To determine the end behavior of the polynomial function [tex]\( f(x) = 3072 - 6x^5 + 78x^4 - 1680x^2 + 1536x - 60x^3 \)[/tex], we need to focus on the term with the highest degree, as it will dominate the behavior of the function as [tex]\( x \)[/tex] approaches infinity or negative infinity.
1. Identify the leading term:
The term with the highest degree in the polynomial is [tex]\( -6x^5 \)[/tex].
2. Analyze the leading term [tex]\( -6x^5 \)[/tex]:
- When [tex]\( x \to \infty \)[/tex]:
- [tex]\( x^5 \)[/tex] approaches infinity.
- Since the coefficient of [tex]\( x^5 \)[/tex] is negative ([tex]\( -6 \)[/tex]), [tex]\( -6x^5 \)[/tex] will approach negative infinity.
- Thus, [tex]\( f(x) \to -\infty \)[/tex].
- When [tex]\( x \to -\infty \)[/tex]:
- [tex]\( x^5 \)[/tex] approaches negative infinity.
- Since the coefficient of [tex]\( x^5 \)[/tex] is negative ([tex]\( -6 \)[/tex]), [tex]\( -6x^5 \)[/tex] will approach positive infinity.
- Thus, [tex]\( f(x) \to \infty \)[/tex].
So the end behavior of the function [tex]\( f(x) \)[/tex] is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex].
Therefore, the correct answer is:
- As [tex]\( x \rightarrow \infty, f(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty, f(x) \rightarrow \infty \)[/tex].
So, the correct option is:
[tex]\[
\text{as } x \rightarrow \infty, f(x) \rightarrow -\infty \text{ and as } x \rightarrow -\infty, f(x) \rightarrow \infty
\][/tex]
