To determine the end behavior of the function [tex]\( f(x) = -4x^6 - 6x^2 - 52 \)[/tex], we need to understand how polynomials behave as [tex]\( x \)[/tex] approaches positive and negative infinity.
1. Identify the Degree and Leading Coefficient:
- The polynomial function [tex]\( f(x) \)[/tex] has several terms, but the term with the highest degree will dominate the end behavior.
- The term with the highest degree is [tex]\( -4x^6 \)[/tex]. Thus, the degree of the polynomial is 6.
- The leading coefficient is the coefficient of this highest degree term, which is [tex]\(-4\)[/tex].
2. Determine Degree and Leading Coefficient Effect:
- The degree of the polynomial (6) is an even number. For polynomial functions, an even degree means that the ends of the polynomial will move in the same direction as [tex]\( x \)[/tex] approaches positive and negative infinity.
- The leading coefficient is [tex]\(-4\)[/tex], which is negative.
3. Analyze the End Behavior:
- When dealing with a polynomial where the leading degree term is of an even degree and has a negative leading coefficient, as [tex]\( x \)[/tex] approaches [tex]\( +\infty \)[/tex] or [tex]\( -\infty \)[/tex], the polynomial will head towards [tex]\( -\infty \)[/tex].
- This means that both ends of the graph will go downward.
Therefore, the correct assessment of the end behavior of the function [tex]\( f(x) = -4x^6 - 6x^2 - 52 \)[/tex] is that both ends of the graph go downwards.
So the correct answer is:
C. The leading coefficient is negative so the left end of the graph goes down.
