To determine the end behavior of the function [tex]\( f(x) = -4x^6 + 6x^2 - 52 \)[/tex], we need to focus on the term with the highest degree, which is [tex]\( -4x^6 \)[/tex].
### Steps to Determine End Behavior:
1. Identify the Degree of the Polynomial:
- The degree of the polynomial is determined by the highest power of [tex]\( x \)[/tex]. Here, the term with the highest power is [tex]\( -4x^6 \)[/tex], so the polynomial is of degree 6.
2. Identify the Leading Term and Coefficient:
- The leading term is [tex]\( -4x^6 \)[/tex].
- The leading coefficient is [tex]\( -4 \)[/tex], which is negative.
3. Understanding the Degree:
- Since the degree (6) is even, we know that for large values of [tex]\( |x| \)[/tex], the ends of the graph will behave similarly because even powers of [tex]\( x \)[/tex] produce the same sign for both large positive and large negative values of [tex]\( x \)[/tex].
4. Impact of the Leading Coefficient:
- Because the leading coefficient [tex]\( -4 \)[/tex] is negative, it flips the usual behavior of an even-degree polynomial. For a positive leading coefficient, both ends of the graph go upwards. However, since our leading coefficient is negative, both ends of the graph will go downwards.
### Conclusion:
- As [tex]\( x \to \infty \)[/tex] (as [tex]\( x \)[/tex] increases positively), since the leading term is negative, [tex]\( f(x) \)[/tex] will tend to [tex]\( -\infty \)[/tex].
- As [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \)[/tex] decreases negatively), similarly, [tex]\( f(x) \)[/tex] will also tend to [tex]\( -\infty \)[/tex].
Therefore, the correct end behavior is that both ends of the graph go in the same direction, which is downward, due to the negative leading coefficient and the even degree.
Hence, the correct option is:
- D. [tex]\( f(x) \)[/tex] is an even function so both ends of the graph go in the same direction.
