To determine the end behavior of the function [tex]\( f(x) = 80x^2 + 640 - 400x - 5x^3 \)[/tex], we focus on the leading term of the polynomial. The end behavior of a polynomial is determined by the term with the highest degree because it dominates the value of the polynomial as [tex]\( x \)[/tex] approaches [tex]\(\pm\infty\)[/tex].
In this function, the term with the highest degree is [tex]\(-5x^3\)[/tex].
Let's analyze the behavior of this term as [tex]\( x \)[/tex] approaches [tex]\(\pm\infty\)[/tex]:
1. As [tex]\( x \rightarrow \infty \)[/tex]:
\begin{align}
-5x^3 \rightarrow -\infty
\end{align}
As [tex]\( x \)[/tex] becomes very large, [tex]\( x^3 \)[/tex] becomes very large, and since it’s multiplied by a negative coefficient [tex]\(-5\)[/tex], the value of [tex]\(-5x^3\)[/tex] becomes very large and negative. Hence,
[tex]\[
\lim_{x \to \infty} f(x) = -\infty
\][/tex]
2. As [tex]\( x \rightarrow -\infty \)[/tex]:
\begin{align}
-5x^3 \rightarrow \infty
\end{align}
As [tex]\( x \)[/tex] becomes very large in the negative direction, [tex]\( (-x)^3 \)[/tex] becomes very large and negative, and since it’s multiplied by a negative coefficient [tex]\(-5\)[/tex], the value of [tex]\(-5(-x)^3\)[/tex] becomes very large and positive. Hence,
[tex]\[
\lim_{x \to -\infty} f(x) = \infty
\][/tex]
Therefore, the end behavior of [tex]\( f(x) \)[/tex] is:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( f(x) \rightarrow \infty \)[/tex]
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( f(x) \rightarrow -\infty \)[/tex]
Thus, the answer is:
[tex]\[
\text{as } x \rightarrow -\infty, f(x) \rightarrow \infty \text{ and as } x \rightarrow \infty, f(x) \rightarrow -\infty.
\][/tex]
