Question
Describe the end behavior of f(x)=1+7x+9x^3+6x^4-2x^5.
Answers
Answered by
GPT-4o mini
To analyze the end behavior of the polynomial function \( f(x) = 1 + 7x + 9x^3 + 6x^4 - 2x^5 \), we focus on the term with the highest degree, which is the leading term. The highest degree term in this polynomial is \(-2x^5\).
The end behavior of a polynomial function is primarily determined by its leading term. Specifically, for a polynomial of the form \( ax^n \), the end behavior can be summarized as follows:
1. If \( a > 0 \) and \( n \) is even, \( f(x) \to +\infty \) as \( x \to +\infty \) and \( f(x) \to +\infty \) as \( x \to -\infty \).
2. If \( a > 0 \) and \( n \) is odd, \( f(x) \to +\infty \) as \( x \to +\infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).
3. If \( a < 0 \) and \( n \) is even, \( f(x) \to -\infty \) as \( x \to +\infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).
4. If \( a < 0 \) and \( n \) is odd, \( f(x) \to -\infty \) as \( x \to +\infty \) and \( f(x) \to +\infty \) as \( x \to -\infty \).
In our case, the leading term is \(-2x^5\):
- Coefficient \( a = -2 \) (which is less than zero)
- Degree \( n = 5 \) (which is odd)
Therefore, we can deduce the end behavior of the function \( f(x) \):
- As \( x \to +\infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to +\infty \).
In summary, the end behavior of \( f(x) = 1 + 7x + 9x^3 + 6x^4 - 2x^5 \) is:
- \( f(x) \to -\infty \) as \( x \to +\infty \)
- \( f(x) \to +\infty \) as \( x \to -\infty \)
The end behavior of a polynomial function is primarily determined by its leading term. Specifically, for a polynomial of the form \( ax^n \), the end behavior can be summarized as follows:
1. If \( a > 0 \) and \( n \) is even, \( f(x) \to +\infty \) as \( x \to +\infty \) and \( f(x) \to +\infty \) as \( x \to -\infty \).
2. If \( a > 0 \) and \( n \) is odd, \( f(x) \to +\infty \) as \( x \to +\infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).
3. If \( a < 0 \) and \( n \) is even, \( f(x) \to -\infty \) as \( x \to +\infty \) and \( f(x) \to -\infty \) as \( x \to -\infty \).
4. If \( a < 0 \) and \( n \) is odd, \( f(x) \to -\infty \) as \( x \to +\infty \) and \( f(x) \to +\infty \) as \( x \to -\infty \).
In our case, the leading term is \(-2x^5\):
- Coefficient \( a = -2 \) (which is less than zero)
- Degree \( n = 5 \) (which is odd)
Therefore, we can deduce the end behavior of the function \( f(x) \):
- As \( x \to +\infty \), \( f(x) \to -\infty \).
- As \( x \to -\infty \), \( f(x) \to +\infty \).
In summary, the end behavior of \( f(x) = 1 + 7x + 9x^3 + 6x^4 - 2x^5 \) is:
- \( f(x) \to -\infty \) as \( x \to +\infty \)
- \( f(x) \to +\infty \) as \( x \to -\infty \)