Let’s work through this problem step by step, based on the function provided:

$f(x) = 8x^4 + 3x^5 - 2x^7 - 6$

1. Degree of the function:

   • The degree of a polynomial is the highest power of $x$. In this case, the highest exponent is 7 (from the term $-2x^7$), so the degree of the polynomial is 7.

2. Leading coefficient:

   • The leading coefficient is the coefficient of the term with the highest power of $x$. Here, the highest power is $-2x^7$, so the leading coefficient is -2.

3. End behavior:

   • For polynomials, the end behavior is determined by the leading term, which in this case is $-2x^7$.
     • As $x \to -\infty$, since the highest power is odd and the coefficient is negative, $f(x) \to \infty$ (positive infinity).
     • As $x \to \infty$, $f(x) \to -\infty$ (negative infinity).

4. Maximum number of intercepts:

   • The number of x-intercepts (real roots) is at most equal to the degree of the polynomial. Since the degree is 7, the maximum number of intercepts is 7.

5. Maximum number of turning points:

   • The number of turning points in a polynomial is at most $n - 1$, where $n$ is the degree of the polynomial. Here, the degree is 7, so the maximum number of turning points is $7 - 1 = 6$.

Summary of the answers:

• Degree of the function: 7
• Leading coefficient: -2
• End behavior:
  • As $x \to -\infty$, $f(x) \to \infty$.
  • As $x \to \infty$, $f(x) \to -\infty$.
• Maximum number of intercepts: 7
• Maximum number of turning points: 6

Would you like further details on any of the points above or a specific explanation?

Here are 5 related questions for more practice:

1. What is the significance of the degree of a polynomial in its graph?
2. How does the sign of the leading coefficient affect the end behavior of a polynomial function?
3. Can a polynomial of degree 7 have exactly 5 turning points? Why or why not?
4. What is the relationship between the degree of a polynomial and its graph's symmetry?
5. How can you find the x-intercepts of a polynomial function?

Tip: To predict the behavior of a polynomial for large values of $x$, always focus on the term with the highest degree, as it dominates the function's growth.