To analyze the end behavior of the function [tex]\( f(x) = -4x^6 + 6x^2 - 52 \)[/tex], we need to consider two main aspects: the degree of the polynomial and the leading coefficient.
### Step-by-Step Solution:
1. Identify the Leading Term:
The leading term of a polynomial is the term with the highest power of [tex]\( x \)[/tex]. For [tex]\( f(x) = -4x^6 + 6x^2 - 52 \)[/tex], the leading term is [tex]\( -4x^6 \)[/tex].
2. Determine the Leading Coefficient:
The leading coefficient is the coefficient of the leading term. In this case, the coefficient of [tex]\( -4x^6 \)[/tex] is [tex]\(-4\)[/tex].
3. Degree of the Polynomial:
The degree of a polynomial is the highest exponent on the variable [tex]\( x \)[/tex]. Here, the degree is [tex]\( 6 \)[/tex].
4. Analyze the Degree and Leading Coefficient:
- The degree of the polynomial is [tex]\( 6 \)[/tex], which is an even number.
- The leading coefficient is [tex]\(-4\)[/tex], which is a negative number.
5. End Behavior Rules:
- For polynomials with an even degree:
- If the leading coefficient is positive, both ends of the graph will go upwards.
- If the leading coefficient is negative, both ends of the graph will go downwards.
6. Conclusion:
Because the degree of the polynomial [tex]\( f(x) = -4x^6 + 6x^2 - 52 \)[/tex] is even, and the leading coefficient [tex]\( -4 \)[/tex] is negative, both ends of the graph will go downwards.
### Final Answer:
D. [tex]\( f(x) \)[/tex] is an even function so both ends of the graph go in the same direction.
