## Reflection: Developing a Conceptual Understanding Discontinuity in Rational Functions - Section 3: Graphing Discontinuities

Why not simply have students graph the function with some sort of technology?

Even at the Algebra 2 level, many students still struggle to understand the connection between the equation and the graph.  By carefully evaluating the equation to produce data, then plotting that data by hand, the students have the chance to make the necessary connections.  That's not to say that the technology is bad, but I don't want to rely on it exclusively.

For instance, after noticing that f(-2) is indeterminate, one student asked today, "So how do we graph that?"

When I asked him to clarify, he said, "Well, I know you can't put anything on the graph for x = -2, but what about the other points.  It's like when we graph parabolas; we just plot a few points and then we fill in the curve to connect them.  But what should we do here?"

I was blown away!  This student had noticed that when we "connect the dots" we're implying that the function is continuous on that interval!

Going further, he pulled out his graphing calculator and showed me the screen.  "See," he said, "the calculator just drew a line across here (pointing to where the vertical asymptote ought to be), but it shouldn't have.  So what should it look like?"

"Great question," I replied, "why don't you evaluate the function for x-values close to that point and see for yourself."

Why graph by hand?
Developing a Conceptual Understanding: Why graph by hand?

# Discontinuity in Rational Functions

Unit 7: Rational Functions
Lesson 4 of 17

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

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Standards:
Subject(s):
Math, Graphing (Algebra), Algebra, Function Operations and Inverses, Algebra 2, master teacher project, rational function, asymptotes, discontinuity
45 minutes

### Jacob Nazeck

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