Learning Objectives
Introduction
This section covers the basic ideas of graphing: rectangular coordinate system, solutions to equations in two variables, and sketching a graph. Graphs are important in giving a visual representation of the correlation between two variables. Even though in this section we are going to look at it generically, using a general x and y variable, you can use two-dimensional graphs for any application where you have two variables. For example, you may have a cost function that is dependent on the quantity of items made. If you needed to show your boss visually the correlation of the quantity with the cost, you could do that on a two-dimensional graph. I believe that it is important for you learn how to do something in general, then when you need to apply it to something specific you have the knowledge to do so. Going from general to specific is a lot easier than specific to general. And that is what we are doing here looking at graphing in general so later you can apply it to something specific, if needed.
Tutorial
It is split into four quadrants which are marked on this graph with Roman numerals.
Each point on the graph is associated with an ordered pair. When dealing with an x, y graph, x is always first and y is always second in the ordered pair (x, y). It is a solution to an equation in two variables. Even though there are two values in the ordered pair, be careful that it associates to ONLY ONE point on the graph, the point lines up with both the x value of the ordered pair (x-axis) and the y value of the ordered pair (y-axis).
Example
1: Plot the ordered pairs and name the
quadrant
or axis in which the point lies. A(2, 3), B(-1,
2), C(-3, -4), D(2, 0), and E(0, 5).
B(-1, 2) lies in quadrant II.
C(-3, -4) lies in quadrant III.
D(2, 0) lies on the x-axis.
E(0, 5) lies on the y-axis.
In other words, if your equation has two variables x and y, and you plug in a value for x and its corresponding value for y and the mathematical statement comes out to be true, then the x and y value that you plugged in would together be a solution to the equation.
Equations in two variables can have more than one solution.
We usually write the solutions to equations in two
variables in ordered
pairs.
Example
2: Determine whether each ordered pair is a solution
of
the given equation. y = 5x - 7; (2, 3), (1, 5), (-1, -12)
Which number is the x value and which one is the y value? If you said x = 2 and y = 3, you are correct!
Let’s plug (2, 3) into the equation and see what we get:
Now let’s take a look at (1, 5).
Which number is the x value and which one is the y value? If you said x = 1 and y = 5, you are right!
Let’s plug (1, 5) into the equation and see what we get:
Now let’s look at (-1, -12).
Which number is the x value and which one is the y value? If you said x = -1 and y = -12, you are right!
Let’s plug (-1, -12) into the equation and see what we get:
Note that you were only given three ordered pairs to check, however, there are an infinite number of solutions to this equation. It would very cumbersome to find them all.
Example
3: Determine whether each ordered pair is a solution
of
the given equation. 
; (0, -3), (1, -3), (-1, -3)Which number is the x value and which one is the y value? If you said x = 0 and y = -3, you are correct!
Let’s plug (0, -3) into the equation and see what we get:
Now, let’s take a look at (1, -3).
Which number is the x value and which one is the y value? If you said x = 1 and y = -3, you are right!
Let’s plug (1, -3) into the equation and see what we get:
. Now, let’s look at (-1, -3).
Which number is the x value
and which one
is the y value? If
you said x = -1 and y = -3, you are right!
Let’s plug (-1, -3) into the equation and see what we get:
.Standard Form:
Ax + By = C
This form is called the standard form of a linear equation.
Example
4: Write the following linear equation in standard
form. 
Step 1: Find three ordered pair
solutions.
Yes, it can be ANY three values you want, 1, -3, or even 10,000. Remember there are an infinite number of solutions. As long as you find the corresponding y value that goes with each x, you have a solution.
Step 2: Plot the points found
in step 1.
The point lines up with both the x value of the ordered pair (x-axis) and the y value of the ordered pair (y-axis).
Step 3: Draw the graph.
If you know it is a linear equation and your points don’t line up, then you either need to check your math in step 1 and/or that you plotted all the points found correctly.
Example
5: Determine whether the equation is linear or
not.
Then graph the equation. y = 5x - 3Since we can write it in the standard form, Ax + By = C, then we have a linear equation.
This means that we will have a line when we go to graph this.
If you do this step the same each time, then it will make it easier for you to remember how to do it.
I usually pick out three points when I know I’m dealing
with a line.
The three x values I’m going to use are
-1,
0, and 1. (Note that you can pick ANY three x values that you
want. You do not have to use the values that I picked.) You
want to keep it as simple as possible. The following is the chart
I ended up with after plugging in the values I mentioned for x.
Step 1: Find six or seven
ordered pair solutions.
Step 2: Plot the points found
in step 1.
Step 3: Draw the graph.
Example
6: Determine whether the equation is linear or
not.
Then graph the equation. 
.
Is this a linear equation? Note how we have an x squared
as opposed to x to the one power.
It looks like we cannot write it in the form Ax + By = C because the x has to be to the one power, not squared. So this is not a linear equation.
However, we can still graph it.
Note that the carrot top (^) represents an exponent. For
example, x squared can be written as x^2. The
second
column is showing you the 'scratch work' of how we got the
corresponding
value of y.
(x, y)
-3
y = (-3)^2 - 4 = 5
(-3, 5)
-2
y = (-2)^2 - 4 = 0
(-2, 0)
-1
y = (-1)^2 - 4 = -3
(-1, -3)
0
y = (0)^2 - 4 = -4
(0, -4)
1
y = (1)^2 - 4 = -3
(1, -3)
2
y = (2)^2 - 4 = 0
(2, 0)
3
y = (3)^2 - 4 = 5
(3, 5)
Example
7: Determine whether the equation is linear or
not.
Then graph the equation. 
In other words, we can’t write it in the form Ax + By = C. This means that this equation is not a linear equation.
If you are unsure that an equation is linear or not, you can ALWAYS plug in x values and find the corresponding y values to come up with ordered pairs to plot.
(x, y)
-3
y = |-3 + 1| = 2
(-3, 2)
-2
y = |-2 + 1| = 1
(-2, 1)
-1
y = |-1 + 1| = 0
(-1, 0)
0
y = |0 + 1| = 1
(0, 1)
1
y = |1 + 1| = 2
(1, 2)
2
y = |2 + 1| = 3
(2, 3)
3
y = |3 + 1| = 4
(3, 4)
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problem 1a: Plot the ordered pairs and name the quadrant or axis in which the point lies.
Need Extra Help on these Topics?
http://www.purplemath.com/modules/plane.htm
This webpage helps you with plotting points.
http://www.math.com/school/subject2/lessons/S2U4L1DP.html
This website helps you with plotting points.
http://www.purplemath.com/modules/graphlin.htm
This webpage helps you with graphing linear equations.
Go to Get Help Outside the Classroom found in Tutorial 1: How to Succeed in a Math Class for some more suggestions.
Last revised on July 3, 2011 by Kim Seward.
All contents copyright (C) 2001 - 2011, WTAMU and Kim Seward. All rights reserved.
