# Equations and Inequalities

## The Rectangular Coordinate Systems and Graphs


### Learning Objectives

1. Plot points on a real number line (IA 1.4.7)
2. Plot points in a rectangular coordinate system (IA 3.1.1)

### Objective 1:  Plot points on a real number line (IA 1.4.7)


### Practice Makes Perfect

Plot each set of points on a real number line.

### Objective 2: Plot points in a rectangular coordinate system (IA 3.1.1)


### Practice Makes Perfect

Plot Points in a Rectangular Coordinate System
In the following exercises, plot each point in a rectangular coordinate system and identify the quadrant in which the point is located.
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 . Laying a rectangular coordinate grid over the map, we can see that each stop aligns with an intersection of grid lines. In this section, we will learn how to use grid lines to describe locations and changes in locations.

### 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  and the vertical axis the .
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 
The center of the plane is the point at which the two axes cross. It is known as the origin, or point 
  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 .
Each point in the plane is identified by its , or horizontal displacement from the origin, and its , or vertical displacement from the origin. Together, we write them as an ordered pair indicating the combined distance from the origin in the form 
  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 
  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 .
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.

### 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 
  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.  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 . 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 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 
  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 
  has been entered in the TI-84 Plus shown in a. In b, 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 
  and 
  See 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 a and b.

### Finding x-intercepts and y-intercepts

The intercepts of a graph are points at which the graph crosses the axes. The  is the point at which the graph crosses the x-axis. At this point, the y-coordinate is zero. The  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 
  
To find the x-intercept, set 
  
To find the y-intercept, set 
  
We can confirm that our results make sense by observing a graph of the equation as in . Notice that the graph crosses the axes where we predicted it would.

### 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, 
  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 .
The relationship of sides 
  and 
  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, 
  ) The symbols 
  and 
  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.

### 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, 
  and 
  the midpoint formula states how to find the coordinates of the midpoint 
  
A graphical view of a midpoint is shown in . Notice that the line segments on either side of the midpoint are congruent.

### Key Concepts

1. We can locate, or plot, points in the Cartesian coordinate system using ordered pairs, which are defined as displacement from the x-axis and displacement from the y-axis. See .
2. An equation can be graphed in the plane by creating a table of values and plotting points. See .
3. Using a graphing calculator or a computer program makes graphing equations faster and more accurate. Equations usually have to be entered in the form y=_____. See .
4. Finding the x- and y-intercepts can define the graph of a line. These are the points where the graph crosses the axes. See .
5. The distance formula is derived from the Pythagorean Theorem and is used to find the length of a line segment. See  and .
6. The midpoint formula provides a method of finding the coordinates of the midpoint dividing the sum of the x-coordinates and the sum of the y-coordinates of the endpoints by 2. See  and .

### Section Exercises


### Verbal


### Algebraic

For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.
For each of the following exercises, solve the equation for y in terms of x.
For each of the following exercises, find the distance between the two points. Simplify your answers, and write the exact answer in simplest radical form for irrational answers.
For each of the following exercises, find the coordinates of the midpoint of the line segment that joins the two given points.

### Graphical

For each of the following exercises, identify the information requested.
For each of the following exercises, plot the three points on the given coordinate plane. State whether the three points you plotted appear to be collinear (on the same line).
For each of the following exercises, construct a table and graph the equation by plotting at least three points. 

### Numeric

For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.
For each of the following exercises, use the graph in the figure below.

### Technology

For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 1:value button, hit enter. At the lower part of the screen you will see “x=” and a blinking cursor. You may enter any number for x and it will display the y value for any x value you input. Use this and plug in x = 0, thus finding the y-intercept, for each of the following graphs.
For the following exercises, use your graphing calculator to input the linear graphs in the Y= graph menu.
After graphing it, use the 2nd CALC button and 2:zero button, hit ENTER. At the lower part of the screen you will see “left bound?” and a blinking cursor on the graph of the line. Move this cursor to the left of the x-intercept, hit ENTER. Now it says “right bound?” Move the cursor to the right of the x-intercept, hit ENTER. Now it says “guess?” Move your cursor to the left somewhere in between the left and right bound near the x-intercept. Hit ENTER. At the bottom of your screen it will display the coordinates of the x-intercept or the “zero” to the y-value. Use this to find the x-intercept.
Note: With linear/straight line functions the zero is not really a “guess,” but it is necessary to enter a “guess” so it will search and find the exact x-intercept between your right and left boundaries. With other types of functions (more than one x-intercept), they may be irrational numbers so “guess” is more appropriate to give it the correct limits to find a very close approximation between the left and right boundaries.

### Extensions


### Real-World Applications
