Discontinuity in Rational Functions

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Objective

SWBAT use the structure of a rational equation to identify points of discontinuity.

Big Idea

Unlike polynomials, rational functions may be discontinuous. Blink and you'll miss it; there's just one point removed!

Definitions

5 minutes

Let's cut to the chase.  The students have been studying some examples of rational functions recently, and you can't really make general claims about something without first defining it, so it's time to give them a formal definition.

When I do, I like to begin by asking if anyone remembers the definition of a polynomial.  They should recall that the operation of division is excluded.  This is just the opening that I'm looking for: when you divide by x, you make a new kind of expression, in the same way that when you start dividing integers by integers you create a new kind of number.

So, a rational expression/equation/function is one that consists of one polynomial divided by another polynomial.  Together, now, we write out and discuss the formal definition.

Getting to Know a Rational Function

15 minutes

Hand out Intro to Rational Functions.

Students should begin by working independently.  At first, they're simply asked to copy down the definition, but then they need to apply it in a very simple way.  These first tasks should take no more than a few minutes.  

Before long, students should be engaged in computing some function values and beginning to wonder why their calculators say "Error" for some of them!  At this point, I'm careful not to tell anyone why their calculators are having trouble.  The answer should be pretty clear, especially if they try to do the computations by hand!  Also, the notion that division by zero is undefined is so crucial to understanding rational functions, that it's very important for students to reason through these problems and really see for themselves why the function must be discontinuous. This will help them to develop a habit of paying attention to the structure of a function.  (MP 7)

If your students are using calculators for the computations, watch out!  Many of them may try to enter everything at once and they usually don't pay as much attention to the order of operations as their calculators do.

Once everyone has the first table of values completed, the individual time ends. 

Graphing Discontinuities

20 minutes

Working in small groups (three students, at most) everyone should now work to make a careful graph of the function.

The focus should be on the behavior of the function around the two discontinuities.  You may find that some students want to connect the two branches of the hyperbola; rather than simply tell them not to, ask them to compute more function values close to the point of discontinuity to get a more precise notion of the behavior of the function.  On their own, they'll come to recognize the existence of the vertical asymptote. (MP 6) For more on this, please see the video, A Common Misconception .

At this point, it's enough for students to recognize the difference between the behavior of the function around the two discontinuities.  Later, we'll focus on understanding why.

Finally, although it isn't explicitly mentioned, students should also be encouraged to consider the end behavior of the function.

Wrapping Up

5 minutes

At the end of class, I will project a graph of this function on the board for everyone to see.  We'll comment on the following:

  • the limited domain of the function (caused by division by zero)
  • the recognizable, hyperbolic shape
  • the end behavior of the function (horizontal asymptote)
  • the existence of two different types of discontinuity (removable vs. vertical asymptote)

Take a look at these samples of student work along with my comments.

To wrap things up, I'll point out that during this unit we're going to spend time getting more familiar with the structure of rational expressions, combining rational expressions in different ways, solving rational equations, and eventually reaching a point at which we'll be able to make accurate predictions about the overall behavior of the function just by examining the equation.