To determine the end behavior of the polynomial [tex]\( f(x) = -6x^7 - 8x^5 - 9x^4 + 2x + 12 \)[/tex], we need to analyze the term with the highest degree, which in this case is [tex]\(-6x^7\)[/tex]. The highest degree term dictates the behavior of the polynomial as [tex]\(x\)[/tex] approaches infinity or negative infinity.
### Analyzing the behavior as [tex]\( x \to -\infty \)[/tex]:
- The highest degree term is [tex]\(-6x^7\)[/tex].
- Since the exponent [tex]\(7\)[/tex] is odd, the behavior of [tex]\(x^7\)[/tex] will mimic the behavior of [tex]\(x\)[/tex] itself.
- As [tex]\( x \to -\infty \)[/tex], [tex]\( (-x)^7 \to -\infty \)[/tex].
- The coefficient [tex]\(-6\)[/tex] is negative, so multiplying it by [tex]\( (-x)^7\)[/tex] will result in [tex]\(-(-\infty) = \infty\)[/tex].
However, due to the negative coefficient:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( -6x^7 \to -\infty \)[/tex].
### Analyzing the behavior as [tex]\( x \to \infty \)[/tex]:
- Similarly, as [tex]\( x \to \infty \)[/tex], [tex]\(-6x^7\)[/tex] will still dominate.
- Since [tex]\(-6\)[/tex] is negative, multiplying it by [tex]\( x^7\)[/tex] as [tex]\( x \to \infty \)[/tex] will result in [tex]\(-\infty\)[/tex].
Thus, we can conclude:
1. As [tex]\( x \to -\infty \)[/tex], [tex]\( y \to -\infty \)[/tex]. Therefore, [tex]\(y\)[/tex] decreases.
2. As [tex]\( x \to \infty \)[/tex], [tex]\( y \to -\infty \)[/tex]. Therefore, [tex]\(y\)[/tex] decreases.
### Summary of correct options:
- As [tex]\( x \to -\infty \)[/tex], [tex]\( y \)[/tex] decreases.
- As [tex]\( x \to \infty \)[/tex], [tex]\( y \)[/tex] decreases.
Thus, the true statements about the polynomial [tex]\( f(x) = -6x^7 - 8x^5 - 9x^4 + 2x + 12 \)[/tex] are:
- As [tex]\(x\)[/tex] approaches negative infinity, [tex]\(y\)[/tex] decreases.
- As [tex]\(x\)[/tex] approaches positive infinity, [tex]\(y\)[/tex] decreases.
