Horizontal and vertical lines
Lesson

Straight lines on the $xy$xy-plane can actually be in any direction and pass through any two points. This means that straight lines can be:

 decreasing horizontal increasing vertical

Horizontal and vertical lines

We can quickly identify that a line is horizontal if it is parallel to the $x$x-axis. Similarly, a line is vertical if it's parallel to the $y$y-axis.

### Horizontal lines

We can draw a set of points that makes a horizontal line in the same way we can draw any set of points as before. Consider the following table of values:

 $x$x $y$y $-5$−5 $-4$−4 $-3$−3 $-2$−2 $2$2 $2$2 $2$2 $2$2

Plotting each column as points on the $xy$xy-plane gives us the following:

 Plotting points from the table of values

Clearly the set of points form a line that is parallel to the $x$x-axis. But just to confirm, we can draw a line through these points to show this.

 Horizontal line passing through all four points

It's also clear from the graph that the distance between any two neighbouring points is exactly $1$1 unit apart. We obtain the distance by referencing the $x$x-axis, and subtracting the $x$x-values of the two points. This means we don't have to plot the points to determine the distance between a pair of points. Consider the table of values earlier.

 $x$x $y$y $-5$−5 $-4$−4 $-3$−3 $-2$−2 $2$2 $2$2 $2$2 $2$2

The distance between the points $\left(-4,2\right)$(4,2) and $\left(-3,2\right)$(3,2) is:

$-3-\left(-4\right)=1$3(4)=1 unit.

Careful!

You may find yourself with a negative distance depending on what order you subtract the two $x$x-values. Make sure to subtract in the correct order or change your final answer to a positive number.

### Vertical lines

Just as we did before, we can plot a set of points obtained from a table of values and show that the points fall on a vertical line. Consider the following table of values:

 $x$x $y$y $2$2 $2$2 $2$2 $2$2 $-3$−3 $-1$−1 $1$1 $3$3

Plotting the points in the table of values gives us the following graph. We can also pass a line through the points to show that the points lie on a vertical line.

 Plotting points from the table of values

By referencing the $y$y-axis we can see that the distance between a pair of neighbouring points is $2$2 units. Alternatively we can refer to the ordered pairs $\left(2,-1\right)$(2,1) and $\left(-1,-3\right)$(1,3) and subtract the two $y$y-values:

$-1-\left(-3\right)=2$1(3)=2 units

We still need to be careful about the order we subtract the two $y$y-values to ensure we obtain a positive distance (or you could change your final answer so that it's positive).

Let's have a look at some examples of questions that might arise when dealing with horizontal or vertical lines.

#### Examples

##### Question 1

Plot the points in the table of values.

 $x$x $y$y $2$2 $3$3 $4$4 $5$5 $5$5 $5$5 $5$5 $5$5

##### Question 2

Consider the points in the plane below.

Which of the following statements is true?

1. The set of points lie on a decreasing line.

A

The set of points lie on an increasing line.

B

The set of points lie on a vertical line.

C

The set of points lie on a horizontal line.

D

The set of points lie on a decreasing line.

A

The set of points lie on an increasing line.

B

The set of points lie on a vertical line.

C

The set of points lie on a horizontal line.

D

##### Question 3

What is the shortest distance between any two of the following points?