To determine the end behavior of the polynomial function [tex]\( y = 10x^9 - 4x \)[/tex], we need to focus on the term with the highest degree in the polynomial, as it will dominate the behavior of the function as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] or [tex]\( -\infty \)[/tex].
1. Identify the leading term:
The leading term in the polynomial [tex]\( y = 10x^9 - 4x \)[/tex] is [tex]\( 10x^9 \)[/tex] because it has the highest degree.
2. Determine the behavior of the leading term:
- For even-degree polynomials, the end behavior depends on the sign of the leading coefficient.
- For odd-degree polynomials, the end behavior is different in the [tex]\( x \to \infty \)[/tex] direction compared to the [tex]\( x \to -\infty \)[/tex] direction.
- If the leading coefficient is positive, as [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex] and as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
- If the leading coefficient is negative, as [tex]\( x \to \infty \)[/tex], [tex]\( y \to -\infty \)[/tex] and as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to \infty \)[/tex].
Since we are dealing with the polynomial [tex]\( 10x^9 \)[/tex], it is an odd-degree polynomial (degree 9) with a positive leading coefficient (10).
Thus, the end behavior is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
So, the correct choice is:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
Thus, the correct end behavior is:
[tex]\[ \text{As } x \to -\infty, y \to -\infty \text{ and as } x \to \infty, y \to \infty. \][/tex]
