Use an end behavior diagram to describe the end behavior of the given function. [tex]P(x)=-\pi x^6+x^5-x^4-x+5[/tex]
Choose the behavior which best models the polynomial.
A.
B.
C.
D.
Answer (2)
Identify the leading term: − π x 6 .
Determine the sign of the leading coefficient: Negative.
Determine the degree of the polynomial: Even (6).
Conclude the end behavior: As x → ± ∞ , P ( x ) → − ∞ . The correct end behavior diagram is C.
Explanation
Identify the Leading Term The given polynomial is P ( x ) = − π x 6 + x 5 − x 4 − x + 5 . To determine the end behavior, we need to identify the leading term, which is the term with the highest degree. In this case, the leading term is − π x 6 .
Determine the Sign of the Leading Coefficient The leading coefficient is the coefficient of the leading term, which is − π . Since π ≈ 3.14159 , the leading coefficient is negative.
Determine the Degree of the Polynomial The degree of the polynomial is the exponent of the leading term, which is 6. This is an even number.
Determine the End Behavior Since the degree is even and the leading coefficient is negative, the end behavior is as follows: As x approaches infinity ( x → ∞ ), P ( x ) approaches negative infinity ( P ( x ) → − ∞ ). As x approaches negative infinity ( x → − ∞ ), P ( x ) approaches negative infinity ( P ( x ) → − ∞ ).
Select the Correct End Behavior Diagram The end behavior diagram that best models this polynomial is the one where both ends point downwards. This corresponds to option C.
Examples
Understanding end behavior of polynomials helps in modeling real-world phenomena. For instance, if P ( x ) represents the profit of a company over x years, a negative leading coefficient and even degree indicate that, in the long run, the profit will decrease regardless of whether we look into the distant future or the distant past. This could signify a product with a limited lifespan or a market with diminishing returns, prompting the company to innovate or adapt.
The leading term of the polynomial P ( x ) = − π x 6 + x 5 − x 4 − x + 5 is − π x 6 with a negative leading coefficient. Therefore, the end behavior shows that as x → + ∞ and x → − ∞ , the polynomial approaches negative infinity. The correct end behavior diagram is option C.
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