Math Problem Statement
Solution
To sketch the graph of the function , let's first analyze its key features.
Step-by-Step Analysis:
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Zeros of the function: The function is factored, so we can determine the zeros by setting each factor equal to zero:
- →
- →
- →
Therefore, the function has zeros at , , and .
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Multiplicity of the roots:
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At , the factor has a multiplicity of 1 (linear).
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At , the factor has a multiplicity of 2 (even).
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At , the factor has a multiplicity of 3 (odd).
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Multiplicity implications:
- A root with odd multiplicity crosses the x-axis.
- A root with even multiplicity touches the x-axis but doesn't cross.
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End behavior:
- As , the leading term is , so .
- As , the leading term is still dominated by , so .
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Behavior at roots:
- At , the graph crosses the x-axis (since the multiplicity is odd).
- At , the graph touches the x-axis and turns back up (since the multiplicity is even).
- At , the graph crosses the x-axis with an "S" shape (since the multiplicity is odd).
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Shape of the graph:
- The graph starts from positive infinity for large negative , comes down to zero at , crosses the axis, touches the x-axis at , and then crosses again at with a steep "S" shape.
Would you like me to plot the actual graph, or would you prefer a detailed numerical analysis for specific values of ? Let me know if you have any questions!
Here are 5 related questions to expand your understanding:
- What is the significance of root multiplicity in determining how the graph behaves at a given point?
- How do you determine the end behavior of a polynomial function?
- How does changing the leading coefficient (e.g., the in this case) affect the graph?
- How would the graph change if had an even exponent instead of odd?
- How can you use the first and second derivatives to analyze the concavity of this function?
Tip: Always pay attention to the multiplicity of roots; even multiplicity means the graph touches but doesn't cross, while odd multiplicity means it crosses the x-axis.
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Math Problem Analysis
Mathematical Concepts
Polynomial Functions
Roots and Multiplicity
End Behavior
Formulas
f(x) = 6x(x-1)^2(x-2)^3
Multiplicity rules: even multiplicity - touch, odd multiplicity - cross
Theorems
Multiplicity Theorem
End Behavior of Polynomial Functions
Suitable Grade Level
Grades 10-12
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