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Cartesian coordinate system

A two-dimensional plane where the

  • x -axis is the horizontal axis
  • y -axis is the vertical axis

A point in the plane is defined as an ordered pair, ( x , y ) , such that x is determined by its horizontal distance from the origin and y is determined by its vertical distance from the origin.

Plotting points in a rectangular coordinate system

Plot the points ( −2 , 4 ) , ( 3 , 3 ) , and ( 0 , −3 ) in the plane.

To plot the point ( −2 , 4 ) , 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 ( 3 , 3 ) , 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 ( 0 , −3 ) , 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] .

This is an image of a graph on an x, y coordinate plane. The x and y axes range from negative 5 to 5.  The points (-2, 4); (3, 3); and (0, -3) are labeled.  Arrows extend from the origin to the points.
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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 y = 2 x 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 .

x y = 2 x 1 ( 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. See [link] . 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.

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.

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 ( 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. See [link]

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).
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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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