Which of the following describes the end behavior of the function [tex]f(x)=-5 x^3+3 x^2 +x-9[/tex]?
A) As [tex]x \rightarrow-\infty, y \rightarrow-\infty[/tex] and as [tex]x \rightarrow+\infty, y \rightarrow+\infty[/tex]
B) As [tex]x \rightarrow-\infty, y \rightarrow+\infty[/tex] and as [tex]x \rightarrow+\infty, y \rightarrow-\infty[/tex]
C) As [tex]x \rightarrow-\infty, y \rightarrow+\infty[/tex] and [tex]25 x \rightarrow+\infty, y \rightarrow+\infty[/tex]
D) As [tex]x \rightarrow-\infty, y \rightarrow-\infty[/tex] and as [tex]x \rightarrow+\infty, y \rightarrow-\infty[/tex]
Answer (1)
The end behavior of a polynomial function is determined by its leading term.
As x approaches − ∞ , − 5 x 3 approaches + ∞ .
As x approaches + ∞ , − 5 x 3 approaches − ∞ .
The end behavior is: As x → − ∞ , y → + ∞ and as x → + ∞ , y → − ∞ , so the answer is B .
Explanation
Identifying the Leading Term We are given the function f ( x ) = − 5 x 3 + 3 x 2 + x − 9 and asked to determine its end behavior. The end behavior of a polynomial is determined by its leading term, which is the term with the highest degree. In this case, the leading term is − 5 x 3 .
End Behavior as x approaches negative infinity As x approaches − ∞ , we have: ( − 5 ) × ( − ∞ ) 3 = ( − 5 ) × ( − ∞ ) × ( − ∞ ) × ( − ∞ ) = ( − 5 ) × ( − ∞ ) = + ∞ Thus, as x → − ∞ , f ( x ) → + ∞ .
End Behavior as x approaches positive infinity As x approaches + ∞ , we have: ( − 5 ) × ( + ∞ ) 3 = ( − 5 ) × ( + ∞ ) × ( + ∞ ) × ( + ∞ ) = ( − 5 ) × ( + ∞ ) = − ∞ Thus, as x → + ∞ , f ( x ) → − ∞ .
Conclusion Therefore, the end behavior of the function is:As x → − ∞ , y → + ∞ and as x → + ∞ , y → − ∞ . This corresponds to option B.
Examples
Understanding the end behavior of functions is crucial in various real-world applications. For instance, in economics, it can help predict long-term trends in market growth or decline. In physics, it can model the behavior of systems approaching extreme conditions. By analyzing the leading term of a polynomial, we can make informed predictions about the overall behavior of the function as the input values become very large or very small, providing valuable insights in diverse fields.
