Graphing Horizontal and Vertical Lines

Among the family of linear relationships, there are some subgroups of lines that are noteworthy. Two such types are horizontal and vertical lines.

Concept

Horizontal Line

A horizontal line is a line that runs parallel to the $x$ -axis. An example can be seen in the diagram below.

Every point on the line above has a $y$ -coordinate of $2 .$ In fact, the rule for this function is $y = 2 .$ Notice that this looks different than most linear functions seen thus far. The slope, $m,$ of a line is given by the relationship $m = \frac{rise}{run} .$ In the graph above, it can be seen that the line does not rise. In other words, the vertical change between any two points is $0 .$ In fact, all horizontal lines have the slope $m = 0 .$ Writing a linear function rule in slope-intercept form gives $y = 0 \cdot x + b \Rightarrow y = b .$

Thus, all horizontal lines can be written in this form where

b

is the

y

-intercept.

Concept

Vertical Line

A vertical line is a line that runs parallel to the $y$ -axis. A vertical line is drawn in the coordinate plane below.

Notice that the line passes through all points where the

x

-coordinate is

3 .

This means the same

x

-value corresponds to infinitely many

y

-values. Thus, it is not a function. In fact, no vertical line is a function. Similar to horizontal lines, vertical lines take the form

$x = a,$

where

a

is the

x

-intercept. The rule for the line above is

x = 3 .

Write the horizontal and the vertical lines that intersect

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Exercise

One horizontal line and one vertical line intersect at the point $(1, - 6) .$ Write the equations of the lines.

Show Solution

Solution

To write the equations of the horizontal and vertical lines that intersect at $(1, - 6),$ it can be helpful to first sketch a graph. We'll plot the point, then draw a vertical and horizontal line through the point.

The horizontal line intersects the $y$ -axis at $- 6$ . Thus, we can write the rule as $y = - 6 .$ The vertical line intersects the $x$ -axis at $1 .$ Thus, its equation is $x = 1 .$ Therefore, the horizontal and vertical lines that intersect at $(1, - 6)$ are $y = - 6 and x = 1 .$

Write the horizontal and the vertical lines that form a square

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Exercise

Together with the axes, a horizontal and a vertical line form a square in Quadrant I of the coordinate plane. Write the rules for the lines so that the square has an area of $16$ square units.

Show Solution

Solution

To begin, let's consider the area of a square. One formula that can be used to calculate the area of a square is $A = s^{2},$ where $s$ is the side length of the square. The square's area, $A,$ is $16$ square units. Therefore, we must find a side that when squared equals $16 .$ We recognize that $16$ is a perfect square and that $16 = 4^{2} .$ Therefore, each side of the square must measure $4$ units. Let's sketch a horizontal and vertical line that, together with the axes, for a square with side length 4. Notice that the $x$ -axis and $y$ -axis will be the bottom and left side of the square, respectively. To ensure that the top and bottom of the square are $4$ units, we can draw a vertical line at $x = 4 .$

Next, to ensure that the sides are also $4$ units, we can draw a horizontal line at $y = 4 .$

The horizontal line intercepts the $y$ -axis at $4 .$ Therefore, the line can be written as $y = 4 .$ The vertical line intercepts the $x$ -axis at $4 .$ Thus, we can write the vertical line as $x = 4 .$ The two lines that form the requested square are $y = 4 and x = 4 .$

Graphing Horizontal and Vertical Lines

Horizontal Line

Vertical Line

Example

Write the horizontal and the vertical lines that intersect

Example

Write the horizontal and the vertical lines that form a square