Identify the end behavior of the function [tex]f(x)=3 x^4+x^3-7 x^2+12[/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 as [tex]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 is determined by its leading term.
As x approaches − ∞ , x 4 approaches + ∞ , so f ( x ) approaches + ∞ .
As x approaches + ∞ , x 4 approaches + ∞ , so f ( x ) approaches + ∞ .
The end behavior is: as x → − ∞ , y → + ∞ , and as x → + ∞ , y → + ∞ , so the answer is As x → − ∞ , y → + ∞ , and as x → + ∞ , y → + ∞ .
Explanation
Understanding the Problem We are asked to identify the end behavior of the polynomial function f ( x ) = 3 x 4 + x 3 − 7 x 2 + 12 . 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 3 x 4 .
Analyzing the End Behavior as x Approaches Negative Infinity As x approaches − ∞ , x 4 approaches + ∞ because any negative number raised to an even power is positive. Therefore, 3 x 4 also approaches + ∞ . This means that as x → − ∞ , f ( x ) → + ∞ .
Analyzing the End Behavior as x Approaches Positive Infinity As x approaches + ∞ , x 4 approaches + ∞ because any positive number raised to any power is positive. Therefore, 3 x 4 also approaches + ∞ . This means that as x → + ∞ , f ( x ) → + ∞ .
Conclusion Therefore, the end behavior of the function f ( x ) = 3 x 4 + x 3 − 7 x 2 + 12 is: as x → − ∞ , y → + ∞ , and as x → + ∞ , y → + ∞ .
Examples
Understanding the end behavior of polynomial functions is crucial in various fields, such as physics and engineering. For example, when modeling the trajectory of a projectile, the end behavior of the polynomial describing its height can tell us whether the projectile will eventually fall back to earth or continue to rise indefinitely (in an idealized, simplified model). Similarly, in economics, polynomial functions can be used to model costs or revenues, and understanding their end behavior can help predict long-term trends.
