What is the end behavior of the graph of the polynomial function [tex]$y=7 x^{12}-3 x^8-9 x^4$[/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.
The leading term of the polynomial y = 7 x 12 − 3 x 8 − 9 x 4 is 7 x 12 .
Since the exponent of the leading term is even and the coefficient is positive, as x approaches − ∞ or ∞ , y approaches ∞ .
Therefore, 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 determine the end behavior of the polynomial function y = 7 x 12 − 3 x 8 − 9 x 4 . The end behavior of a polynomial describes what happens to the y-values as x approaches positive or 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 degree. In this case, the leading term is 7 x 12 .

Determining the End Behavior The exponent of the leading term is 12, which is an even number. The coefficient of the leading term is 7, which is a positive number. When the exponent is even, the end behavior is the same as x approaches both − ∞ and ∞ . Since the coefficient is positive, as x approaches − ∞ or ∞ , y approaches ∞ .

Conclusion Therefore, as x → − ∞ , y → ∞ and as x → ∞ , y → ∞ .


Examples
Understanding the end behavior of polynomial functions is useful in various fields. For example, in physics, when modeling the trajectory of a projectile, the end behavior can help predict the long-term path. In economics, polynomial functions can model cost or revenue, and understanding their end behavior can help forecast long-term profitability or losses. In computer graphics, polynomial functions are used to create curves and surfaces, and their end behavior affects how these shapes appear at the edges of the screen.

Answered by GinnyAnswer | 2025-07-08

The end behavior of the polynomial function y = 7 x 12 − 3 x 8 − 9 x 4 indicates that as x approaches both negative and positive infinity, y approaches infinity. Therefore, the correct answer is D: As x → − ∞ , y → ∞ and as x → ∞ , y → ∞ .
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Answered by Anonymous | 2025-07-09