An old story describes how seventeenth-century philosopher/mathematician René Descartes invented the system that has become the foundation of algebra while sick in bed. According to the story, Descartes was staring at a fly crawling on the ceiling when he realized that he could describe the fly’s location in relation to the perpendicular lines formed by the adjacent walls of his room. He viewed the perpendicular lines as horizontal and vertical axes. Further, by dividing each axis into equal unit lengths, Descartes saw that locating any object in a two-dimensional plane was possible using just two numbers—the displacement from the horizontal axis and the displacement from the vertical axis.

While there is evidence that ideas similar to Descartes’ grid system existed centuries earlier, Descartes introduced the components that comprise the Cartesian coordinate system, a grid system with perpendicular axes. Descartes named the horizontal axis the x-axis and the vertical axis the y-axis.

The Cartesian coordinate system, also called the rectangular coordinate system, is based on a two-dimensional plane consisting of the x-axis and the y-axis. Perpendicular to each other, the axes divide the plane into four sections. Each section is called a quadrant; the quadrants are numbered counterclockwise, as shown in Figure 2.

The center of the plane is the point at which the two axes cross. It is known as the origin, or point [latex]\left(0,0\right).[/latex] From the origin, each axis is further divided into equal units: increasing positive numbers to the right on the x-axis and up the y-axis; decreasing negative numbers to the left on the x-axis and down the y-axis. The axes extend to positive and negative infinity, as shown by the arrowheads in Figure 3.

Each point in the plane is identified by its x-coordinate, or horizontal displacement from the origin, and its y-coordinate, or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form [latex]\,\left(x,y\right).\,[/latex] An ordered pair is also known as a coordinate pair because it consists of x- and y-coordinates. For example, we can represent the point [latex]\,\left(3,-1\right)\,[/latex] in the plane by moving three units to the right of the origin in the horizontal direction and one unit down in the vertical direction. See Figure 4.

When dividing the axes into equally spaced increments, note that the x-axis may be considered separately from the y-axis. In other words, while the x-axis may be divided and labeled according to consecutive integers, the y-axis may be divided and labeled by increments of 2, or 10, or 100. In fact, the axes may represent other units, such as years against the balance in a savings account, or quantity against cost, and so on. Consider the rectangular coordinate system primarily a method for showing the relationship between two quantities.

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 [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 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.

We can plot the points in the table. The points for this particular equation form a line, so we can connect them. See Figure 5. 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 negative and positive values. There is no rule on how many points to plot, although we need at least two to graph a line. However, remember 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 [latex]\,y=-x+2\,[/latex] by plotting points.

Show Solution

First, we construct a table of [latex]x[/latex]  and [latex]y[/latex] values. Choose x values and calculate y.

[latex]x[/latex]	[latex]y=-x+2[/latex]	[latex]\left(x,y\right)[/latex]
[latex]-5[/latex]	[latex]y=-\left(-5\right)+2=7[/latex]	[latex]\left(-5,7\right)[/latex]
[latex]-3[/latex]	[latex]y=-\left(-3\right)+2=5[/latex]	[latex]\left(-3,5\right)[/latex]
[latex]-1[/latex]	[latex]y=-\left(-1\right)+2=3[/latex]	[latex]\left(-1,3\right)[/latex]
[latex]0[/latex]	[latex]y=-\left(0\right)+2=2[/latex]	[latex]\left(0,2\right)[/latex]
[latex]1[/latex]	[latex]y=-\left(1\right)+2=1[/latex]	[latex]\left(1,1\right)[/latex]
[latex]3[/latex]	[latex]y=-\left(3\right)+2=-1[/latex]	[latex]\left(3,-1\right)[/latex]
[latex]5[/latex]	[latex]y=-\left(5\right)+2=-3[/latex]	[latex]\left(5,-3\right)[/latex]

Now, plot the points. Connect them if they form a line. See Figure 6

 

Graphing Equations with a Graphing Utility

Most graphing calculators require similar techniques to graph an equation. The equations sometimes have to be manipulated, so they are written in style [latex]\,y\,[/latex]=_____. The TI-84 Plus and many other calculators have a mode function, which allows the window (the screen for viewing the graph) to be altered so the pertinent parts of a graph can be seen.

For example, the equation [latex]\,y=2x-20\,[/latex] has been entered in the TI-84 Plus shown in Figure 7.a. In Figure 7.b, the resulting graph is shown. Notice that we cannot see where the graph crosses the axes on the screen. The standard window screen on the TI-84 Plus shows [latex]\,-10\le x\le 10,[/latex] and [latex]\,-10\le y\le 10.\,[/latex] See Figure 7.c.

By changing the window to show more of the positive x-axis and more of the negative y-axis, we have a much better view of the graph and the x- and y-intercepts. See Figure 8.a. and Figure 8.b.

Using a Graphing Utility to Graph an Equation

Use a graphing utility to graph the equation: [latex]\,y=-\frac{2}{3}x-\frac{4}{3}.[/latex]

Show Solution

Enter the equation in the y= function of the calculator. Set the window settings so that both the x- and y- intercepts are showing in the window. See Figure 9.

Finding x-intercepts and y-intercepts

The intercepts of a graph are points at which the graph crosses the axes. The x-intercept is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The y-intercept is the point at which 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, let’s find the intercepts of the equation [latex]\,y=3x-1.[/latex]

To find the x-intercept, set [latex]\,y=0.[/latex]

To find the y-intercept, set [latex]\,x=0.[/latex]

We can confirm that our results make sense by observing a graph of the equation as in Figure 10. Notice that the graph crosses the axes where we predicted it would.

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.

Show Solution

Set [latex]\,y=0\,[/latex] to find the x-intercept.

[latex]\begin{array}{l}\phantom{\rule{1em}{0ex}}y=-3x-4\hfill \\ \phantom{\rule{1em}{0ex}}0=-3x-4\hfill \\ \phantom{\rule{1em}{0ex}}4=-3x\hfill \\ -\frac{4}{3}=x\hfill \\ \left(-\frac{4}{3},0\right)\phantom{\rule{3em}{0ex}}x\text{−intercept}\hfill \end{array}[/latex]

Set [latex]\,x=0\,[/latex] to find the y-intercept.

[latex]\begin{array}{l}y=-3x-4\hfill \\ y=-3\left(0\right)-4\hfill \\ y=-4\hfill \\ \left(0,-4\right)\phantom{\rule{3.5em}{0ex}}y\text{−intercept}\hfill \end{array}[/latex]

Plot both points and draw a line passing through them, as in Figure 11.

Access the online resource below for additional instruction and practice with finding [latex]x[/latex] and [latex]x[/latex] intercepts.

Using the Distance Formula

Derived from the Pythagorean Theorem, the distance formula is used to find the distance between two points in the plane. The Pythagorean Theorem,[latex]\,{a}^{2}+{b}^{2}={c}^{2},[/latex] is based on a right triangle where a and b are the lengths of the legs adjacent to the right angle and c is the length of the hypotenuse. See Figure 12.

The relationship of sides[latex]\,|{x}_{2}-{x}_{1}|\,[/latex]and[latex]\,|{y}_{2}-{y}_{1}|\,[/latex]to side d is the same as that of sides a and b to side c. We use the absolute value symbol to indicate that the length is a positive number because the absolute value of any number is positive. (For example,[latex]\,|-3|=3.\,[/latex]) The symbols[latex]\,|{x}_{2}-{x}_{1}|\,[/latex]and[latex]\,|{y}_{2}-{y}_{1}|\,[/latex]indicate that the lengths of the sides of the triangle are positive. To find the length c, take the square root of both sides of the Pythagorean Theorem.

It follows that the distance formula is given as

We do not have to use the absolute value symbols in this definition because any number squared is positive.

Finding the Distance between Two Points

Find the distance between the points[latex]\,\left(-3,-1\right)\,[/latex]and[latex]\,\left(2,3\right).[/latex]

Show Solution

Let us first look at the graph of the two points. Connect the points to form a right triangle, as in Figure 13.

Then, calculate the length of d using the distance formula.

[latex]\begin{array}{l}\\ \begin{array}{l}d=\sqrt{{\left({x}_{2}-{x}_{1}\right)}^{2}+{\left({y}_{2}-{y}_{1}\right)}^{2}}\hfill \\ d=\sqrt{{\left(2-\left(-3\right)\right)}^{2}+{\left(3-\left(-1\right)\right)}^{2}}\hfill \\ \phantom{\rule{.7em}{0ex}}=\sqrt{{\left(5\right)}^{2}+{\left(4\right)}^{2}}\hfill \\ \phantom{\rule{.7em}{0ex}}=\sqrt{25+16}\hfill \\ \phantom{\rule{.7em}{0ex}}=\sqrt{41}\hfill \end{array}\end{array}[/latex]

Access the online resource below for additional instruction and practice with finding distance using the distance formula.

Using the Midpoint Formula

When the endpoints of a line segment are known, we can find the point midway between them. This point is known as the midpoint, and the formula is known as the midpoint formula. Given the endpoints of a line segment, [latex]\,\left({x}_{1},{y}_{1}\right)\,[/latex]and[latex]\,\left({x}_{2},{y}_{2}\right),[/latex] the midpoint formula states how to find the coordinates of the midpoint [latex]\,M.[/latex]

A graphical view of a midpoint is shown in Figure 15. Notice that the line segments on either side of the midpoint are congruent.