To determine the end behavior of the polynomial function [tex]\( y = 10x^9 - 4x \)[/tex], we need to focus on the highest degree term, as it will dominate the behavior of the polynomial function for very large or very small values of [tex]\( x \)[/tex].
1. As [tex]\( x \to -\infty \)[/tex]:
- The dominant term in the polynomial is [tex]\( 10x^9 \)[/tex].
- For large negative [tex]\( x \)[/tex], [tex]\( x^9 \)[/tex] becomes very large and negative because raising a negative number to an odd power results in a negative number.
- Multiplying this by 10, the term [tex]\( 10x^9 \)[/tex] will be very large and negative.
- The term [tex]\( -4x \)[/tex] becomes large and positive as [tex]\( x \to -\infty \)[/tex], but its influence is negligible compared to [tex]\( 10x^9 \)[/tex].
- Therefore, as [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
2. As [tex]\( x \to \infty \)[/tex]:
- Again, the dominant term is [tex]\( 10x^9 \)[/tex].
- For large positive [tex]\( x \)[/tex], [tex]\( x^9 \)[/tex] becomes very large and positive because raising a positive number to an odd power results in a positive number.
- Multiplying this by 10, the term [tex]\( 10x^9 \)[/tex] will be very large and positive.
- The term [tex]\( -4x \)[/tex] becomes large and negative as [tex]\( x \to \infty \)[/tex], but its influence is negligible compared to [tex]\( 10x^9 \)[/tex].
- Therefore, as [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
So, the correct end behavior of the graph of the polynomial function [tex]\( y = 10x^9 - 4x \)[/tex] is:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex].
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \to \infty \)[/tex].
Thus, the correct statement that describes this behavior is:
- As [tex]\( x \rightarrow-\infty, y \rightarrow-\infty \)[/tex] and as [tex]\( x \rightarrow \infty, y \rightarrow \infty \)[/tex].
