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].
Answer (1)
The end behavior of a polynomial is determined by its leading term.
As x approaches − ∞ , x 9 also approaches − ∞ , so y approaches − ∞ .
As x approaches ∞ , x 9 also approaches ∞ , so y approaches ∞ .
Therefore, as x → − ∞ , y → − ∞ and as x → ∞ , y → ∞ , so the answer is \boxed{\text{As } x \rightarrow -\infty, y \rightarrow -\infty \text{ and as } x \rightarrow \infty, y \rightarrow \infty}} .
Explanation
Understanding the Problem We are given the polynomial function y = 10 x 9 − 4 x and asked to determine its end behavior. The end behavior of a polynomial describes what happens to the y values as x approaches positive and negative infinity.
Identifying the Leading Term 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 . The other term, − 4 x , will become insignificant as x approaches infinity or negative infinity.
Analyzing the Behavior as x Approaches Negative Infinity Let's analyze the behavior as x approaches − ∞ . Since the exponent of the leading term is 9, which is odd, when x is a large negative number, x 9 will also be a large negative number. Multiplying by 10, we have that 10 x 9 is a large negative number. Therefore, as x → − ∞ , y → − ∞ .
Analyzing the Behavior as x Approaches Positive Infinity Now let's analyze the behavior as x approaches ∞ . When x is a large positive number, x 9 will also be a large positive number. Multiplying by 10, we have that 10 x 9 is a large positive number. Therefore, as x → ∞ , y → ∞ .
Conclusion Therefore, the end behavior of the graph of the polynomial function y = 10 x 9 − 4 x is: 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, analyzing polynomial models can provide insights into long-term trends in market behavior. By knowing how a function behaves as x approaches extreme values, we can make informed predictions and decisions in real-world scenarios.
