What is the end behavior of the graph of the polynomial function [tex]$y=10 x^9-4 x$[/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].

The end behavior of a polynomial is determined by its leading term.
As x approaches − ∞ , x 9 approaches − ∞ , so y approaches − ∞ .
As x approaches ∞ , x 9 approaches ∞ , so y approaches ∞ .
Therefore, as x i g h t ha r p oo n u p − ∞ , y i g h t ha r p oo n u p − ∞ and as x i g h t ha r p oo n u p ∞ , y i g h t ha r p oo n u p ∞ .

Explanation

Analyzing the Problem We are given the polynomial function y = 10 x 9 − 4 x and asked to determine its end behavior. This means we need to find out what happens to the value of y as x approaches both negative infinity ( − ∞ ) and positive infinity ( ∞ ). The end behavior of a polynomial is determined by its leading term, which is the term with the highest power of x . In this case, the leading term is 10 x 9 .

End Behavior as x Approaches Negative Infinity Let's analyze the behavior of the leading term 10 x 9 as x approaches negative infinity. Since the exponent 9 is odd, when x is a large negative number, x 9 will also be a large negative number. Multiplying this by 10 (a positive number) still results in a large negative number. Therefore, as x → − ∞ , 10 x 9 → − ∞ .

End Behavior as x Approaches Positive Infinity Now let's analyze the behavior of the leading term 10 x 9 as x approaches positive infinity. When x is a large positive number, x 9 will also be a large positive number. Multiplying this by 10 still results in a large positive number. Therefore, as x → ∞ , 10 x 9 → ∞ .

Conclusion The term − 4 x has a lower degree than the leading term 10 x 9 , so it does not affect the end behavior of the function. Therefore, the end behavior of the polynomial function y = 10 x 9 − 4 x is as follows: as x → − ∞ , y → − ∞ , and as x → ∞ , y → ∞ .


Examples
Understanding the end behavior of polynomial functions is crucial in various fields. For instance, in physics, when modeling the trajectory of a projectile, the end behavior helps predict its long-term path. Similarly, in economics, polynomial functions can model growth trends, and analyzing their end behavior provides insights into potential future outcomes. By knowing how a function behaves as x approaches extreme values, we can make informed decisions and predictions in real-world scenarios.

Answered by GinnyAnswer | 2025-07-08

The end behavior of the polynomial function y = 10 x 9 − 4 x is determined by its leading term 10 x 9 . As x \rightarrow -\infty , y \rightarrow -\infty and as x \rightarrow +\infty , y \rightarrow +\infty . Therefore, the correct option is C.
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Answered by Anonymous | 2025-07-14