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Find the zeros of the function, and then describe the behavior of the graph at each zero.

Ask by Ingram Owen. in the United States
Dec 27,2024

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The zeros of the function are and . At , the graph crosses the x-axis, and at , it touches the x-axis and turns around without crossing.

Solución

To find the zeros of the function and describe the behavior of the graph at each zero, follow these steps:

1. Factor the Function

First, factor the given cubic function:
Factor out the greatest common factor (which is ):
Next, factor the quadratic expression :
So, the fully factored form of the function is:

2. Find the Zeros

Set each factor equal to zero to find the zeros:
  • First Zero:
  • Second Zero:
Thus, the zeros of the function are:

3. Determine the Multiplicity of Each Zero

  • Zero at : This zero comes from the factor and has a multiplicity of 1.
  • Zero at : This zero comes from the factor and has a multiplicity of 2.

4. Describe the Behavior at Each Zero

  • At (Multiplicity 1):
    • Multiplicity is odd (1).
    • Behavior: The graph crosses the x-axis at this point.
  • At (Multiplicity 2):
    • Multiplicity is even (2).
    • Behavior: The graph touches the x-axis and turns around at this point without crossing.

5. Summary

  • Zeros:
    • (Multiplicity 1)
    • (Multiplicity 2)
  • Graph Behavior:
    • At : The graph crosses the x-axis.
    • At : The graph touches the x-axis and turns around.

6. Graphical Illustration

Here’s a rough sketch to visualize the behavior:
       |
     f(x)
       |
    *--•--*--•
       |
      -2   0
  • The graph crosses the x-axis at .
  • The graph touches and turns around at .
This analysis fully describes the zeros and the corresponding behavior of the graph for the given function.

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Bonus Knowledge

To find the zeros of the function , we can start by factoring out the greatest common factor, which is :
Next, we can further factor the quadratic part:
Thus, our factored function is:
Setting the function to zero to find the zeros, we have:
From here, we can see that the zeros are:
  1. (with multiplicity 2)
Now, let’s discuss the behavior of the graph at these zeros:
  1. At , the graph crosses the x-axis since it’s a simple zero (multiplicity of 1).
  2. At , because of the multiplicity of 2, the graph touches the x-axis and bounces back up, rather than crossing over.
So, we have one crossing point at and a bouncing point at . Happy graphing!

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