To determine the end behavior of the polynomial function [tex]\( q(x) = -4x^5 + 2x^4 - 3x^2 + 12x \)[/tex], we should focus on the term with the highest power of [tex]\(x\)[/tex] because it will dominate the behavior of the polynomial as [tex]\(x\)[/tex] becomes very large (positive or negative).
The term with the highest degree in this polynomial is [tex]\(-4x^5\)[/tex].
1. Identify the leading term: The leading term of [tex]\( q(x) \)[/tex] is [tex]\(-4x^5\)[/tex].
2. Determine the coefficient: The coefficient of the leading term is [tex]\(-4\)[/tex], which is negative.
3. Determine the exponent of [tex]\(x\)[/tex]: The exponent of the leading term is [tex]\(5\)[/tex], which is an odd number.
For polynomial functions of the form [tex]\( ax^n \)[/tex], where [tex]\( n \)[/tex] is the highest power and [tex]\( a \)[/tex] is the leading coefficient:
- If [tex]\( n \)[/tex] is odd and [tex]\( a \)[/tex] is negative, the end behavior of the polynomial is:
- As [tex]\( x \to \infty \)[/tex], [tex]\( q(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( q(x) \to \infty \)[/tex].
So, for [tex]\( q(x) = -4x^5 + 2x^4 - 3x^2 + 12x \)[/tex]:
- As [tex]\( x \to \infty \)[/tex], [tex]\( q(x) \to -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex], [tex]\( q(x) \to \infty \)[/tex].
Therefore, the correct answer is:
(B) As [tex]\( x \rightarrow \infty, q(x) \rightarrow -\infty \)[/tex], and as [tex]\( x \rightarrow -\infty, q(x) \rightarrow \infty \)[/tex].
