# Engaging students: Finding the asymptotes of a rational function

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Belle Duran. Her topic, from Algebra: finding the asymptotes of a rational function.

How has this topic appeared in high culture?

Although the topic itself has not appeared in high culture, idea of asymptotes brings me the idea of the myth of Tantalus. In a nutshell, Tantalus was always committing crimes against the Gods of Olympus but always going unpunished. One day, he invites the Gods to his home for a feast in which he serves the Gods a rather vile dish. This ultimately angered the Gods to the point of punishing Tantalus by hanging him from a fruit tree amidst a lake, sentencing him to suffer eternal hunger and thirst. Tantalus was always so close to the water and fruits, yet they stayed beyond his reach. In the same way, when a graph has an asymptote then a part of the graph will approach that asymptote without ever touching it or being equal to it.

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The word, “asymptote” derives from the Greek word, “asumptotos” which translates to “not falling together.” The term was first introduced by Apollonius of Perga in his work on conic sections, but used the term to represent a line that will not meet the curve in any finite point. Other achievements by Apollonius includes the introduction of eccentric and epicyclic motion to explain the motion of the planets as well as the hemicyclium which is a sundial with hour lines drawn on the surface of a conic section to give greater accuracy.

How does this topic be used in your students’ future courses in mathematics or science?

One way finding asymptotes can be used in students’ future courses are to understand finding the limits of a function. When it comes to limits, it can be shown that vertical asymptotes are concerned with objectives in which the function is not usually defined and near which the function becomes large positively or negatively, or if a line x=a is called a vertical asymptote for the graph of a function of either the limit to positive infinity as x approaches positive a or negative a. Likewise, horizontal asymptotes are concerned with finite values approached by the function as the independent variable grows large positively or negatively. In other words, a line y=b is a horizontal asymptote for the graph is either the limit of the function is b as x approaches positive infinity or negative infinity.

References

http://www.greekmyths-greekmythology.com/the-myth-of-tantalus/

http://www-history.mcs.st-and.ac.uk/Biographies/Apollonius.html

http://jwilson.coe.uga.edu/emat6680/greene/emat6000/greek%20geom/Apollonius/apollonius.html

http://www.education.com/study-help/article/horizontal-vertical-asymptotes/

http://oregonstate.edu/instruct/mth251/cq/Stage3/Lesson/asymptotes.html