Lesson 6E Sign on the Dotted Line Practice Understanding

Learning Focus

Determine a process for graphing rational functions from an equation.

How can we quickly determine the behavior near the asymptotes of a rational function?

Open Up the Math: Launch, Explore, Discuss

Josue and Francia are working on graphing all kinds of rational functions when they have this little dialogue:

Josue: It’s easy to figure out where the asymptotes and intercepts are on a rational function.

Francia: Yes, and it’s almost like the asymptotes split the graph into sections. All you need to know is what the graph is doing in each section.

Josue: It seems almost easier than that. It’s like all you need to know is whether it’s going up or down either side of the vertical asymptote and then use logic to figure it out from there.

Francia: Don’t overlook the intercepts. They give some pretty important clues.

Josue: Yeah, yeah. I wonder if we can figure out an easy way to determine the behavior near the asymptotes.

Francia: Seems easy enough to just plug in numbers and see what the outputs are, but maybe you don’t even need exact values. Hmmm. We need to think about this.

Josue and Francia are definitely on to something. Everyone wants to find a way to be able to predict and sketch graphs easily. In this task, you’re going to work on just that. Start by finding asymptotes and intercepts, then figure out a strategy that you can use every time to quickly sketch the graph. After using your strategy to graph the function, use technology to check your work and refine your strategy.

The examples you need to develop your strategy are provided. Some of the functions given need to be combined or have common factors divided out to make one rational function. If this is the case, write the equivalent function in the space next to the graph.

1.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

Graph:

a blank 17 by 17 grid

2.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

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3.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

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4.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

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5.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

a blank 17 by 17 grid

6.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

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7.

Vertical Asymptote(s):

Horizontal or Slant Asymptote:

Intercepts:

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8.

Summarize your strategy for graphing rational functions with a step-by-step process.

Ready for More?

Write the equation of a rational function that has a slant asymptote and two vertical asymptotes.

Takeaways

How to Graph a Rational Function

Steps:

Example:

Remember this:

Sketch the function.

a blank 17 by 17 grid

Lesson Summary

In this lesson, we learned to sketch graphs of rational functions by finding the intercepts and the asymptotes, and by determining the behavior near the asymptotes. From this information we sketched the general shape of the graph without calculating exact points.

Retrieval

1.

Determine which of the given value(s) is a solution for the equation.

A.

B.

C.

2.

a.

Graph .

a blank 17 by 17 grid

b.

Describe some of the key features of the function.