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 xand y-values is an ordered pair that can be plotted. The table belowlists 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 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.
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 graphing easier.
- 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: Graphing an Equation in Two Variables by Plotting Points
Graph the equation [latex]y=-x+2[/latex] by plotting points.
Try It
Construct a table and graph the equation by plotting points: [latex]y=\frac{1}{2}x+2[/latex].
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Using Intercepts to Plot Lines in the Coordinate Plane
The intercepts of a graph are points where the graph crosses the axes. The x-intercept is the point where the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point where the graph crosses the y-axis. At this point, the x-coordinate is zero.
To determine the x-intercept, we set y equal to zero and solve for x. Similarly, to determine the y-intercept, we set x equal to zero and solve for y. For example, lets find the intercepts of the equation [latex]y=3x - 1[/latex].
To find the x-intercept, set [latex]y=0[/latex].
[latex]\begin{array}{llllll}y=3x - 1\hfill & \hfill \\ 0=3x - 1\hfill & \hfill \\ 1=3x\hfill & \hfill \\ \frac{1}{3}=x\hfill & \hfill \\ \left(\frac{1}{3},0\right)\hfill & x\text{-intercept}\hfill \end{array}[/latex]
To find the y-intercept, set [latex]x=0[/latex].
[latex]\begin{array}{lllll}y=3x - 1\hfill & \hfill \\ y=3\left(0\right)-1\hfill & \hfill \\ y=-1\hfill & \hfill \\ \left(0,-1\right)\hfill & y\text{-intercept}\hfill \end{array}[/latex]
We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.
How To: Given an equation, find the intercepts
- Find the x-intercept by setting [latex]y=0[/latex] and solving for [latex]x[/latex].
- Find the y-intercept by setting [latex]x=0[/latex] and solving for [latex]y[/latex].
Example: Finding the Intercepts of the Given Equation
Find the intercepts of the equation [latex]y=-3x - 4[/latex]. Then sketch the graph using only the intercepts.
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Try It
Find the intercepts of the equation and sketch the graph: [latex]y=-\frac{3}{4}x+3[/latex].
Show Solution
Using a Graphing Utility to Plot Lines
You can use an online graphing tool to quickly plot lines. Watch this short video Tutorial to learn how.
Try It
Desmos has a helpful feature that allows you to turn a constant (number) into a variable. Follow these steps to learn how:
- Graph the line [latex]y=-\frac{2}{3}x-\frac{4}{3}[/latex].
- On the next line enter[latex]y=-a x-\frac{4}{3}[/latex]. You will see a button pop up that says “add slider: a”, click on the button. You will see the next line populated with the variable a and the interval on which a can take values.
- What part of a line does the variable a represent? The slope or the y-intercept?
Here is ashort tutorial with more information about sliders.
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