One of the most important things mathematicians do is graphing. But the question is: what is a graph? You have already used many types of graphs. You have seen bar graphs, you have probably seen some line graphs in the newspaper. These types of graphs are used to organize pieces of information or data. They are more commonly referred to as charts. Charts are used to help people understand data. Mathematicians use graphs to help them understand equations. While the equations you have worked with so far have had just one answer, you have already graphed inequalities on number lines. Those graphs showed you the values that made the inequality true. Now you will work with equations that contain two variables and learn how to create graphs to help you understand those equations.

Before you begin graphing, it is important to learn about the coordinate plane, also call a Cartesian plane. The coordinate plane is created by combining two number lines. The first is a horizontal number line (which you are used to seeing). This line is called the x-axis because it represents values for the variable x in equations. The second number line is a vertical number line. It is called the y-axis because it represents values for the y variable in equations. The two number lines intersect (cross each other) at 0. This point in the graph is known as the origin. This information is labeled in the graph below.

One of the first things you need to know about graphs is how to plot points. Plotting points is about placing points at specific locations called coordinates on a graph. In order to determine where a point belongs on the graph, you are given an ordered pair. An ordered pair is a pair of numbers that tell where a point should be placed on the graph. (4, 6) is an ordered pair. The first number, 4, is the x-coordinate, also known as the abscissa. The second number, 6, is the y-coordinate or the ordinate. The x-coordinate tells how far the point is to the right (positive) or left (negative) of the y-axis. The y-coordinate tells how far to move up (positive) or down (negative) from the x-axis.

Look at the following point (4, 6). This notation means "move 4 units to the right of the y-axis and up 6 units from the x-axis." The image below shows how this point should be plotted on the graph.

Notice that the axes divide the graph into four different sections. These sections are called quadrants. The quadrants are named using Roman Numerals from I - IV (1 - 4), starting in the upper right hand corner and moving counter-clockwise.

It is often helpful identify which quadrant a point or a certain section of a graph is located in. However, if a point is located on an axis, then it is not in a quadrant and the axis is given for its location. There are two ways to identify the quadrant a point is located in. The first is to plot the point on the plane. The second is to look at the signs on each part of the point. If both signs are positive, it is in the first quadrant. If the x-value is negative and the y-value is positive, it is in the second quadrant. If both signs are negative, it is in the third quadrant. If the x-value is positive and the y-value is negative, it is in the fourth quadrant.

A point that is located on the x-axis will have an x-value, but the y-value will be zero (x, 0). A point that is located on the y-axis will have a y-value, but the x-value will be zero (0, y). If the point is located on both axes (the origin), its coordinates will be (0, 0).

So far you have solved numerous equations and inequalities involving one variable. Now it is time to see what happens when two variables are involved.

How would you solve an equation like the following?

4x - 2y =  16

Now, there is no way to isolate both variables. It seems like there is nothing that can be done to find a solution. What could you do if you were told that x  = 0? Try substituting 0 for x.

4(0) + -2y = 16

-2y = 16

Now you can solve the equation to find a value for y.

-2y	=	16
-2		-2

y = -8

You know that if x = 0, then y = -8. What if you substituted another number in for x? Try substituting 1 for x, then solve for y.

4(1) + -2y = 16

4 + -2y = 16

4 + -2y + -4 = 16 +  -4

-2y = 12

-2y	=	12
-2		-2

y = -6

Now you know that if x = 1, then y = -6. You can keep finding values to substitute for x and y; however, it will be difficult to keep track of all the values. In order to keep things organized, create a table in which you can organize the values of x and y that make the equation true.

x-values	y-values
0	-8
1	-6
0	0
0	0

Now, each time you choose a new x-value to put into the equation you can put it in the chart. Then solve the equation and enter the y-value next to the x-value.

x-values	y-values
0	-8
1	-6
2
0	0
0	0

4(2) + -2y = 16

8 + -2y = 16

8 + -2y + -8 = 16 + -8

-2y = 8

-2y	=	8
-2		-2

y = -4

x-values	y-values
0	-8
1	-6
2	-4
3
0	0

4(3) + -2y = 16

12 + -2y = 16

12 + -2y + -12 = 16  + -12

-2y = 4

-2y	=	4
-2		-2

y = -2

x-values	y-values
0	-8
1	-6
2	-4
3	-2
4

4(4) + -2y = 16

16 + -2y = 16

16 + -2y + -16 = 16  + -16

-2y = 0

-2y	=	0
-2		-2

y = 0

x-values	y-values
0	-8
1	-6
2	-4
3	-2
4	0

The table you just created is known as a table of values. It contains a list of values for x and y that make the equation true when substituted back into the equation.

Tables of values help you understand what is happening in an equation. They also provide a good way to organize the values of x and y to keep them together.

NOTE: Look back at problems 1, 2, and 6. Notice that on problems 1 and 2, the constant was a whole number and the x-values were chosen so that they were clearly divisible by the denominator of the coefficient. That left nice, whole numbers behind. On problem 6, that was not done. However, if you work with that problem and get to choose the values yourself, choosing values that are divisible by 4 will make the problem much easier.

What are graphs? The best way to understand the purpose of graphs is to look at the following equation.

2x + 3y = 12

Solve the equation for y.

2x + 3y + -2x  = 12 + -2x

3y = -2x + 12

3y	=	-2	+	12
3		3		3

y = -	2	x + 4
	3

Start a table of values for this equation. Choose values that will make y a whole number. Since x is being multiplied by ^-2⁄₃, use 0 and numbers that are divisible by 3 for x-values.

x-values	y-values
-3
0
3
6

y = -	2	(-3) + 4 = 6
	3

y = -	2	(0) + 4 = 4
	3

y = -	2	(3) + 4 = 2
	3

y = -	2	(6) + 4 = 0
	3

x-values	y-values
-3	6
0	4
3	2
6	0

Now you have four sets of values for x and y. But, just like the previous problems, there are more sets that you can find. In fact, you cannot find all the values of x and y that solve the equation. There are an infinite number of solutions. Since you cannot find all of the values of x and y that make this equation true, mathematicians decided to draw the solution to these equations on a coordinate plane.

In order to find out what this equation looks like, take each set of x and y values from the table of values and create ordered pairs out of them.

x-values	y-values	Ordered Pair
-3	6	(-3, 6)
0	4	(0, 4)
3	2	(3, 2)
6	0	(6, 0)

Now, plot those ordered pairs on a coordinate plane.

This type of equation is called a linear equation. This means that ALL the ordered pairs that make the equation true come together to form a line. Now that you have four points plotted, you can draw a line through those points. This will give you the graph of the equation 2x + 3y = 12.

The graph of an equation shows all the ordered pairs that make the equation work (satisfy the equation).

Now, you have had a few concepts introduced through the last problem. Below are a few definitions that help explain what you just did.

NOTE: It is common for people to confuse a graph with the coordinate plane. The coordinate plane contains the x-axis and y-axis. A graph is the image that is drawn on the coordinate plane that shows the ordered pairs that satisfy an equation. It is also important to notice that the graph that is drawn on a coordinate plane cannot show ALL of the ordered pairs of most equations, because the graph continues infinitely beyond what can be drawn on a coordinate plane.

This method for graphing is called plotting points.

When graphing by plotting points, it is helpful to choose values that will make both the x- and y-values integers. These are much easier to plot than numbers that include fractions or decimals. If you decide to plot fractions, it is helpful to rewrite improper fractions as mixed numbers.

Sometimes, rather than simply graphing a line, it is nice to know whether a point is on the line without sketching the graph. This is a rather simple process. Remember, if a point is on a line, then its coordinates make the equation true when they are substituted in for x and y.

Write about it. You have just learned to graph lines by plotting points. What is good about graphing lines by plotting points? What is difficult? Do you think there might be a better way to graph lines?

Look back at the tables of values from the lesson. See if you can determine any patterns in the y-values. If so, then choose two problems and explain what patterns you saw.

You have made another big jump in your knowledge of mathematics. You have learned about the coordinate plane, plotted points, and graphed lines. You also worked with equations with two variables and found values for those variables. While these processes are just opening the door to graphing, they are processes that you will continue to use in different ways in your mathematical career.

10-1: This section contained a lot of terms related to the coordinate plane. Make sure you learn them. It will help you in understanding later lessons. You also learned how to plot points. You will do this frequently throughout your time as a math student.

10-2: Creating tables of values is an important part of understanding an equation with two variables. When you encounter new types of equations, it is always helpful to create tables of values to understand how the variables relate to each other. It is also a good start for graphing equations.

10-3: Graphing equations by plotting points is an important skill. Just like creating tables of values, it is often used to explore new types of equations. Plotting points gives an idea of what new graphs look like. Remember, the coordinates of each point on the graph make the equation true when they are substituted into the equation for x and y.