LINR 1 | Lesson 3 | Explore (Vertical Lines)

Vertical Lines

Answer the following questions given the line graphed below.

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

Check your Solutions

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.

https://www.khanacademy.org/math/algebra/introduction-to-algebra/division-by-zero/v/why-dividing-by-zero-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 $$x=2$$.