# Graphs and Functions

## Graph  Linear Equations in Two Variables


### Plot Points on a Rectangular Coordinate System

Just like maps use a grid system to identify locations, a grid system is used in algebra to show a relationship between two variables in a rectangular coordinate system. The rectangular coordinate system is also called the xy-plane or the “coordinate plane.”
The rectangular coordinate system is formed by two intersecting number lines, one horizontal and one vertical. The horizontal number line is called the x-axis. The vertical number line is called the y-axis. These axes divide a plane into four regions, called quadrants. The quadrants are identified by Roman numerals, beginning on the upper right and proceeding counterclockwise. See .
In the rectangular coordinate system, every point is represented by an ordered pair. The first number in the ordered pair is the x-coordinate of the point, and the second number is the y-coordinate of the point. The phrase “ordered pair” means that the order is important.
What is the ordered pair of the point where the axes cross? At that point both coordinates are zero, so its ordered pair is  The point  has a special name. It is called the origin.
We use the coordinates to locate a point on the xy-plane. Let’s plot the point  as an example. First, locate 1 on the x-axis and lightly sketch a vertical line through  Then, locate 3 on the y-axis and sketch a horizontal line through  Now, find the point where these two lines meet—that is the point with coordinates  See .
Notice that the vertical line through  and the horizontal line through  are not part of the graph. We just used them to help us locate the point 
When one of the coordinate is zero, the point lies on one of the axes. In  the point  is on the y-axis and the point  is on the x-axis.
The signs of the x-coordinate and y-coordinate affect the location of the points. You may have noticed some patterns as you graphed the points in the previous example. We can summarize sign patterns of the quadrants in this way:
Up to now, all the equations you have solved were equations with just one variable. In almost every case, when you solved the equation you got exactly one solution. But equations can have more than one variable. Equations with two variables may be of the form  An equation of this form is called a linear equation in two variables.
Here is an example of a linear equation in two variables, x and y.
The equation  is also a linear equation. But it does not appear to be in the form  We can use the Addition Property of Equality and rewrite it in  form.
By rewriting  as  we can easily see that it is a linear equation in two variables because it is of the form  When an equation is in the form  we say it is in standard form of a linear equation.
Most people prefer to have A, B, and C be integers and  when writing a linear equation in standard form, although it is not strictly necessary.
Linear equations have infinitely many solutions. For every number that is substituted for x there is a corresponding y value. This pair of values is a solution to the linear equation and is represented by the ordered pair  When we substitute these values of x and y into the equation, the result is a true statement, because the value on the left side is equal to the value on the right side.
Linear equations have infinitely many solutions. We can plot these solutions in the rectangular coordinate system. The points will line up perfectly in a straight line. We connect the points with a straight line to get the graph of the equation. We put arrows on the ends of each side of the line to indicate that the line continues in both directions.
A graph is a visual representation of all the solutions of the equation. It is an example of the saying, “A picture is worth a thousand words.” The line shows you all the solutions to that equation. Every point on the line is a solution of the equation. And, every solution of this equation is on this line. This line is called the graph of the equation. Points not on the line are not solutions!

### Graph a Linear Equation by Plotting Points

There are several methods that can be used to graph a linear equation. The first method we will use is called plotting points, or the Point-Plotting Method. We find three points whose coordinates are solutions to the equation and then plot them in a rectangular coordinate system. By connecting these points in a line, we have the graph of the linear equation.
The steps to take when graphing a linear equation by plotting points are summarized here.
It is true that it only takes two points to determine a line, but it is a good habit to use three points. If you only plot two points and one of them is incorrect, you can still draw a line but it will not represent the solutions to the equation. It will be the wrong line.
If you use three points, and one is incorrect, the points will not line up. This tells you something is wrong and you need to check your work. Look at the difference between these illustrations.
When an equation includes a fraction as the coefficient of , we can still substitute any numbers for x. But the arithmetic is easier if we make “good” choices for the values of x. This way we will avoid fractional answers, which are hard to graph precisely.

### Graph Vertical and Horizontal Lines

Some linear equations have only one variable. They may have just x and no y, or just y without an x. This changes how we make a table of values to get the points to plot.
Let’s consider the equation  This equation has only one variable, x. The equation says that x is always equal to so its value does not depend on y. No matter what is the value of y, the value of x is always 
So to make a table of values, write  in for all the x-values. Then choose any values for y. Since x does not depend on y, you can choose any numbers you like. But to fit the points on our coordinate graph, we’ll use 1, 2, and 3 for the y-coordinates. See .
Plot the points from the table and connect them with a straight line. Notice that we have graphed a vertical line.
What if the equation has y but no x? Let’s graph the equation  This time the y-value is a constant, so in this equation, y does not depend on x. Fill in 4 for all the y’s in  and then choose any values for x. We’ll use 0, 2, and 4 for the x-coordinates.
In this figure, we have graphed a horizontal line passing through the y-axis at 4.
What is the difference between the equations  and 
The equation  has both x and y. The value of y depends on the value of x, so the y -coordinate changes according to the value of x. The equation  has only one variable. The value of y is constant, it does not depend on the value of x, so the y-coordinate is always 4.
Notice, in the graph, the equation  gives a slanted line, while  gives a horizontal line.

### Find x- and y-intercepts

Every linear equation can be represented by a unique line that shows all the solutions of the equation. We have seen that when graphing a line by plotting points, you can use any three solutions to graph. This means that two people graphing the line might use different sets of three points.
At first glance, their two lines might not appear to be the same, since they would have different points labeled. But if all the work was done correctly, the lines should be exactly the same. One way to recognize that they are indeed the same line is to look at where the line crosses the x-axis and the y-axis. These points are called the intercepts of a line.
Let’s look at the graphs of the lines.
First, notice where each of these lines crosses the x-axis. See .
Now, let’s look at the points where these lines cross the y-axis.
Do you see a pattern?
For each line, the y-coordinate of the point where the line crosses the x-axis is zero. The point where the line crosses the x-axis has the form  and is called the x-intercept of the line. The x-intercept occurs when y is zero.
In each line, the x-coordinate of the point where the line crosses the y-axis is zero. The point where the line crosses the y-axis has the form  and is called the y-intercept of the line. The y-intercept occurs when x is zero.
Recognizing that the x-intercept occurs when y is zero and that the y-intercept occurs when x is zero, gives us a method to find the intercepts of a line from its equation. To find the x-intercept, let  and solve for x. To find the y-intercept, let  and solve for y.

### Graph a Line Using the Intercepts

To graph a linear equation by plotting points, you need to find three points whose coordinates are solutions to the equation. You can use the x- and y- intercepts as two of your three points. Find the intercepts, and then find a third point to ensure accuracy. Make sure the points line up—then draw the line. This method is often the quickest way to graph a line.
The steps to graph a linear equation using the intercepts are summarized here.
When the line passes through the origin, the x-intercept and the y-intercept are the same point.

### Key Concepts

1. Points on the Axes


2. Quadrant



3. Graph of a Linear Equation: The graph of a linear equation  is a straight line.
Every point on the line is a solution of the equation.
Every solution of this equation is a point on this line.
4. How to graph a linear equation by plotting points.


5. 


6. Find the 


7. How to graph a linear equation using the intercepts.



### 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.
In the following exercises, for each ordered pair, decide
ⓐ is the ordered pair a solution to the equation? ⓑ is the point on the line?
Graph a Linear Equation by Plotting Points
In the following exercises, graph by plotting points.
Graph Vertical and Horizontal lines
In the following exercises, graph each equation.
In the following exercises, graph each pair of equations in the same rectangular coordinate system.
Find 
In the following exercises, find the x- and y-intercepts on each graph.
In the following exercises, find the intercepts for each equation.
Graph a Line Using the Intercepts
In the following exercises, graph using the intercepts.
Mixed Practice
In the following exercises, graph each equation.

### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ If most of your checks were:
Confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
With some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Whom can you ask for help?Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
No, I don’t get it. This is a warning sign and you must address it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.