To determine which polynomial function has the end behavior of $y \to -\infty$ as $x \to \infty$ and $y \to \infty$ as $x \to -\infty$, we need to analyze the leading term of each polynomial.

1. End behavior analysis:
   • The end behavior of a polynomial function is determined by its leading term (the term with the highest power of $x$).
   • Specifically, if the leading term is $ax^n$:
     • If $a > 0$ and $n$ is even, as $x \to \pm\infty$, $y \to \infty$.
     • If $a < 0$ and $n$ is even, as $x \to \pm\infty$, $y \to -\infty$.
     • If $a > 0$ and $n$ is odd, as $x \to \infty$, $y \to \infty$ and as $x \to -\infty$, $y \to -\infty$.
     • If $a < 0$ and $n$ is odd, as $x \to \infty$, $y \to -\infty$ and as $x \to -\infty$, $y \to \infty$.

Given the polynomials:

• a) $f(x) = 6x^4 + x^5 - 5x^3 + x^2 + 4x - 9$

  • Leading term: $x^5$ (positive coefficient and odd degree)
  • End behavior: as $x \to \infty$, $y \to \infty$ and as $x \to -\infty$, $y \to -\infty$.

• b) $f(x) = -2x^2 + 4x^3 + x - 7$

  • Leading term: $4x^3$ (positive coefficient and odd degree)
  • End behavior: as $x \to \infty$, $y \to \infty$ and as $x \to -\infty$, $y \to -\infty$.

• c) $f(x) = -6x^4 - x^5 + 5x^3 - x^2 - 4x + 9$

  • Leading term: $-x^5$ (negative coefficient and odd degree)
  • End behavior: as $x \to \infty$, $y \to -\infty$ and as $x \to -\infty$, $y \to \infty$.

• d) $f(x) = 3x^2 + x - 11$

  • Leading term: $3x^2$ (positive coefficient and even degree)
  • End behavior: as $x \to \infty$, $y \to \infty$ and as $x \to -\infty$, $y \to \infty$.

From the analysis above, the correct answer is:

c) $f(x) = -6x^4 - x^5 + 5x^3 - x^2 - 4x + 9$

Would you like any further details or have any questions?

Here are some related questions for further exploration:

1. How does the leading coefficient affect the end behavior of a polynomial?
2. What is the significance of the degree of the polynomial in determining its end behavior?
3. Can a polynomial with an even degree have the same end behavior as an odd-degree polynomial?
4. How do we determine the leading term of a polynomial?
5. How does the end behavior of a polynomial help in sketching its graph?
6. What are some examples of real-world phenomena that can be modeled using polynomials?
7. How do polynomial functions differ from other types of functions, such as exponential or logarithmic functions?
8. What are the steps to factor a polynomial completely?

Tip: When analyzing the end behavior of polynomials, always focus on the leading term since it dominates the behavior as $x$ becomes very large or very small.