2.1 The Rectangular Coordinate Systems and Graphs

Plotting Ordered Pairs in the Cartesian Coordinate System

An old story describes how seventeenth-century philosopher/mathematician René Descartes, while sick in bed, invented the system that has become the foundation of algebra. 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 it was possible to locate any object in a two-dimensional plane 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, it was Descartes who introduced the components that comprise the Cartesian coordinate system, a grid system having 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

Figure 2

The center of the plane is the point at which the two axes cross. It is known as the origin, or point $(0,0).$ 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.

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 $\left(x,y\right).$ 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 $(3,\mathrm{-1})$ 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.

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 as a method for showing the relationship between two quantities.

Example 1

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.

Solution

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 Figure 5.

Figure 5

Analysis

Note that when either coordinate is zero, the point must be on an axis. If the x-coordinate is zero, the point is on the y-axis. If the y-coordinate is zero, the point is on the x-axis.

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. Table 1 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 6. This is not true for all equations.

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.

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 Table 2. Choose x values and calculate y.

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

Table 2

Now, plot the points. Connect them if they form a line. See Figure 7

Figure 7

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 the style $y=\_\_\_\_\_.$ The TI-84 Plus, and many other calculator makes and models, 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 $y=2x-20$ has been entered in the TI-84 Plus shown in Figure 8a. In Figure 8b, the resulting graph is shown. Notice that we cannot see on the screen where the graph crosses the axes. The standard window screen on the TI-84 Plus shows $\mathrm{-10}\le x\le 10,$ and $\mathrm{-10}\le y\le 10.$ See Figure 8c.

Figure 8 a. Enter the equation. b. This is the graph in the original window. c. These are the original settings.

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 9a and Figure 9b.

Figure 9 a. This screen shows the new window settings. b. We can clearly view the intercepts in the new window.

Example 3

Using a Graphing Utility to Graph an Equation

Use a graphing utility to graph the equation: $y=-\frac{2}{3}x+\frac{4}{3}.$

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 10.

Figure 10

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, lets find the intercepts of the equation $y=3x-1.$

To find the x-intercept, set $y=0.$

To find the y-intercept, set $x=0.$

We can confirm that our results make sense by observing a graph of the equation as in Figure 11. Notice that the graph crosses the axes where we predicted it would.

Figure 11

Example 4

Finding the Intercepts of the Given Equation

Find the intercepts of the equation $y=\mathrm{-3}x-4.$ Then sketch the graph using only the intercepts.

Solution

Set $y=0$ to find the x-intercept.

$$\begin{array}{l}y=\mathrm{-3}x-4\hfill \\ 0=\mathrm{-3}x-4\hfill \\ 4=\mathrm{-3}x\hfill \\ -\frac{4}{3}=x\hfill \\ \left(-\frac{4}{3},0\right)x\text{−intercept}\hfill \end{array}$$

Set $x=0$ to find the y-intercept.

$$\begin{array}{l}y=\mathrm{-3}x-4\hfill \\ y=\mathrm{-3}(0)-4\hfill \\ y=\mathrm{-4}\hfill \\ (0,\mathrm{-4})y\text{−intercept}\hfill \end{array}$$

Plot both points, and draw a line passing through them as in Figure 12.

Figure 12

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, $a^2+b^2=c^2,$ 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 13.

Figure 13

The relationship of sides $\left|x_2-x_1\right|$ and $\left|y_2-y_1\right|$ 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, $\left|\mathrm{-3}\right|=3.$ ) The symbols $\left|x_2-x_1\right|$ and $\left|y_2-y_1\right|$ 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.

Example 5

Finding the Distance between Two Points

Find the distance between the points $\left(\mathrm{-3},\mathrm{-1}\right)$ and $\left(2,3\right).$

Solution

Let us first look at the graph of the two points. Connect the points to form a right triangle as in Figure 14.

Figure 14

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

$$\begin{array}{l}\\ \begin{array}{l}d=\sqrt{{(x_2-x_1)}^2+{(y_2-y_1)}^2}\hfill \\ d=\sqrt{{(2-(\mathrm{-3}))}^2+{(3-(\mathrm{-1}))}^2}\hfill \\ =\sqrt{{(5)}^2+{(4)}^2}\hfill \\ =\sqrt{25+16}\hfill \\ =\sqrt{41}\hfill \end{array}\end{array}$$

Example 6

Finding the Distance between Two Locations

Let’s return to the situation introduced at the beginning of this section.

Tracie set out from Elmhurst, IL, to go to Franklin Park. On the way, she made a few stops to do errands. Each stop is indicated by a red dot in Figure 1. Find the total distance that Tracie traveled. Compare this with the distance between her starting and final positions.

Solution

The first thing we should do is identify ordered pairs to describe each position. If we set the starting position at the origin, we can identify each of the other points by counting units east (right) and north (up) on the grid. For example, the first stop is 1 block east and 1 block north, so it is at $\left(1,1\right).$ The next stop is 5 blocks to the east, so it is at $\left(5,1\right).$ After that, she traveled 3 blocks east and 2 blocks north to $\left(8,3\right).$ Lastly, she traveled 4 blocks north to $\left(8,7\right).$ We can label these points on the grid as in Figure 15.

Figure 15

Next, we can calculate the distance. Note that each grid unit represents 1,000 feet.

• From her starting location to her first stop at $\left(1,1\right),$ Tracie might have driven north 1,000 feet and then east 1,000 feet, or vice versa. Either way, she drove 2,000 feet to her first stop.
• Her second stop is at $\left(5,1\right).$ So from $\left(1,1\right)$ to $\left(5,1\right),$ Tracie drove east 4,000 feet.
• Her third stop is at $\left(8,3\right).$ There are a number of routes from $\left(5,1\right)$ to $\left(8,3\right).$ Whatever route Tracie decided to use, the distance is the same, as there are no angular streets between the two points. Let’s say she drove east 3,000 feet and then north 2,000 feet for a total of 5,000 feet.
• Tracie’s final stop is at $\left(8,7\right).$ This is a straight drive north from $\left(8,3\right)$ for a total of 4,000 feet.

Next, we will add the distances listed in Table 3.

From/To	Number of Feet Driven
$\left(0,0\right)$ to $\left(1,1\right)$	2,000
$\left(1,1\right)$ to $\left(5,1\right)$	4,000
$\left(5,1\right)$ to $\left(8,3\right)$	5,000
$\left(8,3\right)$ to $\left(8,7\right)$	4,000
Total	15,000

Table 3

The total distance Tracie drove is 15,000 feet, or 2.84 miles. This is not, however, the actual distance between her starting and ending positions. To find this distance, we can use the distance formula between the points $\left(0,0\right)$ and $\left(8,7\right).$

$$\begin{array}{l}d=\sqrt{{(8-0)}^2+{(7-0)}^2}\hfill \\ =\sqrt{64+49}\hfill \\ =\sqrt{113}\hfill \\ \approx 10.63\text{units}\hfill \end{array}$$

At 1,000 feet per grid unit, the distance between Elmhurst, IL, to Franklin Park is 10,630.14 feet, or 2.01 miles. The distance formula results in a shorter calculation because it is based on the hypotenuse of a right triangle, a straight diagonal from the origin to the point $\left(8,7\right).$ Perhaps you have heard the saying “as the crow flies,” which means the shortest distance between two points because a crow can fly in a straight line even though a person on the ground has to travel a longer distance on existing roadways.

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, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right),$ the midpoint formula states how to find the coordinates of the midpoint $M.$

A graphical view of a midpoint is shown in Figure 16. Notice that the line segments on either side of the midpoint are congruent.

Figure 16