To determine the end behaviors of the polynomial function [tex]\( f(x) = -2(x - 17)^4 \)[/tex], we need to analyze the degree of the polynomial and the sign of the leading coefficient.
### Step-by-Step Solution:
1. Identify the Degree of the Polynomial:
- The polynomial is given as [tex]\( f(x) = -2(x - 17)^4 \)[/tex].
- The expression [tex]\( (x - 17)^4 \)[/tex] indicates that the degree of the polynomial is 4. This is because the highest power of [tex]\( x \)[/tex] is [tex]\( x^4 \)[/tex] when expanded.
2. Determine the Leading Coefficient:
- The leading coefficient is the coefficient of the term with the highest degree.
- In this case, the term with the highest degree (after expanding) would be [tex]\( -2x^4 \)[/tex].
- Thus, the leading coefficient is [tex]\( -2 \)[/tex], which is negative.
3. Analyze the End Behavior Based on the Degree and Leading Coefficient:
- For polynomials with an even degree:
- If the leading coefficient is positive, both ends of the graph go up as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex].
- If the leading coefficient is negative, both ends of the graph go down as [tex]\( x \)[/tex] approaches [tex]\( \infty \)[/tex] and [tex]\( -\infty \)[/tex].
- Since our polynomial has an even degree (4) and a negative leading coefficient (-2), we conclude that both ends of the polynomial function will go down.
### Conclusion:
The end behaviors of the function [tex]\( f(x) = -2(x - 17)^4 \)[/tex] indicate that both ends go down.
Therefore, the correct answer is:
[tex]\[ \boxed{\text{B. Both ends go down.}} \][/tex]
