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Linear Functions

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 y-coordinate 2. In fact, the rule for this function is y=2. Notice that this looks different than most linear functions. The slope m of a line is the quotient between the rise and the run between any two of its points.
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. Therefore, its slope is 0. In fact, all horizontal lines have slope m=0. The slope-intercept form of horizontal lines can be written as follows.
Therefore, 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 that the same x-value corresponds to infinitely many y-values. Therefore, this line does not represent a function. In fact, no vertical line is a function. Similar to horizontal lines, vertical lines take the following form.

x=a

Here, a is the x-intercept. The rule for the line above is x=3.
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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
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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=s2,
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=42.
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
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