To determine the end behavior of the polynomial function [tex]\( y = 10x^9 - 4x \)[/tex], we need to consider the behavior of the function as [tex]\( x \)[/tex] approaches both positive infinity ([tex]\( +\infty \)[/tex]) and negative infinity ([tex]\( -\infty \)[/tex]).
1. As [tex]\( x \rightarrow -\infty \)[/tex]:
- The term [tex]\( 10x^9 \)[/tex] will dominate the behavior of the polynomial because it is the highest-degree term.
- Since [tex]\( x \)[/tex] is approaching [tex]\( -\infty \)[/tex] and the exponent 9 is odd, [tex]\( 10x^9 \)[/tex] will approach [tex]\( -\infty \)[/tex] because raising a negative number to an odd power will yield a negative result.
- The term [tex]\( -4x \)[/tex] will approach [tex]\( +\infty \)[/tex] because multiplying a negative number ([tex]\( x \rightarrow -\infty \)[/tex]) by a negative coefficient [tex]\( -4 \)[/tex] results in a positive product.
- However, for sufficiently large negative values of [tex]\( x \)[/tex], the magnitude of [tex]\( 10x^9 \)[/tex] will far exceed that of [tex]\( -4x \)[/tex], meaning the term [tex]\( 10x^9 \)[/tex] will be the dominant factor driving the behavior of the polynomial, pulling [tex]\( y \)[/tex] towards [tex]\( -\infty \)[/tex].
Therefore, as [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
2. As [tex]\( x \rightarrow \infty \)[/tex]:
- Again, the term [tex]\( 10x^9 \)[/tex] dominates because it is the highest-degree term.
- Since [tex]\( x \)[/tex] is approaching [tex]\( +\infty \)[/tex] and the exponent 9 is odd, [tex]\( 10x^9 \)[/tex] will approach [tex]\( +\infty \)[/tex] because raising a positive number to an odd power yields a positive result.
- The term [tex]\( -4x \)[/tex] will approach [tex]\( -\infty \)[/tex] because multiplying a positive number by a negative coefficient results in a negative product.
- Even though [tex]\( -4x \)[/tex] approaches [tex]\( -\infty \)[/tex], the magnitude of [tex]\( 10x^9 \)[/tex] will be significantly larger for sufficiently large values of [tex]\( x \)[/tex], meaning [tex]\( 10x^9 \)[/tex] will dominate the behavior and pull [tex]\( y \)[/tex] towards [tex]\( +\infty \)[/tex].
Therefore, as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
Putting this all together:
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
The correct choice is:
As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( y \rightarrow -\infty \)[/tex] and as [tex]\( x \rightarrow \infty \)[/tex], [tex]\( y \rightarrow \infty \)[/tex].
