### Vertical Lines

Answer the following questions given the line graphed below.

- Explain why the line graphed above represents a linear relationship.
- Create a table containing all the defined ordered pairs shown in the graph.
- What do you notice about all the \(x\)-values in the table?
- What is the slope of this line? What method did you use to calculate the slope?
- What do you notice about the slope of this line?
- What is the \(y\)-intercept of this line? Name this point as an ordered pair.
- What equation can be used to represent any point on this line?
- Is this equation in the slope-intercept form?

Vertical lines have slopes that are in the form \(\Large \frac{rise}{run}=\Large \frac{a}{0}\), where \(a\) is an element of the complex numbers.

Mathematicians assert that any complex number divided by zero is undefined, and therefore, a slope in the form \(\Large \frac{a}{0}\) is undefined.

Click here to see a discussion between three students on the steepness of vertical cliffs.

Watch the Khan Academy video, linked below, which gives an explanation of why \(\Large \frac{a}{0}\) is undefined.

Visit Wolfram.com to view the explanation of division by zero using limits at: http://mathworld.wolfram.com/DivisionbyZero.html

Since a vertical line does not intersect the \(y\)-axis and has an undefined slope, there is no value for \(b\) (\(y\)-intercept) or value for \(m\) (slope).

As such, vertical lines cannot be represented using the slope-intercept form of a line (\(y=mx+b\)). There are no values of \(x\) that will produce values for \(y\).

Instead, the equation for any vertical line can be written in the form \(x=c\), where \(c\) is the constant.

In the graph shown above, the constant is \(2\), since every coordinate has the \(x\)-value of \(2\) for any value of \(y\). In this example, the equation for this vertical line is \(x=2\).