Let's analyze the polynomial function [tex]\( f(x) = 8x^2 - 3x^3 - 2x^4 - 6 \)[/tex].
### Degree of [tex]\( f(x) \)[/tex]:
The degree of a polynomial is the highest power of [tex]\( x \)[/tex] in the expression. In [tex]\( f(x) = 8x^2 - 3x^3 - 2x^4 - 6 \)[/tex], the term with the highest power of [tex]\( x \)[/tex] is [tex]\( -2x^4 \)[/tex]. Therefore, the degree of [tex]\( f(x) \)[/tex] is:
[tex]\[\boxed{4}\][/tex]
### Leading Coefficient:
The leading coefficient is the coefficient of the term with the highest power of [tex]\( x \)[/tex]. In this polynomial, that term is [tex]\( -2x^4 \)[/tex]. The leading coefficient is:
[tex]\[\boxed{-2}\][/tex]
### End Behavior:
To determine the end behavior, we analyze the term with the highest degree, [tex]\( -2x^4 \)[/tex].
- As [tex]\( x \rightarrow -\infty \)[/tex]:
The leading term [tex]\( -2x^4 \)[/tex] dominates the function. Since [tex]\( x^4 \)[/tex] is always positive and multiplied by [tex]\(-2\)[/tex], it will go to [tex]\(-\infty\)[/tex] as [tex]\( x \rightarrow -\infty \)[/tex]. Thus,
[tex]\[ \boxed{-\infty} \][/tex]
- As [tex]\( x \rightarrow \infty \)[/tex]:
Similarly, the leading term [tex]\( -2x^4 \)[/tex] dominates the function. [tex]\( x^4 \)[/tex] is still positive and multiplied by [tex]\(-2\)[/tex], causing the function to go to [tex]\(-\infty\)[/tex] as [tex]\( x \rightarrow \infty \)[/tex]. Thus,
[tex]\[ \boxed{-\infty} \][/tex]
### Maximum Number of Intercepts:
A polynomial function of degree [tex]\( n \)[/tex] can have at most [tex]\( n \)[/tex] intercepts. Since the degree of [tex]\( f(x) \)[/tex] is 4, the maximum number of intercepts is:
[tex]\[\boxed{4}\][/tex]
### Maximum Number of Turning Points:
A polynomial function of degree [tex]\( n \)[/tex] can have at most [tex]\( n-1 \)[/tex] turning points. For [tex]\( f(x) \)[/tex], the degree is 4. Therefore, it can have at most [tex]\( 4-1 = 3 \)[/tex] turning points:
[tex]\[\boxed{3}\][/tex]
To summarize, the detailed analysis gives us:
- Degree of [tex]\( f(x) \)[/tex]: [tex]\( \boxed{4} \)[/tex]
- Leading coefficient: [tex]\( \boxed{-2} \)[/tex]
- End behavior:
- As [tex]\( x \rightarrow -\infty, f(x) \rightarrow \boxed{-\infty} \)[/tex]
- As [tex]\( x \rightarrow \infty, f(x) \rightarrow \boxed{-\infty} \)[/tex]
- Maximum number of intercepts: [tex]\( \boxed{4} \)[/tex]
- Maximum number of turning points: [tex]\( \boxed{3} \)[/tex]
