To describe the end behavior of a polynomial function \( h(x) \), we first need to consider the degree and leading coefficient of the polynomial.
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Degree of the Polynomial: The degree of the polynomial is the highest power of \( x \) in \( h(x) \). This determines whether the ends of the graph go up or down.
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Leading Coefficient: The leading coefficient is the coefficient of the term with the highest degree. This also influences the direction of the ends of the graph.
End Behavior Descriptions
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If the degree of \( h(x) \) is even:
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If the leading coefficient is positive:
- As \( x \to +\infty \) (x approaches positive infinity), \( h(x) \to +\infty \).
- As \( x \to -\infty \) (x approaches negative infinity), \( h(x) \to +\infty \).
- The ends of the graph rise on both sides.
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If the leading coefficient is negative:
- As \( x \to +\infty \), \( h(x) \to -\infty \).
- As \( x \to -\infty \), \( h(x) \to -\infty \).
- The ends of the graph fall on both sides.
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If the degree of \( h(x) \) is odd:
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If the leading coefficient is positive:
- As \( x \to +\infty \), \( h(x) \to +\infty \).
- As \( x \to -\infty \), \( h(x) \to -\infty \).
- The left end of the graph falls while the right end rises.
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If the leading coefficient is negative:
- As \( x \to +\infty \), \( h(x) \to -\infty \).
- As \( x \to -\infty \), \( h(x) \to +\infty \).
- The left end of the graph rises while the right end falls.
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Example
If, for example, \( h(x) = 2x^4 - 3x^2 + 1 \):
- It is a degree 4 polynomial (even degree).
- The leading coefficient (2) is positive.
Thus, the end behavior of \( h(x) \) would be:
- As \( x \to +\infty \), \( h(x) \to +\infty \).
- As \( x \to -\infty \), \( h(x) \to +\infty \).
In conclusion, to determine the end behavior of any polynomial function \( h(x) \), identify the degree and leading coefficient, and use the rules outlined above.
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