\left(\mathrm{-2},4\right),$ $\left(3,3\right),$ and $\left(0,\mathrm{-3}\right)$ in the plane.

To plot the point $\left(\mathrm{-2},4\right),$ begin at the origin. The x -coordinate is –2, so move two units to the left. The y -coordinate is 4, so then move four units up in the positive y direction.

To plot the point $\left(3,3\right),$ begin again at the origin. The x -coordinate is 3, so move three units to the right. The y -coordinate is also 3, so move three units up in the positive y direction.

To plot the point $\left(0,\mathrm{-3}\right),$ begin again at the origin. The x -coordinate is 0. This tells us not to move in either direction along the x -axis. The y -coordinate is –3, so move three units down in the negative y direction. See the graph in [link] .

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. [link] lists values of x from –3 to 3 and the resulting values for y .

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

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.

Graphing an equation in two variables by plotting points

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

First, we construct a table similar to [link] . Choose x values and calculate y.

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

Now, plot the points. Connect them if they form a line. See [link]