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Types of Line Graphs

Graphs are very important aspects of math and specifically in geometry. The most common of the graphs are graphs of linear equations more commonly known as line graphs. The equations, which when plotted on a grid yield a straight line, are called equations of line graphs or simple linear equations. Line graphs can be of various types.

They are as listed below:

Horizontal line graph
2. Vertical line graph
3. Parallel lines graph
4. Perpendicular lines graph
5. Skewed lines graph

All these types of graphs find application in the real world as well. In general, we see the use of lines in almost each and everything in our homes and outside. Let us look at each of these types of graphs one by one.

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Horizontal Line Graph

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A line graph is said to be a horizontal line graph if when it is plotted on the coordinate grid, turns out to be a horizontal line. A horizontal line is typically the line that is parallel to the x-axis. Now, since the line is parallel to the x-axis, obviously it is not going to intercept the x-axis. It will, however, intercept the y-axis. Suppose the graph intercepts the y-axis in the number b. Since this is a horizontal line graph, the y-coordinates of all the points on this graph would be b. Thus, the graph would look something like this:

Horizontal Line Graph
If we pick any two points on this graph then the coordinates of the two points would be ($x_1$,$b$) and ($x_2$,$b$). The formula for slope of a line is:

$slope$ = $m$ = $\frac{y_2\ -\ y_1}{x_2\ -\ x_1}$

Thus, for this line the slope of the line would be:

$m$ = $\frac{b\ -\ b}{x_2\ -\ x_1}$ = $\frac{0}{x_2\ -\ x_1}$ = $0$

Also, we already know that the y-intercept is $b$.

Thus, using the slope-intercept form of equation of a line we see that the equation of a horizontal line, in 
general, would be:

$y$ = $0\ x\ +\ b$

$y$ = $b$

Vertical Line Graph

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As the name suggests, a vertical line graph is a graph that turns out to be a vertical line when graphed on a coordinate grid. This graph would be parallel to the y-axis. So obviously it would not intercept the y-axis. The graph would, however, intercept the x-axis. Suppose the graph intercepts the x-axis in the point a. In the above section on horizontal line graphs we saw that when the graph is horizontal and intercepts the y axis in b, then the equation of the graph is given by 

$y$ = $b$

Similarly in this case, since the line is vertical, it intercepts the x axis only. The intercept on the x-axis is a, so the equation of this graph would be:

$x$ = $a$

Let’s say that the graph looks as follows:

Vertical Line Graph
Two points on the graph are marked as ($a$,$y_1$) and ($a$,$y_2$). The x co-ordinates of the two points would be same, that is a for obvious reasons. The slope of this line can be found as:

$slope$ = $m$ = $\frac{y_2\ -\ y_1}{x_2\ -\ x_1}$ = $\frac{y_2\ -\ y_1}{a\ -\ a}$ = $\frac{y_2\ -\ y_1}{0}$ = not defined

Since division by zero is not defined, the slope of a vertical line is not defined or we can say that it is equal to infinity.

Parallel Lines Graph

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Two lines in the same plane are said to be parallel if they never meet even if they are extended indefinitely in both the directions. The property of parallel lines is that the equations of two parallel lines would have the same slope. In other words, if the equations of two lines have the same slope then the lines would be parallel. See the following graph:
Parallel Lines Graph

The brown line is the graph of the equation $y$ = $2\ x\ +\ 1$ and the green line is the graph of the equation $y$ = $2\ x\ -\ 1$. Note that these two lines are parallel. The slope of the two lines are both $2$, thus they have equal slopes. In general equations of parallel lines would be of the form:

line 1: $y$ = $mx\ +\ b_1$

line 2: $y$ = $mx\ +\ b_2$

So basically the lines have the same slope but different y-intercepts.

Perpendicular Lines Graph

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Two lines in a plane are said to be perpendicular to each other if they intersect each other at right angles. For the lines to be parallel or perpendicular they need to be in the same plane. The horizontal line and the vertical line that we saw earlier are perpendicular lines. The horizontal line graph would always be perpendicular to the vertical line graph.

Perpendicular Lines Graph

The slopes of perpendicular lines are always negative reciprocal of each other. For example, if lines $l_1$ and $l_2$ are perpendicular lines, and if their slopes are $m_1$ and $m_2$, then:

$m_1$ = $\frac{-\ 1}{m_2}$
This equation can also be written as:

$m_1\ \times\ m_2$ = $-1$
In the above graph, note that the two lines are perpendicular to each other. The equation of the purple line is $y$ = $2\ x\ +\ 1$ and that of the green line is $y$ = $\frac{-\ 1}{2}$ $x\ +\ 1$. Note that the slopes are negative reciprocals of each other.

Skew Lines Graph

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Two lines are said to be skewed when they do not lie in the same plane. The lines cannot be termed as parallel or intersecting as they are not in the same plane. The lines are in different planes. See the picture below:
Skew Lines Graph

The lines in the above picture are in different planes thus they are skewed lines.


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Example 1: A classic example of parallel lines is the railway tracks. The two bars of railway tracks are always parallel to each other however long they may be. 

Types of Line Graphs Examples
Example 2: A set of crossroads is a good example of perpendicular lines. The road that runs north-south is perpendicular to the road that runs east-west.

Example of Perpendicular Lines
Example 3: A good example of skewed lines would be lines drawn on two different pages of your book. Also, lines on the ceiling and floor of your house are examples of skewed lines.
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