## Graphing Rational Functions By Zeros and Asymptotes

1. Start with a simple rational function. Click on the applet to begin. Then slide the x-values of the zero, the asymptote and the scaling factor of the rational function to see how it changes the graph.

What happens when the vertical asympotote moves to the other side of the zero of the function?

What happens when the zero and the asymptote are at the same value?

How does M change the graph?

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2. Now try a more complicated rational function.Click on the applet to begin. Then slide the x-values of the two zeros , the asymptote and the scaling factor of the rational function to see how it changes the graph.

What happens when the vertical asympotote and the two zeros change order?

What happens when the two zeros are at the same value?

How does M change the graph?

What is the end behavior of the graph?

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3. Now try a rational function with two zeros and two asymptotes.

What happens when you change the order of the zeros and asymptotes?

What happens when you have double zeros? Or both asymptotes have the same value?

What is the end behavior of the graph?

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4. Finally, consider a rational function with three zeros and two asymptotes. Explore how the graph changes when you move the zeros and the asymptotes.

What is the end behavior of the graph?

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John Maharry, 1/22/2007, Created with GeoGebra