Graph Lines
The main purpose of graphs is not to plot random points, but
rather to give a picture of the solutions to an equation. A linear
equation will always produce a straight line.
We start by finding possible x and y combinations. Notice that
using 0 is rather convenient. When x=0, this is the point where
the line will cross the y-axis and is called the y-intercept. When
y=0, this is where the line will cross the x-axis and is called the
x-intercept.
Example 1: Find the x-intercept and y-intercept of the linear equation:
3x – 7y = 21
To find the x-intercept, let y=0
3x – 7(0) = 21
3x – 0 = 21
3x = 21
3
3
x=7
The x-intercept is (7, 0)
To find the y-intercept, let x=0
3(0) – 7y = 21
0 – 7y = 21
7y = 21
7
7
y=3
The y-intercept is (0, 3)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
In order to graph a straight line, we need a minimum of two
unique points. A third point is desirable to make sure that the
other points are correct. We will start by finding possible x and y
combinations. We will do this using a table of values. Using the
x-intercept and y-intercept is not only easy to determine but easy
to graph.
Example 2: Graph: 2x – 3y = 6
x
y
0
0
We will complete this table of
values. Note that any values
could be used.
-3
2(0) – 3y = 6
- 3y = 6
-3 -3
y = -2
Substitute 0 for x
2x – 3(0) = 6
2x
=6
2
2
x
=3
Substitute 0 for y
Divide both sides by -3
When x=0, y=-2: (0, -2)
Divide both sides by 2
When y=0, x=3: (3, 0)
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
2(-3) – 3y = 6
- 6 – 3y = 6
+6
+6
- 3y = 12
-3
-3
y = -4
x
y
0
3
-3
-2
0
-4
Substitute -3 for x
Add 6 to both sides
Divide both sides by -3
When x=-3, y=-4: (-3, -4)
The completed table
The points are (0, -2), (3, 0), (-3, -4). Plotting the points
and drawing the line, we have the following graph.
y
6
4
2
6
4
2
2
4
6
x
2
4
6
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Sometimes linear equations are written in function notation.
Recall that when evaluating a function, values for x are put into
the function, and the result is an ordered pair (x, y). In other
words, we find the y value. To graph a function, it is often useful
to re-write it by replacing the f(x) by y.
1
Example 3: Graph: f(x) = x – 3
2
1
Re-write the function: y = x – 3
2
x
y
0
2
-2
1
We will complete this table of
values. Note that in this case,
choosing all x-values is easier.
Since the coefficient of x is a
fraction, we choose multiples of
the denominator.
y = (0) – 3
Substitute 0 for x
y=0–3
y=-3
When x=0, y=-3: (0, -3)
2
1
y = (2) – 3
Substitute 2 for x
y=1–3
y=-2
When x=2, y=-2: (2, -2)
2
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
1
y = (-2) – 3
Substitute -2 for x
y=-1–3
y=-4
When x=-2, y=-4: (-2, -4)
2
x
y
0
2
-2
-3
-2
-4
The completed table
The points are (0, -3), (2, -2), (-2, -4). Plotting the points
and drawing the line, we have the following graph.
y
6
4
2
6
4
2
2
4
6
x
2
4
6
Horizontal and vertical can also be graphed using the table of
values. Since horizontal lines are indicated by y = value, all of
the y’s will be that value while the x’s can be anything. Since
vertical lines are indicated by x = value, all of x’s will be that
value while the y’s can be anything.
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 4: Graph: y = 3
x
We will complete this table of
values. Note that all the y-values
are 3.
y
3
3
3
We can choose any value for x. For example, we choose:
x
y
2
4
-3
3
3
3
The points are (2, 3), (4, 3), (-3, 3). Plotting the points
and drawing the line, we have the following graph.
y
6
4
2
6
4
2
2
4
6
x
2
4
6
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)
Example 5: Graph: x + 7 = 5
x+7=5
-7 -7
x
= -2
x
Subtract 7 from both sides
We will complete this table of
values. Note that all the x-values
are -2.
y
-2
-2
-2
We can choose any value for y. For example, we choose:
x
y
-2
-2
-2
5
0
-3
The points are (-2, 5), (-2, 0), (-2, -3). Plotting the points
and drawing the line, we have the following graph.
y
6
4
2
6
4
2
2
4
6
x
2
4
6
Modified from Beginning and Intermediate Algebra, by Tyler Wallace, CC-BY 2010. Licensed under a
Creative Commons Attribution 3.0 Unported License (http://creativecommons.org/licenses/by/3.0)