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How does the multiplicity of a zero determine the behavior of the graph at that zero?

How does the multiplicity of a zero determine the behavior of the graph at that zero-example-1
User Persephone
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Answer:

  • -6: crosses
  • 0: crosses
  • 1: touches
  • 4: crosses

Explanation:

When the multiplicity is odd, the function changes sign on either side of the zero. Hence the graph crosses the x-axis.

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When the multiplicity is even, the function has the same sign on either side of the zero, so the graph only touches the x-axis, but does not cross.

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For the given zeros, the behavior is ...

-6, multiplicity odd, crosses the x-axis

0, multiplicity odd, crosses the x-axis (curve is flatter than at -6 or 4)

1, multiplicity is even, touches the x-axis

4, multiplicity is odd, crosses the x-axis

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Additional comment

The higher the multiplicity of the root, the "flatter" the curve is at the x-axis.

How does the multiplicity of a zero determine the behavior of the graph at that zero-example-1
User Strahinja Kustudic
by
7.5k points

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