To determine the end behavior of the polynomial function [tex]\( q(x) = -4x^5 + 2x^4 - 3x^2 + 12 \)[/tex], we focus on the leading term of the polynomial, since the leading term dominates the behavior of the polynomial for large values of [tex]\( |x| \)[/tex].
The leading term of the polynomial [tex]\( q(x) \)[/tex] is [tex]\( -4x^5 \)[/tex].
1. Identify the leading coefficient and the leading exponent:
- The leading coefficient is [tex]\(-4\)[/tex].
- The leading exponent is [tex]\(5\)[/tex].
2. Consider the sign of the leading coefficient:
- The leading coefficient [tex]\(-4\)[/tex] is negative.
3. Consider the parity (odd/even) of the leading exponent:
- The leading exponent [tex]\(5\)[/tex] is odd.
Now we analyze how these factors affect the end behavior of the polynomial:
- When the leading exponent is odd:
- For [tex]\( x \to \infty \)[/tex], the value of [tex]\( x^5 \)[/tex] becomes very large and positive. Multiplying by [tex]\(-4\)[/tex] (the negative leading coefficient) makes [tex]\( q(x) \to -\infty \)[/tex].
- For [tex]\( x \to -\infty \)[/tex], the value of [tex]\( x^5 \)[/tex] becomes very large and negative. Again, multiplying by [tex]\(-4\)[/tex] makes [tex]\( q(x) \to \infty \)[/tex].
From this analysis, we can conclude the end behavior of [tex]\( q(x) \)[/tex] is:
- As [tex]\( x \rightarrow \infty \)[/tex], [tex]\( q(x) \rightarrow -\infty \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex], [tex]\( q(x) \rightarrow \infty \)[/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].
